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+{"text":"\\section{Introduction}\n\nIn the 1970s, Wilson introduced and studied lattice versions of Yang-Mills\n(YM) theory, in which the dynamical variables are elements of the gauge\ngroup, defined on the links that connect the adjacent lattice points \\cit\n{Wilson:1974sk}. The choice of the lattice action in Yang-Mills theory was\nan important aspect of the problems considered in the early development of\nlattice gauge theory \\cite{Wilson:1974sk}-\\cite{Onofri:1981qk}. In\nparticular, in the late 1970s and early 1980s, a number of papers studied\nthis possibility, leading to the consideration of several alternatives to\nthe Wilson action \\cite{Wilson:1974sk}, like the heat-kernel action \\cit\n{Susskind:1979up,Stone:1978pe,Menotti:1981ry} or the Manton action \\cit\n{Manton:1980ts}. One of the motivations was the need for a proper\nunderstanding of the transition from strong to weak coupling in lattice\ngauge theories.\n\nA well-known result in the study of non-Abelian two-dimensional Yang-Mills\ntheory with the Wilson action is the third-order phase transition, found by\nGross and Witten \\cite{GW} and Wadia \\cite{Wadia:1980cp}, in a one-plaquette\nmodel, described by a unitary matrix model. This result has turned out to be\nof relevance in many current problems in theoretical physics, like the study\nof Hagedorn and deconfining transitions in weakly coupled Yang-Mills theory \n\\cite{Aharony:2003sx}. The Gross-Witten model has been also recently\ndiscussed in the study of type 0B and 0A fermionic string theories \\cite{Kms}\nand in relation with other solvable models, like the Kontsevich model \\cit\n{Morozov:2009jv}.\n\nWe will first show that the unitary matrix model of $U(N)$\\ Chern-Simons\ntheory on $S^{3}$ \\cite{Okuda:2004mb,Szabo:2010sd,Ooguri:2010yk} is\nintimately related to the Gross-Witten model when one considers it together\nwith the Villain approximation of the XY model \\cit\n{Villain:1974ir,Jose:1977gm}. We introduce the Villain approximation,\ntogether with the two relevant matrix models, in the next section. We will\nshow that both the weak-coupling and strong-coupling regimes of the\nGross-Witten model can be described, using the Villain approximation, by\nanalytically continued $U(N)$ Chern-Simons on $S^{3}.$ Recall that the\nChern-Simons action is given by \\cite{Witten:1988hf\n\\begin{equation}\nS_{\\mathrm{CS}}(A)={\\frac{k}{4\\pi }}\\int_{M}\\mathrm{Tr}(A\\wedge dA+{\\frac{2}\n3}}A\\wedge A\\wedge A),  \\label{cs}\n\\end{equation\nwhere $A$ is the connection, a 1-form valued on the corresponding Lie\nalgebra, and $k\\in \\mathbb{Z}$ is the level. The $q$-parameter is defined in\nterms of the level $k$ by $q=\\exp \\left( 2\\pi i\/(k+N)\\right) $.\n\nWe will see that the weak-coupling limit corresponds to the $q\\rightarrow 1$\nlimit of the Chern-Simons theory, whereas the strong-coupling limit of the\nGross-Witten model corresponds to the opposite limit, $q\\rightarrow 0$. In\nthe matrix model formulation, the $q$ parameter is treated as real and\nwritten in terms of a coupling constant $g_{s\\text{ }}$as $q=\\mathrm{e\n^{-g_{s}}$ \\cite{Tierz:2002jj}. The above characterization of Chern-Simons\ntheory as being analytically continued precisely refers to this treatment of \n$q$ as a real parameter.\n\nIndeed, actual computations with the matrix model are carried out with $q$\nreal, using for example the associated $q$-orthogonal polynomials \\cit\n{Tierz:2002jj}, and the identification $g_{s}=2\\pi i\/(k+N)$ at the end,\nallows us to make contact with the well-known expressions for the\nChern-Simons observables \\cite{Witten:1988hf}. See for example \\cit\n{Tierz:2002jj}, where the simple case of the $U(N)$ Chern-Simons partition\nfunction on $S^{3}$ is computed with the Stieltjes-Wigert polynomials with a \n$q$ parameter, $q=\\mathrm{e}^{-g_{s}}$.\n\nThe correspondence between the Gross-Witten model and the Chern-Simons\nmatrix model is of a rather different nature in the two opposite limits \nq\\rightarrow 0$ and $q\\rightarrow 1$. In the weak-coupling limit $g_{\\mathrm\n\\ YM}}^{2}\\rightarrow 0$, as shown already in \\cite{Szabo:2010qv}, the\ncoupling constants are related by $g_{\\mathrm{YM}}^{2}=2g_{s},$ whereas in\nthe strong-coupling limit we will have that $g_{\\mathrm{s}}=2\\ln \\left( g_\n\\mathrm{YM}}^{2}\\right) $ or, equivalently, $q=1\/g_{\\mathrm{YM}}^{4},$ as we\nshall see in Sec. 2 in detail.\n\nWe will end Sec. 2 by exploring some consequences of these relationships\nbetween the models. In particular, in Sec. 2.1 we show that the Gross-Witten\nmodel at weak coupling is a Gaussian matrix model whose free energy has an\nexpansion that can be interpreted in terms of closed strings.\n\nThe relationship between the Gross-Witten model and the Chern-Simons matrix\nmodel, based on the application of the Villain approximation to the Wilson\naction, indicates that the direct consideration of the Abelian Villain\naction in lattice two-dimensional Yang-Mills theory, should describe\nChern-Simons theory. Two-dimensional Yang-Mills theory was also studied with\nthe Manton action \\cite{Lang:1980sz,Lang:1980ws} and the heat-kernel action \n\\cite{Menotti:1981ry}. We shall see that, indeed, the straightforward\ngeneralization of the Abelian $U(1)$ lattice action, which is just a theta\nfunction \\cite{Villain:1974ir,Banks:1977cc,Peskin:1977kp\n\\begin{equation}\n\\exp (-S_{\\mathrm{V}}\\left( \\phi \\right) )=\\sum_{l=-\\infty }^{\\infty \n\\mathrm{e}^{-\\frac{1}{g^{2}}\\left( \\phi +2\\pi l\\right) ^{2}},\n\\label{A-Villain}\n\\end{equation\nto the non-Abelian case, in the setting of $U(N)$ two-dimensional Yang-Mills\ntheory, directly gives $U(N)$ Chern-Simons theory on $S^{3}$. This\nstraightforward extension of the Villain lattice action to the non-Abelian\ncase was explored by Onofri, shortly after the study of the heat-kernel case \n\\cite{Menotti:1981ry}, in a less well-known work \\cite{Onofri:1981qk}. The\ndescription of pure Chern-Simons theory by such a model has not hitherto\nbeen realized.\n\nIt is well known that the Villain model arises in the Kogut-Susskind\nHamiltonian lattice gauge theory \\cite{Kogut:1974ag} in the Abelian case,\nwhich leads to a direct correspondence with the planar Heisenberg (or XY)\nmodel \\cite{Polyakov:1978vu,Susskind:1979up}. The non-Abelian case leads to\nthe heat kernel \\cite{Susskind:1979up,Menotti:1981ry}, and we shall see that\nthe Chern-Simons matrix model follows from Abelianization of the heat-kernel\npropagator in the context of two-dimensional Yang-Mills theory. This is the\ncontent of Sec. 3 and, in particular, we show in Sec. 3.1 that this Abelian\nprojection is equivalent to a $q$ deformation of 2d Yang-Mills, in\nconsistency with \\ the known relationship between Chern-Simons theory and a \nq$ deformation of 2d Yang-Mills theory \\cite{Aganagic:2004js}. Recall that\nChern-Simons theory is known to be explained in terms of an Abelian\ntwo-dimensional Yang-Mills theory, as was shown at the level of the path\nintegral, first in the case of manifolds of the type $S^{1}\\times \\Sigma _{h}\n$, where $\\Sigma _{h}$ denotes a Riemann surface of genus $h$ \\cit\n{Blau:1993tv} and, more recently, for Seifert fibrations over $\\Sigma _{h}$ \n\\cite{Blau:2006gh}, which contains the $S^{3}$ case, the one studied in this\npaper at the level of the matrix model.\n\nTo conclude, we study in the Appendix the precise relationship between the\nunitary and the Hermitian versions of the Chern-Simons matrix model focusing\nalso in the rotation of the contours of integration.\\newline\n\n\\section{Gross-Witten model and the Villain approximation}\n\nThe approximation devised by Villain in 1975 in the study of the\ntwo-dimensional XY\\ model \\cite{Villain:1974ir} is based on the simple\nobservation that the term $\\exp \\left( \\beta \\cos \\theta \\right) $ that\nappears in the 2d XY model can be well approximated for large $\\beta $ by a\nperiodic Gaussian with minima in the same locations and with the same\ncurvature. That is \n\\begin{equation}\n\\exp \\left( \\beta \\cos \\theta \\right) \\sim \\mathrm{e}^{\\beta\n}\\sum_{n=-\\infty }^{\\infty }\\mathrm{e}^{-\\frac{1}{2}\\beta \\left( \\theta\n-2\\pi n\\right) ^{2}}\\text{ for }\\beta \\rightarrow \\infty .  \\label{Villain}\n\\end{equation\nBut the l.h.s. term is of course also the weight function of the matrix\nmodel description of the one-plaquette model of Yang-Mills theory based on\nthe Wilson action (namely, the Gross-Witten model \\cite{GW}). The r.h.s is a\ntheta function and then the Villain approximation applied to the\nGross-Witten model leads to the relationship with a unitary matrix model\nwith a theta function as weight function.\n\nPrecisely, the unitary matrix model that describes $U(N)$ Chern-Simons\ntheory on $S^{3\\text{ }}$ \\cite{Okuda:2004mb,Szabo:2010sd,Ooguri:2010yk} is\ngiven by \\footnote\nThe unitary matrix model (\\ref{UCS}) also describes Chern-Simons theory if\nthe weight function is $\\Theta ^{-1}({\\,-\\mathrm{e}}\\,^{{\\,\\mathrm{i}\\,\n\\theta _{j}}|q)$ \\cite{Szabo:2010sd}. This possibility has also been noticed\nin \\cite{Ooguri:2010yk}. The nonuniqueness description of the Chern-Simons\nmatrix models is described in \\cite{Tierz:2002jj}. See the Appendix for its\nprecise relationship with the Hermitian matrix model} \n\\begin{equation}\nZ_{\\mathrm{CS}}^{U(N)}\\left( S^{3}\\right) =\\int_{0}^{2\\pi\n}\\prod\\limits_{j=1}^{N}\\,\\frac{\\mathrm{d}\\theta _{j}}{2\\pi }~\\Theta ({\\\n\\mathrm{e}}\\,^{{\\,\\mathrm{i}\\,}\\theta _{j}}|q)~\\prod\\limits_{k<l}\\,\\big\\vert\n\\,\\mathrm{e}}\\,^{{\\,\\mathrm{i}\\,}\\theta _{k}}-{\\,\\mathrm{e}}\\,^{{\\,\\mathrm{i\n\\,}\\theta _{l}}\\big\\vert^{2},  \\label{UCS}\n\\end{equation\nwhere the weight function of the matrix model is a Jacobi third theta\nfunctio\n\\begin{equation}\n\\omega \\left( \\theta \\right) =\\Theta ({\\,\\mathrm{e}}\\,^{{\\,\\mathrm{i}\\,\n\\theta _{j}}|q)=\\sum_{n=-\\infty }^{\\infty }\\mathrm{q}^{n^{2}\/2}\\mathrm{e\n^{in\\theta }.  \\label{weight}\n\\end{equation\nWe show now how this model follows from the Gross-Witten model \\cit\n{GW,Wadia:1980cp}, by using the Villain approximation \\cite{Jose:1977gm}.\nAspects of the relationship between the two models, especially in the\nweak-coupling regime, have already been studied in \\cite{Szabo:2010qv},\nusing orthogonal polynomials. The results of the seminal works \\cit\n{Villain:1974ir,Jose:1977gm} also allow us to extend the relationship\nbetween the Gross-Witten and the Chern-Simons models to the strong-coupling\nregime.\n\nRecall that the Gross-Witten model is a unitary one-matrix model which\narises as the one-plaquette reduction of the combinatorial quantization of\nYang-Mills theory. In two dimensions the reduction is exact and described by\nthe partition function~\\cite{GW} \n\\begin{align}\nZ_{N}^{\\mathrm{GW}}(\\beta )& :=\\int_{U(N)}\\,\\mathrm{d}U~{\\exp }\\left( \\frac\n\\beta }{2}\\,\\mathrm{Tr}\\big(U+U^{\\dag }\\big)\\right)   \\label{GW} \\\\[4pt]\n& =\\int_{0}^{2\\pi }~\\prod\\limits_{i=1}^{N}\\,\\mathrm{d}\\theta _{i}~\\mathrm{e\n^{\\beta \\,\\cos \\theta _{i}}~\\prod\\limits_{i<j}\\,\\sin ^{2}\\left( \\frac{\\theta\n_{i}-\\theta _{j}}{2}\\right) \\ ,  \\notag\n\\end{align\nwhere $\\mathrm{d}U$ denotes the bi-invariant Haar measure for integration\nover the unitary group $U(N)$. The $\\beta $ parameter is usually written in\nterms of the gauge coupling constant as $\\beta =2\/g_{\\mathrm{YM}}^{2}$ \\cit\n{GW} and hence, the strong coupling limit is given by $\\beta \\rightarrow 0$\nand the weak coupling limit by $\\beta \\rightarrow \\infty $.\n\nThe planar or XY model is characterized by a coupling between\nnearest-neighbor spins which has the same analytical form as the potential\nin the Gross-Witten model \\cite{Villain:1974ir,Jose:1977gm} \n\\begin{equation}\nV_{\\beta }=-\\beta \\left[ 1-\\cos \\left( \\theta -\\theta ^{\\prime }\\right)\n\\right] .  \\label{full-V}\n\\end{equation\nThe coefficients of the Fourier expansion of this potential are given b\n\\begin{equation}\n\\mathrm{e}^{\\widetilde{V}(s)}=I_{s}\\left( \\beta \\right) ,  \\label{Bessel}\n\\end{equation\nwhere $I_{n}(z)$ is the modified Bessel function of order $n$. The partition\nfunction of the Gross-Witten model can also be written as the determinant of\na Toeplitz matrix with (\\ref{Bessel}) as entries of the matrix \\cit\n{Bars:1979xb\n\\begin{equation}\nZ_{N}(\\beta )=\\det_{1\\leq i,j\\leq N}\\,\\big[I_{i-j}(2\\beta )\\big]\\ .\n\\label{IT}\n\\end{equation\nOn the other hand, for $U(N)$\\ Chern-Simons theory on $S^{3}$ the\ncorresponding Toeplitz determinant is \\cite{Szabo:2010sd}\\footnote\nSuch a determinant was already considered in \\cite{Onofri:1981qk}. See Sec.\n3.\n\\begin{equation}\nZ_{\\mathrm{CS}}^{U(N)}\\left( S^{3}\\right) =\\det_{1\\leq i,j\\leq N}\\,\\big\na_{i-j}(q)\\big]\\ ,\\qquad \\text{ with }\\quad a_{j}=q^{{j^{2}}\/{2}}\\ .\n\\label{D}\n\\end{equation\nIn the weak-coupling limit $\\beta \\rightarrow \\infty ,$ the Fourier\ncoefficient i\n\\begin{equation}\n\\lim_{\\beta \\rightarrow \\infty }\\mathrm{e}^{\\widetilde{V}(s)}=\\mathrm{e\n^{-s^{2}\/2\\beta }=\\mathrm{e}^{-s^{2}g_{\\mathrm{YM}}^{2}\/4}.  \\label{Gaussian}\n\\end{equation\nIn this limit, the Toeplitz determinant (\\ref{IT}) has the Gaussian\ncoefficients (\\ref{Gaussian}) as entries, and hence it coincides with (\\re\n{D}). Of course, $g_{\\mathrm{YM}}^{2}\\rightarrow 0$ in (\\ref{Gaussian}) and\nthis implies $g_{s}\\rightarrow 0$ on the Chern-Simons theory side as well.\n\nThe prescription in \\cite{Villain:1974ir}, namely (\\ref{Villain}), is valid\nfor both the opposite $\\beta \\rightarrow 0$ and $\\beta \\rightarrow \\infty $\nlimits, using always periodic Gaussians, but including a renormalization\nscale $R_{V}\\left( \\beta \\right) $ and a rescaled inverse temperature $\\beta\n_{V}=f\\left( \\beta \\right) .$ This implies that a correspondence between the\nGross-Witten model and the Chern-Simons matrix model holds for both the\nweak-coupling and the strong-coupling regimes of the model. As seen above\nwith the Toeplitz determinant representation of the matrix model, and also\nin \\cite{Szabo:2010qv} using orthogonal polynomials, the weak-coupling limit\nfollows in a straightforward way. This result, as shown in \\cite{Jose:1977gm\n, also follows by considering decimation \\cite{Kadanoff:1976jb}. It is\nexplained in \\cite{Jose:1977gm} that, after a few iterations of the\nKadanoff-Migdal decimation procedure, any interaction function at reasonably\nlow temperatures generates a new interaction of the Villain typ\n\\begin{equation*}\n\\mathrm{e}^{V_{V}\\left( \\theta -\\theta ^{\\prime }\\right) }=\\sum_{m=-\\infty\n}^{\\infty }\\mathrm{e}^{-\\beta _{V}(\\theta -\\theta ^{\\prime }-2\\pi m)^{2}\/2}.\n\\end{equation*\nIn addition, the critical properties of the two models could be identical by\nconveniently choosing $\\beta _{V}=f\\left( \\beta \\right) $ a function of \n\\beta $ \\cite{Jose:1977gm}. In particular, a comparison of the interaction\nform for weak and strong coupling showed equivalence if \n\\begin{eqnarray}\nf\\left( \\beta \\right) &=&\\beta \\qquad \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\text\nfor\\ \\ \\ }\\beta \\rightarrow \\infty  \\notag \\\\\nf\\left( \\beta \\right) &=&\\left[ 2\\ln \\left( 2\/\\beta \\right) \\right]\n^{-1}\\qquad \\text{for\\ \\ \\ }\\beta \\rightarrow 0.  \\label{limit}\n\\end{eqnarray\nNotice that to compare with the theta function as it appears in the matrix\nmodel (\\ref{UCS}), we have to take into account tha\n\\begin{equation}\n\\sum_{n=-\\infty }^{\\infty }\\mathrm{e}^{-\\beta \\left( \\theta +2\\pi n\\right)\n^{2}}=\\frac{1}{\\sqrt{4\\pi \\beta }}\\sum_{n=-\\infty }^{\\infty }\\mathrm{e\n^{-n^{2}\/(4\\beta )}\\mathrm{e}^{in\\theta }.  \\label{inv}\n\\end{equation\nThe r.h.s. of (\\ref{inv}) is the series expansion of the theta function in \n\\ref{UCS}). Thus, the decimation of the weight function of the Gross-Witten\nmodel coincides with the Villain approximation and it leads to the weight\nfunction of the unitary Chern-Simons matrix model (\\ref{UCS}). In the\nweak-coupling limit $\\beta \\rightarrow \\infty $, we obtain that \ng_{s}=1\/\\beta $ and hence that $g_{s}=g_{\\mathrm{YM}}^{2}\/2.$ This also\nfollows from (\\ref{Gaussian}).\n\nThe strong-coupling regime $\\beta \\rightarrow 0$ is specially interesting.\nNotice that taking $\\beta =0$ ($g_{YM}^{2}=\\infty $) directly leads to the\ncircular ensemble \\cite{Mehta\n\\begin{equation*}\nZ_{N}^{\\mathrm{GW}}(\\beta =0)=\\int_{0}^{2\\pi }\\prod\\limits_{i<j}\\,\\sin\n^{2}\\left( \\frac{\\theta _{i}-\\theta _{j}}{2}\\right) \\prod\\limits_{i=1}^{N}\\\n\\mathrm{d}\\theta _{i}\\ .\n\\end{equation*\nIn \\cite{Szabo:2010qv}, the relationship between this model and Chern-Simons\ntheory was studied. The result (\\ref{limit}) leads to a refined\nunderstanding of this limit $\\beta \\rightarrow 0.$ In particular, the\nrelationship between the coupling constants is now logarithmic $g_{s}=2\\ln\n\\left( 2\/\\beta \\right) =2\\ln \\left( g_{\\mathrm{YM}}^{2}\\right) $ and hence,\nthe coupling constant of the Gross-Witten model is related to the $q$\nparameter of Chern-Simons theory $q=1\/g_{\\mathrm{YM}}^{4}$. To summarize,\nincluding the prefactor\n\\begin{eqnarray}\n\\mathrm{e}^{\\beta \\,\\cos \\theta } &\\approx &\\frac{\\mathrm{e}^{\\beta }}{\\sqrt\n2\\pi \\beta }}\\Theta ({\\,\\mathrm{e}}\\,^{{\\,\\mathrm{i}\\,}\\theta _{j}}|{\\mathrm\ne}}\\,^{{-1\/\\beta }}),\\text{ with }g_{s}=g_{\\mathrm{YM}}^{2}\/2\\text{ for }g_\n\\mathrm{YM}}\\rightarrow 0,  \\label{result} \\\\\n\\mathrm{e}^{\\beta \\,\\cos \\theta } &\\approx &\\Theta ({\\,\\mathrm{e}}\\,^{{\\\n\\mathrm{i}\\,}\\theta _{j}}|\\,{\\mathrm{e}}^{{-1\/\\beta }_{\\mathrm{V}}{}}),\\text{\nwith }g_{s}=2\\log (g_{\\mathrm{YM}}^{2})\\text{ for }g_{\\mathrm{YM\n}\\rightarrow \\infty .  \\notag\n\\end{eqnarray\nIt is also worth mentioning that, already in the original paper on the XY\nmodel \\cite{Villain:1974ir}, an approximation valid for both limits was also\ngive\n\\begin{equation}\n\\mathrm{e}^{\\beta \\,\\cos \\theta }\\approx R_{V}\\left( \\beta \\right)\n\\sum_{m=-\\infty }^{\\infty }\\mathrm{e}^{-\\beta _{V}(\\theta -2\\pi m)^{2}\/2},\n\\label{general}\n\\end{equation\nwit\n\\begin{eqnarray}\nR_{V}\\left( \\beta \\right)  &=&I_{0}\\left( \\beta \\right) \\sqrt{2\\pi \\beta _{V\n}  \\label{general2} \\\\\n\\mathrm{e}^{-\\beta _{V}\/2} &=&I_{1}\\left( \\beta \\right) \/I_{0}(\\beta ), \n\\notag\n\\end{eqnarray\nwhere $I_{1}\\left( \\beta \\right) $ and $I_{0}(\\beta )$ are, as in (\\re\n{Bessel}), Bessel functions. These are the first two Fourier coefficients of\nthe expansion of the l.h.s. of (\\ref{general}). The values for the prefactor \n$R_{V}\\left( \\beta \\right) $ and the coupling constant $\\beta _{V}$ are then\nfound by imposing the first two Fourier coefficients of the two expressions\nin (\\ref{general}) to coincide. The limits $\\beta \\rightarrow 0$ and $\\beta\n\\rightarrow \\infty $ of (\\ref{general2}) coincide with the previous results.\nIt was shown in \\cite{Villain:1974ir} that this approximation is rather\ngood, even for values close to the critical temperature. This approximation\nhas been studied in further detail in \\cite{Janke:1986ej}, where it was\nfound that the convergence is better for the strongly coupled regime $\\beta\n\\rightarrow 0.$ In addition, the result (\\ref{general}) can be extended,\nwith good convergence in both limits, to the case where the original\nfunction is $\\omega \\left( \\theta \\right) =\\beta \\cos \\theta +\\gamma \\cos\n\\left( 2\\theta \\right) $ \\cite{Janke:1986ej}. This suggests that not only\nthe Gross-Witten model, but more complex multicritical unitary matrix models \n\\cite{Periwal}, can also be expressed in terms of the unitary Chern-Simons\nmatrix model, only with a more sophisticated expression for the coupling\nconstant and the prefactor.\n\n\\subsection{Gaussian behavior and closed string interpretation at weak\ncoupling}\n\nThe relationship between the Gross-Witten model in the weak-coupling limit\nand the semiclassical limit of the Chern-Simons matrix model, indicates that\nthe Gross-Witten model in this regime should be related to closed\ntopological strings \\cite{Ooguri:2002gx}. The reason is that the\nsemiclassical limit ($k\\rightarrow \\infty $) of the $U(N)$ Chern-Simons on \nS^{3}$ free energy also coincides with the nonperturbative part of the total\nfree energy. The Chern-Simons free energy can be suitably expressed in terms\nof nonperturbative and perturbative contribution\n\\begin{equation}\nF_{\\mathrm{CS}}=\\log Z_{\\mathrm{CS}}=F_{\\mathrm{np}}+F_{\\mathrm{p}}\\ .\n\\label{np+p}\n\\end{equation\nThe nonperturbative contribution $F_{\\mathrm{np}}$ is the logarithm of the\nmeasure factor in the path integral, which is not captured by Feynman\ndiagrams, and it gives the exact Chern-Simons partition function in the\nsemiclassical limit $k\\rightarrow \\infty $~\\cite[eq.~(2.8)]{Ooguri:2002gx}.\nIt has the explicit expressio\n\\begin{equation}\nF_{\\mathrm{np}}=\\log \\Big(\\,\\frac{\\left( 2\\pi \\,g_{s}\\right) ^{N^{2}\/2}}\n\\mathrm{vol}\\big(U(N)\\big)}\\,\\Big)\\ .  \\label{FNP}\n\\end{equation\nThis free energy (\\ref{FNP}), given by a Hermitian Gaussian matrix model,\nhas an expansion which can be interpreted in terms of closed topological\nstring theory on the resolved conifold geometry~\\cite{Ooguri:2002gx}. \nSee~\\cite{Ooguri:2002gx} for equivalent string theory and gauge theory\ninterpretations.\n\nDue to the correspondence between the Chern-Simons and the Gross-Witten\nmatrix models, the latter should have (\\ref{FNP}) as free energy in the\nweak-coupling limit $\\beta \\rightarrow \\infty $. Consider the expression for\nthe free energy of the Gross-Witten model for finite $N$ and $g_{\\mathrm{YM\n}\\rightarrow 0$ \\cite{Neuberger:1980qh\n\\begin{equation*}\nF_{\\mathrm{GW}}\\simeq \\frac{2N}{g_{\\mathrm{YM}}^{2}}-\\frac{N}{2}\\ln (2\\pi )\n\\frac{N^{2}}{2}\\ln (\\frac{2}{g_{\\mathrm{YM}}^{2}})+\\sum_{j=1}^{N-1}\\ln j!\n\\mathcal{O}(g_{YM}^{2}).\n\\end{equation*\nThis coincides with the free energy of a Hermitian Gaussian matrix model \n\\cite{Mehta\n\\begin{equation}\nZ_{\\mathrm{G}}\\left( \\beta \\right) =\\mathrm{e}^{\\beta N}\\int_{-\\infty\n}^{\\infty }\\mathrm{e}^{-\\sum_{j=1}^{N}\\beta x_{j}^{2}\/2}\\prod_{j<k}\\left(\nx_{j}-x_{k}\\right) ^{2}.\n\\end{equation\nThis relationship between the Gross-Witten and the Gaussian matrix model\nagrees with \\cite{Baik:2007}. Notice that the term $\\mathrm{e}^{\\beta N}$\nactually corresponds to the $\\mathrm{e}^{\\beta }$ term in the Villain\napproximation (\\ref{result}) and implies that one has to consider the\nGross-Witten model with the potential (\\ref{full-V})\\footnote\nThis form of the potential corresponds exactly to the Wilson lattice action.}\n, instead of the one in (\\ref{GW}).\n\n\\section{The Villain lattice action and Abelian\/$q$-deformed 2d Yang-Mills\ntheory}\n\nWe begin by discussing the heat-kernel action, which was introduced in\nlattice gauge theory, at least in part, as an alternative to the Wilson\naction that also provided a natural extension to the non-Abelian case \\cit\n{Susskind:1979up,Stone:1978pe,Menotti:1981ry} of the Villain approximation\nof the two-dimensional XY model \\cite{Villain:1974ir}, which had been\ncrucially used in the study of $U(1)$ lattice gauge theories \\cit\n{Banks:1977cc,Peskin:1977kp}.\n\nThe setting is quantum Yang-Mills theory with gauge group $U(N)$ on an\noriented closed Riemann surface $\\Sigma _{h}$ of genus $h$ and unit area\nform $\\mathrm{d}\\mu $~ \\cite{review}. The action i\n\\begin{equation}\nS_{\\mathrm{YM}}=-\\frac{1}{4g_{YM}}\\,\\int_{\\Sigma _{h}}\\,\\mathrm{d}\\mu \n\\mathrm{Tr}\\,F^{2}\\ ,  \\label{continuum}\n\\end{equation\nwhere $g_{s}$ plays the role of the coupling constant, $F$ is the field\nstrength of a matrix gauge connection, and $\\mathrm{Tr}\\,$ is the trace in\nthe fundamental representation of $U(N)$. A lattice regularization of the\ngauge theory relies on a triangulation of the two-dimensional manifold \n\\Sigma $ with group matrices situated along the edges~\\cite{Migdal75}. The\npath integral is then approximated by the finite-dimensional unitary matrix\nintegra\n\\begin{equation}\n\\mathcal{Z}_{\\mathrm{M}}=\\int \\,\\prod_{\\mathrm{edges}~\\ell }\\,\\mathrm{d\nU_{\\ell }~\\prod\\limits_{\\mathrm{plaquettes}~P}\\,Z_{P}\\left[ U_{P}\\right] \\ ,\n\\label{latticeint}\n\\end{equation\nwhere $\\mathrm{d}U_{\\ell }$ denotes Haar measure on $SU(N)$ and the holonomy \n$U_{P}=\\prod\\nolimits_{\\ell \\in P}\\,U_{\\ell }$ is the ordered product of\ngroup matrices along the links of a given plaquette. The local factor $Z_{P\n\\left[ U_{P}\\right] $ is a suitable gauge invariant lattice weight that\nconverges in the continuum limit to the Boltzmann weight for the Yang-Mills\naction (\\ref{continuum}). This expression for $\\mathcal{Z}_{\\mathrm{M}}$\nleads to the matrix models presented in the previous section after the\nsuitable choice of lattice action.\n\nLet us discuss now the use of the heat kernel for the lattice weight $Z_{P\n\\left[ U_{P}\\right] $. In this case, the lattice action has many interesting\nfeatures~\\cite{Menotti:1981ry} and is the usual choice in two-dimensional\nYang-Mills theory~\\cite{review}. It leads to the well-known group theory\nexpansion of the partition function~\\cite{Migdal75,Rusakov\n\\begin{equation}\n\\mathcal{Z}_{\\mathrm{M}}=\\sum_{\\lambda }\\,\\left( \\dim \\lambda \\right)\n^{2-2h}\\,\\exp \\big(-g_{YM}^{2}\\,C_{2}(\\lambda )\\big)\\ ,  \\label{HK1}\n\\end{equation\nwhere the sum runs through all isomorphism classes $\\lambda $ of irreducible\nrepresentations of the $SU(N)$ gauge group, $\\dim \\lambda $ is the dimension\nof the representation $\\lambda $, and $C_{2}(\\lambda )$ is the quadratic\nCasimir invariant of $\\lambda $. This expression for the partition function\nis a particular case of the propagator \\cite{Menotti:1981ry,review\n\\begin{equation*}\n\\exp \\left( -S_{\\mathrm{HK}}\\left( U\\right) \\right) =\\left\\langle \\mathbb{I\n\\right\\vert \\exp \\left( \\frac{1}{2}g_{YM}^{2}\\Delta \\right) \\left\\vert\nU\\right\\rangle \\equiv K(U,\\frac{g_{YM}^{2}}{2}),\n\\end{equation*\n\\newline\nwhich can be written as \\cite{Menotti:1981ry\n\\begin{equation}\nK(U,\\frac{g_{YM}^{2}}{2})=\\sum_{\\lambda }\\dim \\lambda \\chi _{\\lambda }\\left(\nU\\right) \\exp \\left( -g_{YM}^{2}C_{2}(\\lambda \\right) ),  \\label{K}\n\\end{equation\n\\newline\nwhere the sum runs over all irreducible unitary representations of the gauge\ngroup, $\\chi _{\\lambda }\\left( U\\right) $ is the character of such a\nrepresentation, $\\dim \\lambda =$ $\\chi _{\\lambda }\\left( \\mathbb{I}\\right) $\nits dimension and $C_{2}(\\lambda )$ the Casimir of the representation.\nWriting (\\ref{K}) in terms of the elements of the Young tableaux that labels\nthe representation $\\lambda $ leads to a discrete matrix model\nrepresentation \\cite{review}.\n\n\\subsubsection{$q$ deformation}\n\nIn recent years, the relationship between two-dimensional Yang-Mills theory\nand Chern-Simons theory on Seifert manifolds has been understood in further\ndetail \\cite{Aganagic:2004js,Beasley:2005vf,Beasley:2009mb} (see also \\cit\n{Blau:2006gh,Arsiwalla:2005jb,Caporaso:2005ta}). Seifert manifolds $M(h,p)$\nare nontrivial circle bundles (of monopole degree $p$) over two-dimensional\nsurfaces of genus $h$. The simplest case, the trivial fibration, \nM(h,0)=\\Sigma _{h}\\times S^{1}$ was studied in detail in \\cit\n{Witten:1988hf,Blau:1993tv}.\n\nChern-Simons theory on Seifert manifolds has been the subject of much\ninterest in the study of topological strings \\cite{Marino:2004uf} and has a\ndirect relationship with a $q$ deformation of the two-dimensional Yang-Mills\ntheory discussed above, when the manifold is a sphere, $\\Sigma _{0}=S^{2}$.\nIn particular, the partition function of $q$-deformed 2d YM on a closed\nRiemann surface of genus $h$ is given by \\cite{Aganagic:2004js\n\\begin{equation}\nZ_{YM}^{q}(\\Sigma _{h})=\\sum_{\\lambda }\\left( \\dim _{q}\\left( \\lambda\n\\right) \\right) ^{2-2h}q^{\\frac{p}{2}C_{2}(\\lambda )},  \\label{q}\n\\end{equation\nwhere $\\dim _{q}\\left( \\lambda \\right) $ is the $q$ deformation of the\ndimensions of $sl_{n}$ representations, i.e. the quantum dimensions $\\dim\n_{q}(\\lambda )$~\\cite{fuchs}, $p$ is a positive integer parameter and, as\nusual, $C_{2}(\\lambda )$ is the Casimir of the representation $\\lambda $.\nThis is related to the partition function of Chern-Simons theory on a circle\nfibration (with Chern class $p$) over $\\Sigma _{h}$, which is a Seifert\nspace. In the case $\\Sigma _{h}=S^{2}$ and $p>1$, the Seifert manifold is\nthe lens space $S^{3}\/\\mathbb{Z}_{p}.$ If $p=1,$ then the connection is with\nChern-Simons theory on $S^{3}$, the case studied here.\n\n\\subsection{Abelianization and q deformation}\n\nThe propagator (\\ref{K}) can be alternatively written in terms of the\nelements of $U(N)$, and then one obtains a unitary matrix model expression.\nWhen written in terms of the invariant angles of the gauge group, it is\ngiven by \\cite{Menotti:1981ry\n\\begin{equation}\n\\exp \\left( -S_{\\mathrm{HK}}\\left( \\theta _{1},...\\theta _{N}\\right) \\right)\n=\\mathcal{N}\\sum_{\\left\\{ l\\right\\} =-\\infty }^{\\infty }\\prod\\limits_{i<j\n\\frac{\\theta _{i}-\\theta _{j}+2\\pi \\left( l_{i}-l_{j}\\right) }{2\\sin \\left[\n\\theta _{i}-\\theta _{j}+2\\pi \\left( l_{i}-l_{j}\\right) \\right] }\\exp \\left[ \n\\frac{1}{g_{YM}^{2}}\\sum_{j=1}^{N}\\left( \\theta _{j}+2\\pi l_{j}\\right) ^{2\n\\right] ,  \\label{HK}\n\\end{equation\nwhere $\\mathcal{N}$ stands for some normalization. This is a rather complex\nmodel, and therefore the heat-kernel case, in contrast to the two cases\nstudied in the previous section, is not studied with a unitary matrix model\nbut rather with a discrete matrix model that follows from (\\ref{HK1}). We\nshow now that a simplification of (\\ref{HK}) leads to the Chern-Simons\nmatrix model. The exponential part in the r.h.s. is the Abelian $U(1)$\nVillain action, which is just a theta function (\\ref{Villain}\n\\begin{equation}\n\\exp (-S_{\\mathrm{V}}\\left( \\theta \\right) )=\\sum_{l=-\\infty }^{\\infty \n\\mathrm{e}^{-\\frac{1}{g_{YM}^{2}}\\left( \\theta +2\\pi l\\right) ^{2}},\n\\label{A-Villain2}\n\\end{equation\nand the straightforward generalization to $U(N)$ gives a propagato\n\\begin{eqnarray}\n\\overline{K}(U,\\frac{g_{YM}^{2}}{2}) &=&\\sum_{\\{l\\}=-\\infty }^{\\infty }\\exp\n\\left[ -\\frac{1}{g_{YM}^{2}}\\sum_{j=1}^{N}\\left( \\theta _{j}+2\\pi\nl_{j}\\right) ^{2}\\right]   \\label{V-action} \\\\\n&=&\\prod\\limits_{i=1}^{N}\\exp (-S_{\\mathrm{V}}(\\theta _{i}))=\\exp \\left( -S_\n\\mathrm{Villain}}\\left( \\theta _{1},...\\theta _{N}\\right) \\right)   \\notag\n\\end{eqnarray\nthat defines an action which is just the direct product of the Abelian\nVillain action (\\ref{A-Villain2}). The corresponding one-plaquette model is\ngiven by a $U(N)$ matrix integral, which is the unitary integration of the\npropagator. The partition function is then given by the matrix mode\n\\begin{equation}\nZ_{N}=\\int_{0}^{2\\pi }\\prod\\limits_{i=1}^{N}\\,\\frac{\\mathrm{d}\\theta _{i}}\n2\\pi }~\\sum_{n=-\\infty }^{\\infty }\\mathrm{e}^{-\\frac{1}{g_{YM}^{2}}\\left(\n\\theta _{i}+2\\pi n\\right) ^{2}}\\prod\\limits_{k<l}\\,\\big\\vert{\\,\\mathrm{e}\n\\,^{{\\,\\mathrm{i}\\,}\\theta _{k}}-{\\,\\mathrm{e}}\\,^{{\\,\\mathrm{i}\\,}\\theta\n_{l}}\\big\\vert^{2},  \\label{UCSb}\n\\end{equation\nwhich, recalling the identity (\\ref{inv}), is the unitary matrix model\ndescription of $U(N)$ Chern-Simons theory on $S^{3}$ (\\ref{UCS}) discussed\nso far and previously studied in \\cit\n{Okuda:2004mb,Szabo:2010sd,Ooguri:2010yk}. Hence, $Z_{N}=Z_{\\mathrm{CS\n}^{U(N)}\\left( S^{3}\\right) $, by identifying $g_{s}=g_{YM}^{2}\/2$. Thus,\nthe Abelianization of the heat kernel (\\ref{HK}), given by the\nstraightforward extension of the $U(1)$ Abelian Villain action to the \nU(1)^{N}$ case (\\ref{V-action}), leads to $U(N)$ Chern-Simons theory on \nS^{3}$.\n\nAt the level of the matrix model, this straightforward generalization of the\nVillain action was already considered in an interesting paper by Onofri \\cit\n{Onofri:1981qk}, that analyzed this problem and presented a detailed study\nof the matrix model (\\ref{UCSb}). It is remarkable that the results in that\npaper describe pure Chern-Simons theory on $S^{3}$, with $U(N)$ and $SU(N)$,\na fact that has not been pointed out so far. Some expressions in \\cit\n{Onofri:1981qk} are manifestly Chern-Simons observables, like the partition\nfunction of $U(N)$ Chern-Simons theory on $S^{3}$.\n\nWe have seen the Abelianization of the heat kernel, using its expression in\nterms of the invariant angles of the gauge group (\\ref{HK}). Let us now look\nat the relationship between the choice of the Villain lattice action and the \n$q$ deformation of the heat-kernel 2d Yang-Mills theory, using instead the\ncharacter expansion (\\ref{K}).\n\nNotice that the $q$-deformed 2d Yang-Mills (\\ref{q}) is just given by the\nusual expression for two-dimensional Yang-Mills theory based on the\nheat-kernel action and defined on a compact manifold of genus $h$, but with \nq$ dimensions instead of ordinary dimensions, $\\dim (\\lambda )\\rightarrow\n\\dim _{q}(\\lambda )$. We will now show that choosing (\\ref{V-action}) with\nthe Villain action (\\ref{A-Villain}) as a lattice action is actually\nequivalent to $q$-deformed 2d Yang-Mills. Since choosing the Villain action\nis tantamount to an Abelian projection of the two-dimensional Yang-Mills\ntheory with the heat-kernel lattice action, this implies that the\nAbelianization and the $q$ deformation of the latter lead to the same theory.\n\nAn expression of the type (\\ref{q}) is valid for 2d Yang-Mills on a compact\nmanifold of genus $h$ and it can be constructed from the central heat\nkernel, which is also the propagator on a cylinder \\cite{review\n\\begin{equation}\nK(\\frac{g_{YM}^{2}}{2},U,U^{\\prime })=\\sum_{\\lambda }\\chi _{\\lambda }\\left(\nU\\right) \\chi _{\\lambda }\\left( U^{\\prime }\\right) \\exp \\left(\n-g_{YM}^{2}C_{2}(\\lambda \\right) ),  \\label{hk}\n\\end{equation\nwith holonomies $U$ and $U^{\\prime }$ on the two disks of the cylinder and\nwhere $\\chi _{\\lambda }\\left( U\\right) $ is the character of the irreducible\nrepresentation $\\lambda ,$ as in (\\ref{K}) above. Recall that $\\chi\n_{\\lambda }\\left( U=\\mathbb{I}\\right) =\\dim (\\lambda )$. Hence, if \nU=U^{\\prime }=\\mathbb{I}$, (\\ref{hk})\\ is the expression for the partition\nfunction of two-dimensional Yang-Mills theory on $S^{2}$. If only $U^{\\prime\n}=\\mathbb{I},$ then it describes two-dimensional Yang-Mills on a dis\n\\begin{equation}\nK(\\frac{g_{YM}^{2}}{2},U,\\mathbb{I})=\\sum_{\\lambda }\\chi _{\\lambda }\\left(\nU\\right) \\dim \\lambda \\exp \\left( -g_{YM}^{2}C_{2}(\\lambda \\right) ).\n\\label{disk}\n\\end{equation\nThis is the heat kernel and it is also the amplitude of a plaquette that\nleads, by gluing, to (\\ref{HK}) \\cite{review}. Since (\\ref{disk}) is also\nthe fundamental solution of the diffusion equation, then, as is well-known,\ntwo-dimensional Yang-Mills theory based on the heat kernel can be understood\nas a diffusion process on the gauge group manifold.\n\nThe relationship between the Chern-Simons model constructed with the Villain\naction and the $q$ deformation of two-dimensional Yang-Mills theory can be\nreadily seen by considering the character expansion of the $U(N)$ Villain\naction (\\ref{V-action}) given in \\cite{Onofri:1981qk\n\\begin{equation}\n\\frac{\\mathrm{e}^{-S_{\\mathrm{Villain}}(U)}}{Z}=\\sum_{m_{1}\\geq m_{2}\\geq\n\\ldots \\geq m_{N}=0}\\chi _{(m_{1},\\ldots ,m_{N})}(U)q^{\\frac{1}{2\n\\sum_{i=1}^{N}m_{i}^{2}}\\prod_{j>i}\\left( \\frac{q^{m_{i}-m_{j}+j-i}-1}\nq^{j-i}-1}\\right) ,  \\label{ch}\n\\end{equation\nwhere the sum is over the integers $\\left\\{ m_{i}\\right\\} $ with $i=1,...,N$\nand $Z$ is the partition function of the unitary matrix model, which has the\nexplicit for\n\\begin{equation*}\nZ=\\left( \\frac{g_{\\mathrm{YM}}^{2}}{8\\pi }\\right) ^{\\frac{N}{2\n}\\prod_{k=1}^{N-1}\\left( 1-q^{N-k}\\right) ^{k},\\qquad q=\\mathrm{e}^{-\\frac\ng_{\\mathrm{YM}}^{2}}{2}}.\n\\end{equation*\nThis character expansion of the Villain action suggests that such an action\nleads to the usual propagator of two-dimensional Yang-Mills theory based on\nthe heat kernel (\\ref{disk}) but with $q$ dimensions instead of dimensions,\nsince the last term in (\\ref{ch}) gives an explicit expression for quantum\ndimensions. An elementary manipulation of the product shows this explicitl\n\\begin{equation*}\n\\prod_{j>i}\\left( \\frac{q^{m_{i}-m_{j}+j-i}-1}{q^{j-i}-1}\\right)\n=\\prod_{j>i}q^{\\frac{m_{j}-m_{i}}{2}}\\frac{[{m_{i}-m_{j}+j-i}]_{q}}{[j-i]_{q\n}=q^{\\frac{1}{2}\\sum_{l=1}^{N}(N-2l+1)m_{l}}\\mathrm{dim}_{q}(\\lambda ),\n\\end{equation*\nwhere $\\lambda $ denotes the unitary irreducible representation of the gauge\ngroup, characterized by a partition whose Young tableaux has columns with \n\\left\\{ m_{i}\\right\\} $ boxes. The explicit expression for the quantum\ndimensions is given, as in \\cite{Aganagic:2004js}, b\n\\begin{equation*}\n\\mathrm{dim}_{q}(\\lambda )=\\prod_{j>i}\\frac{[{m_{i}-m_{j}+j-i}]_{q}}\n[j-i]_{q}},\\text{ with the }q\\text{-number }[x]_{q}=\\frac{q^{\\frac{x}{2\n}-q^{-\\frac{x}{2}}}{q^{\\frac{1}{2}}-q^{-\\frac{1}{2}}}.\n\\end{equation*\nPutting all together, we see that the Casimir term is manifest and appears\nexactly as in \\cite{Aganagic:2004js\n\\begin{equation*}\nC_{2}(\\lambda\n)=\\sum_{i=1}^{N}m_{i}^{2}+(N-2i+1)m_{i}=\\sum_{i=1}^{N}m_{i}(m_{i}-2i+1)+\n\\sum_{i=1}^{N}m_{i},\n\\end{equation*\nand hence we obtai\n\\begin{equation}\n\\frac{\\mathrm{e}^{-S_{\\mathrm{Villain}}(U)}}{Z}=\\sum_{\\lambda }\\chi\n_{\\lambda }(U)q^{\\frac{1}{2}C_{2}(\\lambda )}\\mathrm{dim}_{q}(\\lambda ).\n\\label{propagator}\n\\end{equation\nThus, the r.h.s of (\\ref{ch}) is the disk amplitude for $q$-deformed\ntwo-dimensional Yang-Mills theory with $p=1$, which, by the same procedure\ndiscussed above, and explained in detail in \\cite{review}, leads to the\nexpression for the partition function (\\ref{q}). This identity is actually a\nparticular case of the Kostant identity, which gives a character expansion\nof theta functions of a lattice \\cite{Kostant}. This specific form of the\nidentity was then rediscovered later on, in \\cite{Onofri:1981qk}, in the\ncontext of the Villain lattice action, and it also appears much later in \n\\cite{Aganagic:2005dh}, where, working from the r.h.s. of (\\ref{propagator})\na theta function expression was found, which is, as we have seen here, the\nVillain lattice action. In this paper, we have also shown, by using the\nequivalent formulation of the lattice action in terms of the invariant\nangles (\\ref{HK}), that it follows from taking only the Abelian part of the\nheat kernel.\n\nThe fact that the Abelianization and the $q$ deformation described here are\nequivalent is qualitatively consistent with the fact, explained for example\nin \\cite{Witten:1989rw}, that a $q$ deformation of a Lie group $G$ is not a\ngroup and lacks its symmetry, whereas the maximal torus $T$ of $G$ remains\nan ordinary symmetry group after the symmetry breaking inherent in the\ntransformation of a Lie group into a quantum group.\n\nTo conclude, let us mention that inspection of the analogous expression for \nSU(N)$ in \\cite{Onofri:1981qk} shows that $\\dim _{q}\\lambda $ can also be\nwritten as the character $\\chi _{\\lambda }\\left( T_{q}\\right) $, where \nT_{q}\\in SL\\left( N,\\mathbb{C}\\right) $ is given by $\\mathrm{diag}\\left[\nq^{N-1},q^{N-3},...,q^{3-N},q^{1-N}\\right] .$ Hence, the propagator is now\nof the type $K(\\frac{g^{2}}{2},U,T_{q})$ instead of (\\ref{disk}), which\nsuggests that diffusion does not take place in the whole gauge group.\nIndeed, the explicit form of the matrix model indicates diffusion on the\nmaximal torus $U(1)\\times ...\\times U(1)$ of the gauge group $U(N)$. This is\nin agreement with a previous result that related Chern-Simons theory on \nS^{3}$ with Brownian motion on the Weyl chamber of the gauge group \\cite{dHT\n.\n\n\\section{Conclusions and Outlook}\n\nWe have seen how the unitary matrix model that describes $U(N)$\\\nChern-Simons theory on $S^{3}$ arises from studying two-dimensional\nYang-Mills theory with the Villain lattice action and we have compared it\nboth with the Wilson and the heat-kernel lattice action cases.\n\nRegarding the former, we have seen that the Gross-Witten model is related to\nthe Chern-Simons matrix model both in the weak-coupling and the\nstrong-coupling regimes. As we have seen in Sec. 2.1., one of the\nimplications is that the Gross-Witten model, which describes two-dimensional\nYang-Mills theory on $\\mathbb{R}^{2}$, coincides in the weak-coupling limit\nwith the nonperturbative part of Chern-Simons theory on $S^{3}$, and\nconsequently has the same string theory interpretation \\cite{Ooguri:2002gx}.\nIn both cases, the free energy is given by a Hermitian Gaussian matrix\nmodel. In spite of the apparent simplicity of such a matrix model, it is\nactually relevant in the study of subsectors of $\\mathcal{N}$=4\nsupersymmetric gauge theory and their relationship with two-dimensional\nYang-Mills theory (see \\cite{Giombi:2009ms}, for example). It is possible\nthat taking into account Wilson loops in our discussion would lead to a\nrelationship with that line of research.\n\nNotice also that the general approximation of the Gross-Witten weight (\\re\n{general}) implies that a small modification of the Gross-Witten model\npotential leads to the Chern-Simons matrix model. The interest of this\nresult lies in a possible connection between the Chern-Simons matrix model\nand the unitary matrix models that appear in the study of phase transitions\nof weakly coupled gauge theories \\cite{Aharony:2003sx}.\n\nThe heat-kernel has a character expansion which is the basis of the study of\n2d Yang-Mills theory. However, it also can be expressed in the invariant\nangles of the gauge group (the unitary group in our case), as pointed out in \n\\cite{Menotti:1981ry}. The unitary matrix model that follows from this\nrepresentation is not used in the heat-kernel case, due to its complexity.\nHowever, we also have seen that an Abelian projection of the heat-kernel\nlattice action leads to a $U(1)^{N}$ lattice action, which is the Villain\nlattice action. After unitary integration of the resulting propagator, the\ncorresponding matrix model is now the $U(N)$ Chern-Simons matrix model for \nS^{3}$. On the other hand, since the character expansion of the Villain\nlattice action gives the $q$-deformed 2d Yang-Mills propagator\n(Kostant-Onofri identity), we see that the Abelianization of (\\ref{HK})\ncoincides with the $q$ deformation of (\\ref{disk}), given by (\\re\n{propagator}). In addition, it shows how the $q$ propagator directly leads\nto the unitary Chern-Simons matrix model, instead of the (equivalent)\nChern-Simons Hermitian matrix model.\n\nPrecisely, and to conclude, in the Appendix, the relationship between the\nunitary and the Hermitian versions of the Chern-Simons matrix model is\nstudied in detail, focusing also in the rotation of the contours of\nintegration.\n\n\\subsection*{Acknowledgments}\n\nThanks to Sergio Iguri and Fokko van de Bult for comments and\ncorrespondence. The work of MT has been supported by the project\n\\textquotedblleft Probabilistic approach to finite and infinite dimensional\ndynamical systems\\textquotedblright\\ (PTDC\/MAT\/104173\/2008) at the\nUniversidade de Lisboa.\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and main results}\n\nWe consider a Galton-Watson branching process with $N$ types of particles\nlabelled $1,2,...,N$ \\ and denote by\n\\begin{equation*}\n\\mathbf{Z}(n)=(Z_{1}(n),...,Z_{N}(n)),\\quad \\mathbf{Z}(0)=(1,0,...,0)\n\\end{equation*\nthe population vector at time $n\\in \\mathbb{Z}_{+}=\\left\\{ 0,1,...\\right\\} \n. Along with $\\mathbf{Z}(n)$ we deal with the process \\\n\\begin{equation*}\n\\mathbf{Z}(m,n)=(Z_{1}(m,n),...,Z_{N}(m,n)),\n\\end{equation*\nwhere $Z_{i}(m,n)$ is the number of type $i$ particles existing in $\\mathbf{\n}(\\cdot )$ at moment $m<n$ and having nonempty number of descendants at\nmoment $n$. We agree to write $Z_{i}(n,n)=Z_{i}(n)$.\n\nThe process $\\mathbf{Z}( \\cdot ,n) $ is called a reduced branching process\nand can be thought of as the family tree relating the individuals alive at\ntime $n$. An important characteristic of the reduced process is the birth\nmoment $\\beta _{n}$ of the most recent common ancestor (MRCA) of all\nindividuals existing in the population at moment $n$ defined as \\\n\\begin{equation*}\n\\beta _{n}=\\max \\left\\{ m\\leq n-1:Z_{1}( m,n) +Z_{2}( m,n)\n+...+Z_{N}(m,n)=1\\right\\} .\n\\end{equation*}\n\nThe structure of the family tree and the asymptotic distribution of the\nbirth moment of the MRCA for single-type Galton-Watson branching processes\nhave been studied in \\cite{FP},\\cite{FZ},\\cite{LS} and \\cite{Zub}. The case\nof multitype indecomposable critical Markov branching processes was\nconsidered in \\cite{Yak83}. Family trees for more general models of\nbranching processes were investigated in \\cite{BorVat}, \\cite{FVat99},\\cit\n{Sag85},\\cite{Sag87},\\cite{Sag95},\\cite{Vat2003},\\cite{VD06},\\cite{VV79}.\nHowever, the reduced processes for decomposable branching processes have not\nbeen analyzed yet. We fill this gap in the present paper and study various\nproperties of the family tree for a particular case of the decomposable\nGalton-Watson branching processes. Namely, we consider the Galton-Watson\nbranching process with $N$ types of particles labelled $1,2,...,N$ in which\na type $i$ parent may produce individuals of types $j\\geq i$ only. This\nmodel may be viewed as a stochastic model for the sizes of a geographically\nstructured population occupying $N$ islands, the location of a particle\nbeing considered as its type. The reproduction laws of particles depend on\nthe island on which the particles are located. The newborn particles of\nisland $i\\leq N-1$ either stay at the same island or migrate, just after\ntheir birth to the islands $i+1,i+2,...,N$. Particles of island $N$ do not\nmigrate.\n\nWe investigate the structure of the family tree of this process, the\ndistributions of the birth moment $\\beta _{n}$ and the type $\\zeta _{n}$ of\nthe MRCA. It is shown, in particular, that, as $n\\rightarrow \\infty $ the\nconditional reduced process\n\\begin{equation*}\n\\left\\{ \\mathbf{Z}(n^{t}\\log n,n),0\\leq t<1|\\mathbf{Z}(n)\\neq \\mathbf{0\n\\right\\}\n\\end{equation*\nconverges in a certain sense to an $N-$dimensional inhomogeneous branching\nprocess $\\left\\{ \\mathbf{R}(t),0\\leq t<1\\right\\} $ which, for $t\\in \\lbrack\n0,2^{-(N-1)})$ consists of a single particle of type $1$ only and for $t\\in\n\\lbrack 2^{-(N-i+1)},2^{-(N-i)}),i=2,...,N$ consists of type $i$ particles\nonly. These particles are born at moment $t=2^{-(N-i+1)}$ and die at moment \nt=2^{-(N-i)}$ producing at this moment a random number of descendants having\ntype $\\min (i+1,N)$. This gives a macroscopic view on the structure of the\nfamily tree of the process.\n\nOn the other hand, for each $i=1,2,...,N-1$ the conditional process\n\\begin{equation*}\n\\left\\{ \\mathbf{Z}((y+(\\log n)^{-1})n^{2^{-(N-i)}},n),0<y<\\infty \\big|\\\n\\mathbf{Z}(n)\\neq \\mathbf{0}\\,\\right\\}\n\\end{equation*\nconverges in a certain sense, as $n\\rightarrow \\infty $ to a continuous-time\nhomogeneous Markov branching process $\\left\\{ \\mathbf{U}_{i}(y),0\\leq\ny<\\infty \\right\\} $ which is initiated at time $y=0$ by a random number of\ntype $i$ particles. These type $i$ particles have an exponential life-length\ndistribution. Dying each of them produces either two particles of type $i$\nor one particle of type $i+1$ (both options with probability 1\/2). Particles\nof type $i+1$ in this process are immortal and produce no offspring. This\nprovides a microscopic view on the structure of the family tree.\n\nTo present our results in a more formal way we need some notation. Let \n\\mathbf{e}_{i}$ be a vector whose $i$-th component is equal to one while the\nremaining are zeros. The first moments of the components of $\\mathbf{Z}(n)$\nwill be denoted a\n\\begin{equation*}\nm_{ij}(n)=\\mathbf{E}\\left[ Z_{j}(n)|\\mathbf{Z}\\left( 0\\right) =\\mathbf{e}_{i\n\\right]\n\\end{equation*\nwith $m_{ij}=m_{ij}(1)$ \\thinspace being the average number of children of\ntype $j$ produced by a particle of type $i$.\n\nSince $m_{ij}=0$ if $i>j$, the mean matrix $\\mathbf{M}$ of the decomposable\nGalton-Watson branching process has the form\n\\begin{equation}\n\\mathbf{M=}(m_{ij})_{i,j=1}^{N}=\\left(\n\\begin{array}{ccccc}\nm_{11} & m_{12} & ... & ... & m_{1N} \\\\\n0 & m_{22} & ... & ... & m_{2N} \\\\\n0 & 0 & m_{33} & ... & ... \\\\\n... & ... & ... & ... & ... \\\\\n... & ... & ... & ... & ... \\\\\n0 & 0 & ... & 0 & m_{NN\n\\end{array\n\\right) .  \\label{matrix1}\n\\end{equation}\n\nTo go further it is convenient to deal with the probability generating\nfunctions for the reproduction laws of particles\n\n\\begin{equation}\nh_{i}(s_{1},...,s_{N})=\\mathbf{E}\\left[ s_{i}^{\\eta _{ii}}...\\,s_{N}^{\\eta\n_{iN}}\\right] ,\\ i=1,2,...,N,  \\label{DefNONimmigr}\n\\end{equation\nwhere $\\eta _{ij}$ represent the numbers of daughters of type $j$ a mother\nof type $i$.\n\nWe say that \\textbf{Hypothesis A} is valid if the $N-$type decomposable\nprocess is strongly critical, i.e. (see \\cite{FN2}),\n\\begin{equation}\nm_{ii}=\\mathbf{E}\\left[ \\eta _{ii}\\right] =1,\\ i=1,2,...,N,  \\label{Matpos}\n\\end{equation\nand, in addition\n\\begin{equation}\nm_{i,i+1}=\\mathbf{E}\\left[ \\eta _{i,i+1}\\right] \\in ( 0,\\infty ) ,\\\ni=1,2,...,N-1,  \\label{Maseq}\n\\end{equation\nan\n\\begin{equation}\n\\mathbf{E}\\left[ \\eta _{ij}\\eta _{ik}\\right] <\\infty ,\\,i=1,...,N;\\\nk,j=i,i+1,...,N  \\label{FinCovar}\n\\end{equation\nwit\n\\begin{equation}\nb_{i}=\\frac{1}{2}Var\\left[ \\eta _{ii}\\right] \\in ( 0,\\infty ) ,\\ i=1,2,...,N.\n\\label{FinVar}\n\\end{equation\nThus, a particle of the process under consideration is able to produce the\ndirect descendants of its own type, of the next in the order type, and (not\nnecessarily, as direct descendants) of all the remaining in the order types,\nbut not any preceding ones.\n\nTo simplify the presentation we fix, from now on $N\\geq 2$ and use, when it\nis convenient the notatio\n\\begin{equation*}\n\\gamma _{0}=0,\\ \\gamma _{i}=\\gamma _{i}(N)=2^{-(N-i)},\\ i=1,2,...,N.\n\\end{equation*}\nWe also suppose (if otherwise is not stated) that $\\mathbf{Z}(0)=\\mathbf{e\n_{1}$, i.e., assume that the branching process under consideration is\ninitiated at time zero by a single particle of type $1$.\n\nLet $\\xi ^{(i)}(j),i=1,2,...,N;j=1,2,...$ be a tuple of independent\nidentically distributed random variables with probability generating functio\n\\begin{equation*}\nf(s)=\\mathbf{E}\\left[ s^{\\xi ^{(i)}(j)}\\right] =1-\\sqrt{1-s}.\n\\end{equation*\nBy means of the tuple we give a detailed construction of an $N-$type\ndecomposable branching process $\\mathbf{R}(t)=(R_{1}(t),...,R_{N}(t)),0\\leq\nt<1,$ where $R_{i}(t)$ is the number of type $i$ individuals in the\npopulation at moment~$t$. It is this process describes the macroscopic\nstructure of the family tree $\\left\\{ \\mathbf{Z}(m,n),0\\leq m\\leq n\\right\\}$\nas $n\\rightarrow \\infty $.\n\nLet $\\mathbf{R}(t)=\\mathbf{e}_{1}$ for $\\gamma _{0}\\leq t<\\gamma _{1}$\nmeaning that the branching process $\\mathbf{R}(t)$ starts at $t=0$ by a\nsingle individual of type $1$ which survives up to (but not at) moment \n\\gamma _{1}$ without reproduction. If $\\gamma _{i}\\leq t<\\gamma\n_{i+1},i=1,2,...,N-1$ the\n\\begin{equation*}\nR_{k}(t)=\\left\\{\n\\begin{array}{ccc}\n\\sum_{j=1}^{R_{i}(\\gamma _{i}-0)}\\xi ^{(i)}(j) & \\text{if} & k=i+1 \\\\\n&  &  \\\\\n0 & \\text{if} & k\\neq i+\n\\end{array\n\\right. .\n\\end{equation*\nThus, within the interval $\\gamma _{i}\\leq t<\\gamma _{i+1}$ the population\nconsists of type $i+1$ particles only. These particles were born at moment \n\\gamma _{i}-0$ by particles of type $i$ evolving without reproduction within\nthe interval $\\gamma _{i-1}\\leq t<\\gamma _{i}$. More precisely, the $j-$th\nparticle of type $i$ produces at its death moment $\\gamma _{i}-0$ a random\nnumber $\\xi ^{(i)}(j)$ children of type $i+1$ and no particles of other\ntypes.\n\nIn what follows we use the symbol $\\Longrightarrow $ to denote convergence\nin the space $D_{[a,b)}(\\mathbb{Z}_{+}^{N})$ of cadlag functions $\\mathbf{x\n(t),a\\leq t<b$ with values in \\ $\\mathbb{Z}_{+}^{N}$ endowed with the metric\nof Skorokhod topology. Besides, we agree to consider $\\mathbf{Z}(x,n)$ as \n\\mathbf{Z}([x],n),$ where $[x]$ is the integer part of $x$.\n\nFor $0\\leq t\\leq 1$ pu\n\\begin{equation*}\ng_{n}(t)=1_{\\{0\\leq t<\\gamma _{1}\\}}+g_{n}1_{\\{\\gamma _{1}\\leq t\\leq 1\\}}\n\\end{equation*\nwhere $g_{n}$ is a positive monotone increasing sequence such that\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }g_{n}=\\infty \\text{ and }\\lim_{n\\rightarrow\n\\infty }n^{-\\varepsilon }g_{n}=0\\text{ for any }\\varepsilon >0.\n\\end{equation*}\n\n\\begin{theorem}\n\\label{T_SkorohConst}Let Hypothesis A be valid. Then, as $n\\rightarrow\n\\infty $\n\n1) the finite-dimensional distributions of the process\n\\begin{equation*}\n\\left\\{ (\\mathbf{Z}(n^{t}g_{n}(t),n),0\\leq t<1)|\\mathbf{Z}(n)\\neq \\mathbf{0\n\\right\\}\n\\end{equation*\nconverge to the finite-dimensional dis\\-tri\\-bu\\-tions of $\\left\\{ \\mathbf{R\n(t),0\\leq t<1\\right\\};$\n\n2) for any $i=0,1,2,...,N-1$\n\\begin{equation*}\n\\mathcal{L}\\left\\{ (\\mathbf{Z}(n^{t}g_{n}(t),n),\\gamma _{i}\\leq t<\\gamma\n_{i+1})\\,|\\,\\mathbf{Z}(n)\\neq \\mathbf{0}\\right\\} \\Longrightarrow \\mathcal{L\n\\left\\{ \\mathbf{R}(t),\\gamma _{i}\\leq t<\\gamma _{i+1}\\right\\} .\n\\end{equation*}\n\\end{theorem}\n\n\\textbf{Remark 1.} Theorem \\ref{T_SkorohConst} shows that the passage to\nlimit under the macroscopic time-scaling $n^{t}g_{n}(t)$ transforms the\nreduced process into an inhomogeneous branching process which consists at\nany given moment of particles of a single type only. In particular, the\nphase transition from type $i$ to type $i+1$ in the prelimiting process\nhappens, roughly speaking, at moment $n^{\\gamma _{i}}$. This gives a\nmacroscopic view on the family tree of the reduced process. The microscopic\nstructure of the family tree described by Theorem \\ref{T_Skhod1} below\nclarifies the nature of the revealed phase transition.\n\nLet $c_{ji},1\\leq j\\leq i\\leq N$ be a tuple of positive numbers in which \nc_{ii}=b_{i}^{-1}$ for $i=1,2,...,N$ an\n\\begin{equation}\nc_{ji}=\\sqrt{b_{j}^{-1}m_{j,j+1}c_{j+1,i}}\\text{ for }\\ j\\leq i-1,\\quad\nC_{i}=c_{1i}.  \\label{Const1}\n\\end{equation}\n\nIt is not difficult to check tha\n\\begin{equation}\nc_{iN}=\\left( \\frac{1}{b_{N}}\\right) ^{1\/2^{N-i}}\\prod_{j=i}^{N-1}\\left(\n\\frac{m_{j,j+1}}{b_{j}}\\right) ^{1\/2^{j-i+1}}.  \\label{Const2}\n\\end{equation}\n\nWe now define a tuple of continuous time Markov processe\n\\begin{eqnarray*}\n\\mathbf{U}_{i}(y) &=&(U_{i1}(y),...,U_{iN}(y)),\\ 0\\leq y<\\infty ,\\\ni=1,2,...,N-1, \\\\\n&& \\\\\n\\mathbf{U}_{N}(x) &=&(U_{N1}(x),...,U_{NN}(x)),\\ 0\\leq x<1.\n\\end{eqnarray*}\n\nFirst we describe the structure of the processes $\\mathbf{U}_{i}(y),1\\leq\ni\\leq N-1$. In this case $U_{ij}(y)\\equiv 0,~0\\leq y<\\infty ,~j\\neq i,i+1,$\nwhile the pair\n\\begin{equation*}\n(U_{ii}(y),U_{i,i+1}(y)),0\\leq y<\\infty ,\n\\end{equation*\nconstitutes a two-type continuous-time homogeneous Markov branching process\nwith particles of types $i$ and $i+1$. This two-type process is initiated at\ntime $y=0$ by a random number $R_{i}$ of type $i$ particles whose\ndistribution is specified by the probability generating functio\n\\begin{equation}\n\\mathbf{E}\\left[ s_{i}^{R_{i}}\\right] =\\mathbf{E}\\left[ s_{i}^{U_{ii}(0)\n\\right] =1-(1-s_{i})^{1\/2^{i-1}}  \\label{Dist_ro}\n\\end{equation\n(in particular, $U_{11}(0)=1$ with probability 1). The life-length\ndistribution of type $i$ particles is exponential with parameter \n2b_{i}c_{iN}$. Dying each particle of type $i$ produces either two particles\nof its own type or one particle of type $i+1$ (each option with probability\n1\/2). Particles of type $i+1$ of $\\mathbf{U}_{i}(\\cdot )$ are immortal and\nproduce no children.\n\nThe structure of the $N-$ dimensional process $\\mathbf{U}_{N}(x),0\\leq x<1$\nis different. If $j<N$ then $U_{Nj}(x)\\equiv 0,~0\\leq x<1,$ while the\ncomponent $U_{NN}(\\cdot )$ is a single-type inhomogeneous Markov branching\nprocess initiated at time $x=0$ by a random number $R_{N}$ of type $N$\nindividuals distributed in accordance with probability generating function\n\n\\begin{equation}\n\\mathbf{E}\\left[ s_{N}^{R_{N}}\\right] =\\mathbf{E}\\left[ s_{N}^{U_{NN}(0)\n\\right] =1-(1-s_{N})^{1\/2^{N-1}}.  \\label{DefroN}\n\\end{equation\nThe life-length of each of $R_{N}$ type $N$ initial particles is uniformly\ndistributed on the interval $\\left[ 0,1\\right] $. Dying such a particle\nproduces exactly two children of type~$N$ and nothing else. If the death\nmoment of the parent particle is $x$ then the life length of each of its\noffspring has the uniform distribution on the interval $[x,1]$\n(independently of the behavior of other particles and the prehistory of the\nprocess). Dying each particle of the process produces exactly two\nindividuals of type $N$ and so on...\\thinspace .\n\nWe are now ready to formulate one more important result of the paper,\ndescribing the microscopic structure of the family tree.\n\nLet $l_{n}$ be a monotone decreasing sequence such that\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }l_{n}=0\\text{ and }\\lim_{n\\rightarrow \\infty\n}n^{\\varepsilon }l_{n}=\\infty \\text{ for any }\\varepsilon >0.\n\\end{equation*}\n\n\\begin{theorem}\n\\label{T_Skhod1}Let Hypothesis A be valid. Then, as $n\\rightarrow \\infty $\n\n1) for each $i=1,2,...,N-1$\n\\begin{equation*}\n\\mathcal{L}\\left\\{ (\\mathbf{Z}\\left( (y+l_{n})n^{\\gamma _{i}},n\\right)\n,0\\leq y<\\infty )\\big|\\,\\mathbf{Z}(n)\\neq \\mathbf{0}\\right\\} \\Longrightarrow\n\\mathcal{L}_{R_{i}}\\left\\{ \\mathbf{U}_{i}(y),0\\leq y<\\infty \\,\\right\\} ,\n\\end{equation*\nwhere $\\mathcal{L}_{R_{i}}$ means that $\\mathbf{U}_{i}(\\cdot )$ is initiated\nat time $y=0$ by a random number $R_{i}$ particles of type $i$ (with \nR_{1}\\equiv 1)$;\n\\begin{equation*}\n\\mathit{2)}\\,\\mathcal{L}\\left\\{(\\mathbf{Z}((x+l_{n})n,n),0\\leq x<1)\\,|\\\n\\mathbf{Z}(n)\\neq \\mathbf{0}\\right\\} \\Longrightarrow \\mathcal{L\n_{R_{N}}\\left\\{ \\mathbf{U}_{N}(x),0\\leq x<1\\,\\right\\},\n\\end{equation*\nwhere $\\mathcal{L}_{R_{N}}$ means that $\\mathbf{U}_{N}(\\cdot)$ is initiated\nat time $x=0$ by a random number $R_{N}$ particles of type $N.$\n\\end{theorem}\n\n\\textbf{Remark 2.} Theorems \\ref{T_SkorohConst} and \\ref{T_Skhod1} reveal an\ninteresting phenomenon in the development of the critical decomposable\nbranching processes which may be expressed in terms of the \"island\"\ninterpretation of the processes as follows: If the population survives up to\na distant moment $n$, then all surviving individuals are located at this\nmoment on island $N$ and, moreover, at each moment in the past their\nancestors were (asymptotically) located not more than on two specific\nislands.\n\nBasing on the conclusions of Theorems \\ref{T_SkorohConst} and \\ref{T_Skhod1}\nwe give in the next theorem an answer to the following important question:\nwhat is the asymptotic distribution of the birth moment of the MRCA for the\npopulation survived up to a distant moment $n$?\n\n\\begin{theorem}\n\\label{T_mrcaMany}Let Hypothesis A be valid. Then\n\n1)\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }\\mathbf{P}\\left( \\beta _{n}\\ll n^{\\gamma _{1}\n\\big|\\,\\mathbf{Z}(n)\\neq \\mathbf{0}\\right) =0;\n\\end{equation*\n2) if $y\\in (0,\\infty )$ then for $i=1,2,...,N-1$\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }\\mathbf{P}\\left( \\beta _{n}\\leq yn^{\\gamma _{i}\n\\big|\\,\\mathbf{Z}(n)\\neq \\mathbf{0}\\right) =1-\\frac{1}{2^{i}}-\\frac{1}{2^{i}\ne^{-2b_{i}c_{iN}y};\n\\end{equation*\n3) for $i=1,2,...,N-1$\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty }\\mathbf{P}\\left( \\beta _{n}\\ll n^{\\gamma _{i}\n\\big|\\,\\mathbf{Z}(n)\\neq \\mathbf{0}\\right) =1-\\frac{1}{2^{i-1}};\n\\label{recent_i}\n\\end{equation\n3a) for $i=1,2,...,N-1$\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty }\\mathbf{P}\\left( n^{\\gamma _{i}}\\ll \\beta _{n}\\ll\nn^{\\gamma _{i+1}}\\big|\\,\\mathbf{Z}(n)\\neq \\mathbf{0}\\right) =0;\n\\label{NoMRCA}\n\\end{equation\n4) for any $x\\in (0,1)$\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }\\mathbf{P}(\\beta _{n}\\leq xn|\\mathbf{Z}(n)\\neq\n\\mathbf{0})=1-\\frac{1}{2^{N-1}}(1-x).\n\\end{equation*}\n\\end{theorem}\n\n\\textbf{Remark 3.} As we see by (\\ref{NoMRCA}), there are time-intervals of\nincreasing orders within each of which the probability to find the MRCA of\nthe population survived up to moment $n\\rightarrow \\infty $ is negligible\ncompared to the probability for the population to survive up to this moment.\nMoreover, these time-intervals are separated from each other by the\ntime-intervals of increasing orders within each of which the probability to\nfind the MRCA is strictly positive. Such a phenomena has no analogues for\nthe indecomposable Galton-Watson processes.\n\nAlong with the distribution of the birth moment of the MRCA, the type $\\zeta\n_{n}$ of the MRCA of the population survived up to moment $n$ is of\ninterest. The distribution of this random variable is described by the\nfollowing theorem.\n\n\\begin{theorem}\n\\label{T_type}Let Hypothesis A be valid. Then, for $i=1,2,...,N$\n\\begin{equation*}\np_{i}=\\lim_{n\\rightarrow \\infty }\\mathbf{P}(\\zeta _{n}=i|\\mathbf{Z}(n)\\neq\n\\mathbf{0})=\\frac{1}{2^{i}}(1-\\delta _{iN})+\\frac{1}{2^{N-1}}\\,\\delta _{iN},\n\\end{equation*\nwhere $\\delta _{ij}$ is the Kroneker symbol.\n\\end{theorem}\n\nObserve that $p_{N-1}=p_{N}$.\n\n\\textbf{Remark 4.} The authors of paper \\cite{FN2}, which contains several\nresults used in the proofs of our Theorems \\ref{T_SkorohConst}-\\ref{T_type},\nconsidered a more general case of the strongly critical branching processes.\nNamely, they prove a number of conditional limit theorems for the case when\nby a suitable labelling the types of the multitype Galton-Watson process can\nbe grouped into $N\\geq 2$ partially ordered classes $\\mathcal{C\n_{1}\\rightarrow \\mathcal{C}_{2}\\rightarrow ...\\rightarrow \\mathcal{C}_{N}$\npossessing the following properties:\n\n1) particle types belonging to any given class, say $\\mathcal{C}_{i},$\nconstitute an indecomposable critical branching process with $r_{i}\\geq 1$\ntypes;\n\n2) each class $\\mathcal{C}_{i}$ contains a type whose representatives are\nable to produce offspring in the next class in the order with a positive\nprobability;\n\n3) particles with types from $\\mathcal{C}_{i},i\\geq 2,$ are unable to\nproduce offspring belonging to the classes $\\mathcal{C}_{1},...,\\mathcal{C\n_{i-1}$.\n\nThe methods used in the present paper may be applied to investigate, for\ninstance, the asymptotic distribution of $\\beta _{n}$ for such processes.\nSince the needed arguments are too cumbersome and contain no new ideas, we\nprefer to concentrate on the case when each class $\\mathcal{C}_{i}$ includes\na single type only.\n\nThe remainder of the paper is organized as follows. Section \\ref{Sec2}\ncontains some preliminary results. In particular, we recall the statements\nfrom \\cite{FN} and \\cite{FN2} describing the asymptotic behavior of the\nsurvival probability and the distribution of the number of particles in a\nstrongly critical decomposable branching process. Section \\ref{Sec3} gives a\ndetailed description of the limiting processes. In Sections \\ref{Sec4} and\n\\ref{Sec5} we check convergence of one-dimensional and finite-dimensional\ndistributions of the prelimiting processes to the limiting ones. Section \\re\n{Sec6} contains the proofs of Theorems \\ref{T_SkorohConst} and \\ref{T_Skhod1\n. Finally, Section \\ref{Sec7} is devoted to the proofs of Theorems \\re\n{T_mrcaMany} and \\ref{T_type}.\n\n\\section{Auxiliary results\\label{Sec2}}\n\nFor any vector $\\mathbf{s}=(s_{1},...,s_{p})$ (the dimension will usually be\nclear from the context) and an integer valued vector $\\mathbf{k\n=(k_{1}.....k_{p})$ defin\n\\begin{equation*}\n\\mathbf{s}^{\\mathbf{k}}=s_{1}^{k_{1}},...,\\,s_{p}^{k_{p}}.\n\\end{equation*\nFurther, let $\\mathbf{1}=(1,...,1)$ be a vector of units. It will be\nsometimes convenient to write $\\mathbf{1}^{(i)}$ for the $i-$dimensional\nvector with all its components equal to one.\n\nLet\n\\begin{equation*}\nH_{n}^{(i,N)}(\\mathbf{s})=\\mathbf{E}\\left[ \\mathbf{s}^{\\mathbf{Z}(n)}\n\\mathbf{Z}(0)=\\mathbf{e}_{i}\\right] =\\mathbf{E}\\left[ s_{i}^{Z_{i}(n)}..\n\\,s_{N}^{Z_{N}(n)}|\\mathbf{Z}(0)=\\mathbf{e}_{i}\\right]\n\\end{equation*\nbe the probability generating function for $\\mathbf{Z}(n)$ given the process\nis initiated at time zero by a single particle of type $i\\in \\left\\{\n1,2,...,N\\right\\}.$ Clearly (recall (\\ref{DefNONimmigr})), $H_{1}^{(i,N)}\n\\mathbf{s})=h_{i}(\\mathbf{s}),\\ i=1,...,N$. Denote\n\\begin{equation*}\nQ_{n}^{(i,N)}(\\mathbf{s})=1-H_{n}^{(i,N)}(\\mathbf{s\n),~Q_{n}^{(i,N)}=1-H_{n}^{(i,N)}(\\mathbf{0}),\n\\end{equation*}\nput\n\\begin{equation*}\n\\mathbf{H}_{n}(\\mathbf{s})=(H_{n}^{(1,N)}(\\mathbf{s}),...,H_{n}^{(N,N)}\n\\mathbf{s})),~\\mathbf{Q}_{n}(\\mathbf{s})=(Q_{n}^{(1,N)}(\\mathbf{s\n),...,Q_{n}^{(N,N)}(\\mathbf{s}))\n\\end{equation*\nand se\n\\begin{equation*}\nb_{jk}(n)=\\mathbf{E}\\left[ Z_{j}(n)Z_{k}(n)-\\delta _{jk}Z_{j}(n)|\\mathbf{Z\n(0)=\\vec{e}_{j}\\right].\n\\end{equation*}\n\nThe starting point of our arguments is the following theorem being a\nsimplified combination of the respective results from \\cite{FN} and \\cit\n{FN2}:\n\n\\begin{theorem}\n\\label{T_Foster}Let $\\mathbf{Z}(n),n=0,1,..$ be a strongly critical\ndecomposable multitype branching process satisfying (\\ref{matrix1}), (\\re\n{Matpos}), (\\ref{Maseq}), and (\\ref{FinCovar}). Then, as $n\\rightarrow\n\\infty \n\\begin{eqnarray}\nm_{jj}(n) &=&1,\\ m_{ij}(n)\\sim a_{ij}n^{j-i},\\ i<j,  \\label{MomentSingle3} \\\\\nb_{jk}(n) &\\sim &\\hat{a}_{jk}n^{k-j+1},\\ j\\leq k,  \\label{Momvariance}\n\\end{eqnarray\nwhere $a_{ij}$ and $\\hat{a}_{jk}$ are positive constants known explicitly\n(see \\cite{FN2}, Theorem 1).\n\nBesides \\textit{(}see \\cite{FN}, Theorem 1), as $n\\rightarrow \\infty $\n\\begin{equation}\nQ_{n}^{(i,N)}=1-H_{n}^{(i,N)}(\\mathbf{0})=\\mathbf{P}(\\mathbf{Z}( n) \\neq\n\\mathbf{0}|\\mathbf{Z}( 0) =\\mathbf{e}_{i})\\sim c_{iN}n^{-1\/2^{N-i}},\n\\label{SurvivSingle}\n\\end{equation\nwhere the constants $c_{iN}$ are the same as in (\\ref{Const2}).\n\\end{theorem}\n\nIn the sequel we prove the following Yaglom-type limit theorem being a\ncompliment to Theorem \\ref{T_Foster}.\n\n\\begin{theorem}\n\\label{T_Yaglom}Under the conditions of Theorem \\ref{T_Foster}, for any \n\\lambda >0\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty }\\mathbf{E}\\left[ \\exp \\left\\{ -\\lambda \\frac\nZ_{N}(n)}{b_{N}n}\\right\\} \\Big|\\mathbf{Z}( n) \\neq \\mathbf{0};\\mathbf{Z}( 0)\n=\\mathbf{e}_{i}\\right] =1-\\Big( \\frac{\\lambda }{1+\\lambda }\\Big)\n^{1\/2^{N-i}}.  \\label{Yag}\n\\end{equation}\n\\end{theorem}\n\nSet $d_{ii}=\\sqrt{b_{i}^{-1}m_{i,i+1}}$ , $i=1,2,...,N-1$\\ and, for \nj=1,2,...,i-1$ le\n\\begin{equation}\nd_{ji}=\\sqrt{b_{j}^{-1}m_{j,j+1}d_{j+1,i}},\\quad D_{i}=d_{1i}.\n\\label{Dreccur}\n\\end{equation\nObserve that (see (\\ref{Const1})) for $k=0,1,2,...,i-1$\n\\begin{equation}\nd_{i-k,i}=(b_{i}m_{i,i+1})^{1\/2^{k+1}}c_{i-k,i},\\quad\nD_{i}=(b_{i}m_{i,i+1})^{1\/2^{i}}c_{1i}=(b_{i}m_{i,i+1})^{1\/2^{i}}C_{i}.\n\\label{DcConnection}\n\\end{equation}\n\nLet $\\mathbf{Z}(0)=\\mathbf{e_{1}}$ and denote by\n\\begin{equation*}\nT_{i}=\\min \\left\\{ n\\geq 1:Z_{1}(n)+Z_{2}(n)+...+Z_{i}(n)=0\\right\\}\n\\end{equation*\nthe extinction moment of the population generated by the particles of the\nfirst $i$ in order types. Let $\\eta _{rj}\\left( k,l\\right) $ be the number\nof daughters of type $j$ of the $l-$th mother of type $r$ belonging to the \nk-$th generation and\n\\begin{equation*}\nW_{pij}=\\sum_{r=p}^{i}\\sum_{k=0}^{T_{i}}\\sum_{q=1}^{Z_{r}(k)}\\eta\n_{rj}\\left( k,q\\right)\n\\end{equation*\nbe the total amount of daughters of type $j\\geq i+1$ produced by all\nparticles of types $p,p+1,...,i$ ever born in the process if the process is\ninitiated at time $n=0$ by a single particle of type $p\\leq i.$ Finally, pu\n\\begin{equation*}\nW_{pi}=\\sum_{j=i+1}^{N}W_{pij}=\\sum_{j=i+1}^{N}\\sum_{r=p}^{i\n\\sum_{k=0}^{T_{i}}\\sum_{q=1}^{Z_{r}(k)}\\eta _{rj}\\left( k,q\\right) .\n\\end{equation*}\n\nWe know by (\\ref{SurvivSingle}) tha\n\\begin{equation}\nQ_{n}^{(1,i)}=\\mathbf{P}( T_{i}>n) \\sim c_{1i}n^{-2^{-(i-1)}}.  \\label{Asqq}\n\\end{equation}\n\nThe next lemma describes the tail distributions of $W_{1i,i+1}$ and $W_{1i}$.\n\n\\begin{lemma}\n\\label{L_Laplace} Let Hypothesis A be valid. Then, as $\\lambda \\downarrow 0\n\\begin{equation}\n1-\\mathbf{E}\\left[ e^{-\\lambda W_{1i,i+1}}\\,|\\mathbf{Z}(0)=\\mathbf{e}_{1\n\\right] \\sim d_{1i}\\lambda ^{1\/2^{i}}=D_{i}\\lambda ^{1\/2^{i}}  \\label{Tot1}\n\\end{equation\nand there exists a constant $F_{i}>0$ such tha\n\\begin{equation}\n1-\\mathbf{E}\\left[ e^{-\\lambda W_{1i}}|\\mathbf{Z}(0)=\\mathbf{e}_{1}\\right]\n\\sim F_{i}\\lambda ^{1\/2^{i}}.  \\label{Tot2}\n\\end{equation}\n\\end{lemma}\n\n\\textbf{Proof.} Set\n\\begin{equation*}\nW_{pi,i+1}(n)=\\sum_{r=p}^{i}\\sum_{k=0}^{n}\\sum_{q=1}^{Z_{r}(k)}\\eta\n_{rj}\\left( k,q\\right) ,\n\\end{equation*\ndenot\n\\begin{equation*}\nK_{pi,n}(\\mathbf{s};t)=\\mathbf{E}\\left[\ns_{p}^{Z_{p}(n)}...s_{i}^{Z_{i}(n)}t^{W_{pi,i+1}(n)}|\\mathbf{Z}(0)=\\mathbf{e\n_{p}\\right] ,\\,K_{pi,n}(t)=K_{pi,n}(\\mathbf{1}^{(i-p+1)};t)\n\\end{equation*\nand put\n\\begin{equation*}\nK_{pi}(t)=\\mathbf{E}\\left[ t^{W_{pi,i+1}}\\big|\\,\\mathbf{Z}(0)=\\mathbf{e}_{p\n\\right] =\\lim_{n\\rightarrow \\infty }K_{pi,n}(t)\n\\end{equation*\n(this limit exists since the random variables $W_{pi,i+1}(n),p=1,2,...,i$\nare nondecreasing in $n$). Clearly, to prove the lemma it is sufficient to\nshow that, as~$t\\uparrow 1\n\\begin{equation*}\n1-K_{1i}(t)=1-\\mathbf{E}\\left[ t^{W_{1i,i+1}}\\,|\\mathbf{Z}(0)=\\mathbf{e}_{1\n\\right] {}\\mathbf{\\sim {}}d_{1i}(1-t)^{1\/2^{i}}.\n\\end{equation*}\n\nUsing properties of branching processes it is not difficult to check tha\n\\begin{equation*}\nK_{pi,n+1}(\\mathbf{s};t)=h_{p}\\left( K_{pi,n}(\\mathbf{s};t),...,K_{ii,n}\n\\mathbf{s};t),t,\\mathbf{1}^{(N-i-1)}\\right)\n\\end{equation*\nimplyin\n\\begin{equation*}\nK_{pi,n+1}(t)=h_{p}\\left( K_{pi,n}(t),...,K_{ii,n}(t),t,\\mathbf{1\n^{(N-i-1)}\\right) .\n\\end{equation*\nan\n\\begin{equation*}\nK_{pi}(t)=h_{p}\\left( K_{pi}(t),...,K_{ii}(t),t,\\mathbf{1}^{(N-i-1)}\\right) .\n\\end{equation*\nIn particular\n\\begin{equation*}\nK_{ii}(t)=h_{i}\\left( K_{ii}(t),t,\\mathbf{1}^{(N-i-1)}\\right) .\n\\end{equation*\nSince $\\mathbf{E}\\eta _{ii}=1$ and $b_{i}=\\frac{1}{2}Var\\eta _{ii}\\in\n(0,\\infty )$, it follows that, as $t\\uparrow 1\n\\begin{eqnarray*}\n1-K_{ii}(t) &=&1-h_{i}\\left( K_{ii}(t),t,\\mathbf{1}^{(N-i-1)}\\right) \\\\\n&=&1-K_{ii}(t)-b_{i}(1-K_{ii}(t))^{2}(1+o(1))+m_{i,i+1}(1-t)\n\\end{eqnarray*\no\n\\begin{equation*}\n1-K_{ii}(t)\\sim \\sqrt{b_{i}^{-1}m_{i,i+1}(1-t)}.\n\\end{equation*\nThis, in particular, proves the statement of the lemma for $i=1$.\n\nNow we use induction and assume tha\n\\begin{equation*}\n1-K_{qi}(t)\\sim d_{qi}(1-t)^{1\/2^{i-q+1}},q=p+1,...,i.\n\\end{equation*\nThe\n\\begin{eqnarray*}\n1-K_{pi}(t) &=&1-h_{p}\\left( K_{pi}(t),...,K_{ii}(t),t,\\mathbf{1\n^{(N-i-1)}\\right) \\\\\n&=&1-K_{pi}(t)-b_{p}(1-K_{pi}(t))^{2}(1+o(1)) \\\\\n&&+(1+o(1))\\left( m_{p,p+1}(1-K_{p+1,i}(t))+\\sum_{q=p+2}^{i}m_{pq}\\left(\n1-K_{qi}(t)\\right) \\right) \\\\\n&&+(1+o(1))m_{p,i+1}(1-t)\n\\end{eqnarray*\nimplyin\n\\begin{eqnarray*}\n1-K_{pi}(t) &\\sim &\\sqrt{b_{p}^{-1}m_{p,p+1}(1-K_{p+1,i}(t))} \\\\\n&\\sim &\\sqrt{b_{p}^{-1}m_{p,p+1}d_{p+1,i}}\\left( 1-t\\right)\n^{1\/2^{i-p+1}}=d_{pi}\\left( 1-t\\right) ^{1\/2^{i-p+1}}\n\\end{eqnarray*\nand proving (\\ref{Tot1}).\n\nTo prove (\\ref{Tot2}) it is necessary to use similar arguments. We omit the\ndetails.\n\nLemma \\ref{L_Laplace} is proved.\\\n\nFrom now on and till the end of this section we suppose tha\n\\begin{equation}\ns_{k}=\\exp (-\\lambda _{k}n^{-2^{-(N-k)}})=\\exp (-\\lambda _{k}n^{-\\gamma\n_{k}}),\\lambda _{k}>0,\\,k=1,2,...,N  \\label{Sasymp}\n\\end{equation\nand, keeping in mind this assumption, study in Lemmas \\ref{L_OneOnly}-\\re\n{L_MultiSharp} the asymptotic behavior of the difference $1-H_{m}^{(j,N)}\n\\mathbf{s})$ when $m,n\\rightarrow \\infty .$\n\n\\begin{lemma}\n\\label{L_OneOnly}I\n\\begin{equation}\nm\\ll n^{2^{-(N-j)}}=n^{\\gamma _{j}}  \\label{Mnegl}\n\\end{equation\nthen for $N>j\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }n^{\\gamma _{j}}Q_{m}^{(j,N)}(\\mathbf{s})=\\lambda\n_{j}.\n\\end{equation*}\n\\end{lemma}\n\n\\textbf{Proof.} Clearly, it is sufficient to prove the statement for $j=1$\nonly. Let $r$ be a positive integer such tha\n\\begin{equation*}\n1-H_{r}^{(1,1)}(0)\\leq 1-s_{1}\\leq 1-H_{r-1}^{(1,1)}(0).\n\\end{equation*\nSince $1-s_{1}\\sim \\lambda _{1}n^{-\\gamma _{1}}$ and $1-H_{r}^{(1,1)}(0)\\sim\n(b_{1}r)^{-1}\\,$as $n,r\\rightarrow \\infty $, it follows that $r\\sim\n(b_{1}\\lambda _{1})^{-1}n^{\\gamma _{1}}$. By the branching property of\nprobability generating functions we have for $m\\ll n^{\\gamma _{1}}:$\n\\begin{eqnarray*}\nQ_{m}^{(1,N)}(\\mathbf{s}) &\\geq &1-H_{m}^{(1,1)}(s_{1})\\geq\n1-H_{m}^{(1,1)}(H_{r}^{(1,1)}(0)) \\\\\n&=&1-H_{m+r}^{(1,1)}(0)\\sim b_{1}^{-1}(m+r)^{-1}\\sim \\lambda _{1}n^{-\\gamma\n_{1}}.\n\\end{eqnarray*\nBesides\n\\begin{eqnarray*}\nQ_{m}^{(1,N)}(\\mathbf{s}) &\\leq &1-H_{m}^{(1,1)}(s_{1})+\\mathbf{E}\\left[\n\\left( 1-s_{2}^{Z_{2}(m)}...\\,s_{N}^{Z_{N}(m)}\\right) |\\mathbf{Z}(0)=\\mathbf\ne}_{1}\\right] \\\\\n&\\leq &1-H_{m+r-1}^{(1,1)}(0)+\\sum_{k=2}^{N}(1-s_{k})\\mathbf{E}\\left[\nZ_{k}(m)|\\mathbf{Z}(0)=\\mathbf{e}_{1}\\right] .\n\\end{eqnarray*\nWe know by (\\ref{MomentSingle3}) and (\\ref{Sasymp}) that, for a positive\nconstant $C$\n\\begin{equation*}\n\\sum_{k=2}^{N}(1-s_{k})\\mathbf{E}\\left[ Z_{k}(m)|\\mathbf{Z}(0)=\\mathbf{e}_{1\n\\right] \\leq C\\sum_{k=2}^{N}\\lambda _{k}n^{-\\gamma _{k}}m^{k-1}\n\\end{equation*\nwhich, in view of (\\ref{Mnegl}) is negligible with respect t\n\\begin{equation*}\nC\\max_{2\\leq i\\leq N}\\lambda _{i}\\times \\sum_{k=2}^{N}n^{-\\gamma\n_{k}}(n^{\\gamma _{1}})^{k-1}=C\\max_{2\\leq i\\leq N}\\lambda _{i}\\times\n\\sum_{k=2}^{N}n^{(k-1)2^{-(N-1)}-2^{-(N-k)}}.\n\\end{equation*\nSince $k2^{-(N-1)}-2^{-(N-k)}=2^{-(N-1)}(k-2^{k-1})\\leq 0$ for $k\\geq 2,$ we\nhave\n\\begin{equation*}\nn^{2^{-(N-1)}}\\sum_{k=2}^{N}n^{(k-1)2^{-(N-1)}-2^{-(N-k)}}\n\\sum_{k=2}^{N}n^{k2^{-(N-1)}-2^{-(N-k)}}\\leq N-1.\n\\end{equation*\nConsequently, $Q_{m}^{(1,N)}(\\mathbf{s})\\sim 1-H_{m}^{(1,1)}(s_{1})\\sim\n\\lambda _{1}n^{-\\gamma _{1}}$ as $n\\rightarrow \\infty $.\n\nThis proves the lemma.\n\nIn order to formulate the next lemma we introduce a tuple of functions $\\phi\n_{i}=\\phi _{i}( \\lambda _{1},\\lambda _{2}) ,\\ i=1,2,...,N-1$ solving in the\ndomain $\\left\\{ \\lambda _{1}\\geq 0,\\lambda _{2}\\geq 0\\right\\} $ the\ndifferential equation\n\\begin{equation*}\n\\lambda _{1}\\frac{\\partial \\phi _{i}}{\\partial \\lambda _{1}}+2\\lambda _{2\n\\frac{\\partial \\phi _{i}}{\\partial \\lambda _{2}}=-b_{i}\\phi _{i}^{2}+\\phi\n_{i}+m_{i,i+1}\\lambda _{2}\n\\end{equation*\nwith the initial conditions\n\\begin{equation*}\n\\phi _{i}(\\mathbf{0})=0,\\ \\frac{\\partial \\phi _{i}( \\mathbf{0}) }{\\partial\n\\lambda _{1}}=1,\\ \\frac{\\partial \\phi _{i}( \\mathbf{0}) }{\\partial \\lambda\n_{2}}=m_{i,i+1}.\n\\end{equation*}\n\nOne may check that, for any $y>0$\n\\begin{equation}\n\\frac{\\phi _{i}(\\lambda _{1}y,\\lambda _{2}y^{2})}{y}=\\sqrt{\\frac\nm_{i,i+1}\\lambda _{2}}{b_{i}}}\\frac{b_{i}\\lambda _{1}+\\sqrt\nb_{i}m_{i,i+1}\\lambda _{2}}\\tanh (y\\sqrt{b_{i}m_{i,i+1}\\lambda _{2}})}\nb_{i}\\lambda _{1}\\tanh (y\\sqrt{b_{i}m_{i,i+1}\\lambda _{2}})+\\sqrt\nb_{i}m_{i,i+1}\\lambda _{2}}}.  \\label{DefSimpl}\n\\end{equation}\n\n\\begin{lemma}\n\\label{L_twoOnly}Let condition (\\ref{Sasymp}) be valid. If $m\\sim yn^{\\gamma\n_{i}},$ $y>0$ the\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }n^{\\gamma _{i}}Q_{m}^{(i,N)}(\\mathbf{s\n)=y^{-1}\\phi _{i}(\\lambda _{i}y,\\lambda _{i+1}y^{2}).\n\\end{equation*}\n\\end{lemma}\n\n\\textbf{Proof.} As in the previous lemma, it is sufficient to consider the\ncase $i=1$ only. It follows from Theorem 2 in \\cite{FN2} that for $\\lambda\n_{k}\\geq 0,k=1,2,...,N\n\\begin{eqnarray*}\n&&\\lim_{m\\rightarrow \\infty }m\\left( 1-\\mathbf{E}\\left[ \\exp \\left\\{\n-\\sum_{k=1}^{N}\\lambda _{k}\\frac{Z_{k}(m)}{m^{k}}\\right\\} \\right] \\right) \\\\\n&&\\qquad \\qquad =\\lim_{m\\rightarrow \\infty }m(1-H_{m}^{(1,N)}(e^{-\\lambda\n_{1}\/m},e^{-\\lambda _{2}\/m^{2}},...,e^{-\\lambda _{N}\/m^{N}})) \\\\\n&&\\qquad \\qquad =\\Phi (\\lambda _{1},\\lambda _{2},...,\\lambda _{N}),\n\\end{eqnarray*\nwhere $\\Phi =\\Phi (\\lambda _{1},\\lambda _{2},...,\\lambda _{N})$ solves the\ndifferential equatio\n\\begin{equation*}\n\\sum_{k=1}^{N}k\\lambda _{k}\\frac{\\partial \\Phi }{\\partial \\lambda _{k}\n=-b_{1}\\Phi ^{2}+\\Phi +\\sum_{k=2}^{N}f_{k}\\lambda _{k}\n\\end{equation*\nwith the initial condition\n\\begin{equation*}\n\\Phi (\\mathbf{0})=0,\\ \\frac{\\partial \\Phi (\\mathbf{0})}{\\partial \\lambda _{1\n}=1,\\ \\frac{\\partial \\Phi (\\mathbf{0})}{\\partial \\lambda _{k}}=\\frac{1}{k-1\nf_{k},\\ k=2,...,N\n\\end{equation*\nand\n\\begin{equation*}\nf_{k}=\\frac{1}{(k-2)!}\\prod_{j=1}^{k-1}m_{j,j+1},\\ k=2,...,N.\n\\end{equation*\nSince $m^{2^{k-1}}=m^{k}$ for $k=1,2$ and $m^{2^{k-1}}\\gg m^{k}$ for $k>2,$\nwe conclude by the continuity of $\\Phi $ at point $\\mathbf{0}$ tha\n\\begin{eqnarray*}\n&&\\lim_{n\\rightarrow \\infty }n^{\\gamma _{1}}Q_{m}^{(1,N)}(\\mathbf{s\n)=y^{-1}\\lim_{m\\rightarrow \\infty }mQ_{m}^{(1,N)}(\\mathbf{s}) \\\\\n&&\\quad =y^{-1}\\lim_{m\\rightarrow \\infty }m\\left( 1-\\mathbf{E}\\left[ \\exp\n\\left\\{ -\\sum_{k=1}^{N}\\lambda _{k}\\frac{Z_{k}(m)}{n^{1\/2^{N-k}}}\\right\\}\n\\right] \\right) \\\\\n&&\\quad =y^{-1}\\lim_{m\\rightarrow \\infty }m\\left( 1-\\mathbf{E}\\left[ \\exp\n\\left\\{ -\\sum_{k=1}^{N}\\lambda _{k}y^{2^{k-1}}\\frac{Z_{1}(m)}{m^{2^{k-1}}\n\\right\\} \\right] \\right) \\\\\n&&\\quad =y^{-1}\\Phi (\\lambda _{1}y,\\lambda _{2}y^{2},0,...,0)=y^{-1}\\phi\n_{1}(\\lambda _{1}y,\\lambda _{2}y^{2}).\n\\end{eqnarray*}\n\nLemma \\ref{L_twoOnly} is proved.\\\n\n\\begin{lemma}\n\\label{L_inbetween}Let condition (\\ref{Sasymp}) be valid. If, for some \ni\\leq N-1$\n\\begin{equation}\nn^{\\gamma _{i}}\\ll m\\ll n^{\\gamma _{i+1}}  \\label{msmall}\n\\end{equation\nthen\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }n^{\\gamma _{1}}Q_{m}^{(1,N)}(\\mathbf{s\n)=D_{i}\\left( \\lambda _{i+1}\\right) ^{1\/2^{i}}.\n\\end{equation*}\n\\end{lemma}\n\n\\textbf{Proof.} It follows from (\\ref{Asqq}) and (\\ref{msmall}) that\n\\begin{equation*}\n\\mathbf{P}(T_{i}>m)\\sim c_{1i}m^{-2^{-(i-1)}}=o(n^{-\\gamma _{1}}).\n\\end{equation*\nTherefore\n\\begin{eqnarray*}\nQ_{m}^{(1,N)}(\\mathbf{s}) &=&\\mathbf{E}\\left[\n1-s_{1}^{Z_{1}(m)}s_{2}^{Z_{2}(m)}...\\,s_{N}^{Z_{N}(m)}\\right] \\\\\n&=&\\mathbf{E}\\left[ \\left(\n1-s_{i+1}^{Z_{i+1}(m)}...\\,s_{N}^{Z_{N}(m)}\\right) ;T_{i}\\leq m\\right]\n+o(n^{-\\gamma _{1}}) \\\\\n&=&1-H_{m}^{(1,N)}\\left( \\mathbf{1}^{(i)},s_{i+1},...,s_{N}\\right)\n+o(n^{-\\gamma _{1}}).\n\\end{eqnarray*\nIt is not difficult to check that for our decomposable branching process\n\\begin{eqnarray*}\n&&H_{m}^{(1,N)}\\left( \\mathbf{1}^{(i)},s_{i+1},...,s_{N}\\right) \\\\\n&&\\quad =\\mathbf{E}\\left[ \\prod_{k=0}^{m-1}\\prod_{r=1}^{i\n\\prod_{l=1}^{Z_{r}(k)}\\prod_{j=i+1}^{N}\\left( H_{m-k}^{(j,N)}(\\mathbf{s\n)\\right) ^{\\eta _{rj}\\left( k,l\\right) }\\right] \\\\\n&&\\quad =\\mathbf{E}\\left[ \\prod_{k=0}^{m-1}\\prod_{r=1}^{i\n\\prod_{l=1}^{Z_{r}(k)}\\prod_{j=i+1}^{N}\\left( H_{m-k}^{(j,N)}(\\mathbf{s\n)\\right) ^{\\eta _{rj}\\left( k,l\\right) };T_{i}\\leq \\sqrt{mn^{\\gamma _{i}}\n\\right] \\\\\n&&\\qquad +O\\left( \\mathbf{P}\\left( T_{i}> \\sqrt{mn^{\\gamma _{i}}}\\right)\n\\right) .\n\\end{eqnarray*}\n\nObserving that $\\lim_{m\\to\\infty}H_{m-k}^{(j,N)}(\\mathbf{s})\\rightarrow 1$\nfor $j\\geq i+1$ and $k\\leq T_{i}\\leq \\sqrt{mn^{\\gamma _{i}}}=o(m),$ we get\non the set $T_{i}\\leq \\sqrt{mn^{\\gamma _{i}}}$\n\\begin{eqnarray*}\n&&\\prod_{k=0}^{m-1}\\prod_{r=1}^{i}\\prod_{l=1}^{Z_{r}(k)}\\prod_{j=i+1}^{N\n\\left( H_{m-k}^{(j,N)}(\\mathbf{s})\\right) ^{\\eta _{rj}\\left( k,l\\right) } \\\\\n&&\\quad =\\exp \\left\\{\n-\\sum_{r=1}^{i}\\sum_{k=0}^{T_{i}}\\sum_{l=1}^{Z_{r}(k)}\\sum_{j=i+1}^{N}\\eta\n_{rj}\\left( k,l\\right) Q_{m-k}^{(j,N)}(\\mathbf{s})(1+o(1))\\right\\} .\n\\end{eqnarray*\nIf $j\\geq i+1$ then Lemma \\ref{L_OneOnly} and the estimates $m\\ll n^{\\gamma\n_{i+1}}\\leq n^{\\gamma _{j}}$ yield\n\\begin{equation*}\nQ_{m-k}^{(j,N)}(\\mathbf{s})\\sim Q_{m}^{(j,N)}(\\mathbf{s})\\sim \\lambda\n_{j}n^{-\\gamma _{j}}.\n\\end{equation*\nHence it follows that on the set $T_{i}\\leq \\sqrt{mn^{\\gamma _{i}}\n=o(m)=o(n^{\\gamma _{i+1}})\n\\begin{eqnarray*}\n&&\\sum_{r=1}^{i}\\sum_{k=0}^{T_{i}}\\sum_{l=1}^{Z_{r}(k)}\\sum_{j=i+1}^{N}\\eta\n_{rj}\\left( k,l\\right) Q_{m-k}^{(j,N)}(\\mathbf{s}) \\\\\n&&\\quad =(1+o(1))\\sum_{j=i+1}^{N}Q_{m}^{(j,N)}(\\mathbf{s})\\sum_{r=1}^{i\n\\sum_{k=0}^{T_{i}}\\sum_{l=1}^{Z_{r}(k)}\\eta _{rj}\\left( k,l\\right) \\\\\n&&\\quad =(1+o(1))\\sum_{j=i+1}^{N}W_{1ij}Q_{m}^{(j,N)}(\\mathbf{s}) \\\\\n&&\\quad =(1+o(1))W_{1i,i+1}Q_{m}^{(i+1,N)}(\\mathbf{s})+O\\left(\nQ_{m}^{(i+2,N)}(\\mathbf{s})\\right) \\sum_{j=i+2}^{N}W_{1ij} \\\\\n&&\\quad =(1+o(1))W_{1i,i+1}\\lambda _{i+1}n^{-\\gamma _{i+1}}+O_{n}(n^{-\\gamma\n_{i+2}}W_{1i}).\n\\end{eqnarray*\nUsing the estimates\n\\begin{eqnarray*}\n0 &\\leq &\\mathbf{E}\\left[ \\exp \\left\\{ -(1+o(1))W_{1i,i+1}\\lambda\n_{i+1}n^{-\\gamma _{i+1}}\\right\\} \\right] \\\\\n&&-\\mathbf{E}\\left[ \\exp \\left\\{ -(1+o(1))W_{1i,i+1}\\lambda _{i+1}n^{-\\gamma\n_{i+1}}-O(n^{-\\gamma _{i+2}}W_{1i})\\right\\} \\right] \\\\\n&\\leq &1-\\mathbf{E}\\left[ \\exp \\left\\{ -O(n^{-\\gamma _{i+2}}W_{1i})\\right\\}\n\\right] =O\\left( \\left( n^{-\\gamma _{i+2}}\\right) ^{1\/2^{i}}\\right) =O\\left(\nn^{-\\gamma _{2}}\\right)\n\\end{eqnarray*\nwhere, for the penultimate equality we applied (\\ref{Tot2}), we conclude by \n\\ref{Tot1}) tha\n\\begin{eqnarray*}\n&&1-H_{m}^{(1,N)}\\left( \\mathbf{1}^{(i)},s_{i+1},...,s_{N}\\right) \\\\\n&&\\quad =(1+o(1))\\mathbf{E}\\left[ 1-\\exp \\left\\{ -(1+o(1))W_{1i,i+1}\\lambda\n_{i+1}n^{-\\gamma _{i+1}}\\right\\} \\right] \\\\\n&&\\qquad +O\\left( \\mathbf{P}\\left( T_{i}>\\sqrt{mn^{\\gamma _{i}}}\\right)\n\\right) \\\\\n&&\\quad =(1+o(1))D_{i}\\left( \\lambda _{i+1}n^{-\\gamma _{i+1}}\\right)\n^{1\/2^{i}}+o(n^{-\\gamma _{1}})\\sim D_{i}(\\lambda _{i+1})^{1\/2^{i}}n^{-\\gamma\n_{1}}\n\\end{eqnarray*\nas desired. \\\n\n\\begin{lemma}\n\\label{L_MultiSharp}If $m\\sim yn^{\\gamma _{i}}$ for some $i\\in \\left\\{\n2,3,...,N-1\\right\\} $ the\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }n^{\\gamma _{1}}Q_{m}^{(1,N)}(\\mathbf{s\n)=D_{i-1}(y^{-1}\\phi _{i}(\\lambda _{i}y,\\lambda _{i+1}y^{2}))^{1\/2^{i-1}}.\n\\end{equation*}\n\\end{lemma}\n\n\\textbf{Proof.} If $m\\sim yn^{\\gamma _{i}}$ and $j\\geq i$ then $n^{\\gamma\n_{j}}\\sim (y^{-1}m)^{2^{j-i}}$ and, therefore\n\\begin{equation*}\ns_{j}=\\exp \\left\\{ -\\lambda _{j}n^{-\\gamma _{j}}\\right\\} =\\exp \\left\\{\n-(1+o(1))\\lambda _{j}y^{2^{j-i}}m^{-2^{j-i}}\\right\\} .\n\\end{equation*\nHence we may apply Lemma \\ref{L_twoOnly} to get, as $n\\rightarrow \\infty \n\\begin{equation*}\nn^{\\gamma _{i}}Q_{m}^{(i,N)}(\\mathbf{s})\\sim\ny^{-1}mQ_{m}^{(i,N)}(s_{i},s_{i+1},...,s_{N})\\sim y^{-1}\\phi _{i}(\\lambda\n_{i}y,\\lambda _{i+1}y^{2}).\n\\end{equation*\nFurther, as in the previous lemma we have\n\\begin{equation*}\nQ_{m}^{(1,N)}(\\mathbf{s})=1-H_{m}^{(1,N)}\\left( \\mathbf{1\n^{(i-1)},s_{i},...,s_{N}\\right) +o(n^{-\\gamma _{1}})\n\\end{equation*\nand on the set $T_{i-1}\\leq \\sqrt{mn^{\\gamma _{i-1}}}\\ll m\\sim yn^{\\gamma\n_{i}}\n\\begin{eqnarray*}\n&&\\sum_{r=1}^{i-1}\\sum_{k=0}^{T_{i-1}}\\sum_{l=1}^{Z_{r}(k)}\\sum_{j=i}^{N\n\\eta _{rj}\\left( k,l\\right) Q_{m-k}^{(j,N)}(\\mathbf{s}) \\\\\n&&\\quad \\quad =(1+o(1))\\sum_{j=i}^{N}W_{1,i-1,j}Q_{m}^{(j,N)}(\\mathbf{s}) \\\\\n&&\\quad =(1+o(1))W_{1,i-1,i}Q_{m}^{(i,N)}(\\mathbf{s})+O\\left(\nQ_{m}^{(i+1,N)}(\\mathbf{s})\\right) \\sum_{j=i+1}^{N}W_{1,i-1,j} \\\\\n&&\\quad =(1+o(1))W_{1,i-1,i}(y^{-1}\\phi _{i}(\\lambda _{i}y,\\lambda\n_{i+1}y^{2}))^{1\/2^{i-1}}n^{-\\gamma _{i+1}} \\\\\n&&+O_{n}(n^{-\\gamma _{i+2}}W_{1,i-1}).\n\\end{eqnarray*\nTherefore\n\\begin{eqnarray*}\n&&1-H^{(1,N)}_{m}\\left( \\mathbf{1}^{(i-1)},s_{i},...,s_{N}\\right) \\\\\n&&\\quad =\\mathbf{E}\\left[ 1-\\exp \\left\\{ -(1+o(1))W_{1,i-1,i}y^{-1}\\phi\n_{i}(\\lambda _{i}y,\\lambda _{i+1}y^{2})\\,n^{-\\gamma _{i}}\\right\\} \\right] \\\\\n&&\\qquad +O\\left( \\mathbf{P}\\left( T_{i-1}\\geq \\sqrt{mn^{\\gamma _{i+1}}\n\\right) \\right) \\\\\n&&\\quad =(1+o(1))D_{i-1}\\big(y^{-1}\\phi _{i}(\\lambda _{i}y,\\lambda\n_{i+1}y^{2})\\,n^{-\\gamma _{i}}\\big)^{1\/2^{i-1}}+o(n^{-\\gamma _{1}}) \\\\\n&&\\quad \\sim D_{i-1}(y^{-1}\\phi _{i}(\\lambda _{i}y,\\lambda\n_{i+1}y^{2}))^{1\/2^{i-1}}n^{-\\gamma _{1}}.\n\\end{eqnarray*}\n\nThe lemma is proved.\n\n\\begin{lemma}\n\\label{L_DC}For all $i=1,2,...,N-1\n\\begin{equation}\nC_{N}=C_{i}(m_{i,i+1}b_{i}c_{i+1,N})^{1\/2^{i}}=D_{i}(c_{i+1,N})^{1\/2^{i}}.\n\\label{CD}\n\\end{equation}\n\\end{lemma}\n\n\\textbf{Proof.} Using (\\ref{Const1}) we have\n\\begin{equation*}\nc_{iN}=\\sqrt{b_{i}^{-1}m_{i,i+1}c_{i+1,N}}=b_{i}^{-1}\\sqrt\nb_{i}m_{i,i+1}c_{i+1,N}}=c_{ii}\\sqrt{b_{i}m_{i,i+1}c_{i+1,N}}\n\\end{equation*\nleading in view of (\\ref{Const2}) and (\\ref{DcConnection}) t\n\\begin{eqnarray*}\nC_{N} &=&c_{1N}=\\left( \\frac{1}{b_{N}}\\right)\n^{1\/2^{N-1}}\\prod_{j=1}^{N-1}\\left( \\frac{m_{j,j+1}}{b_{j}}\\right)\n^{1\/2^{j}}= \\\\\n&=&c_{1i}(b_{i}m_{i,i+1})^{1\/2^{i}}\\left( \\left( \\frac{1}{b_{N}}\\right)\n^{1\/2^{N-i}}\\prod_{j=i+1}^{N-1}\\left( \\frac{m_{j,j+1}}{b_{j}}\\right)\n^{1\/2^{j-i}}\\right) ^{1\/2^{i}} \\\\\n&=&c_{1i}(b_{i}m_{i,i+1}c_{i+1,N})^{1\/2^{i}}=D_{i}(c_{i+1,N})^{1\/2^{i}}\n\\end{eqnarray*\nas desired.\n\n\\section{Properties of the limiting processes\\label{Sec3}}\n\nIn this section we give a more detailed description of the properties of the\nlimiting processes. It follows from the definition of $\\mathbf{R}(t)$ that\nif\n\\begin{equation*}\n\\mathbf{S}_{i}=(s_{i1},s_{i2},...,s_{iN})\\in \\left[ 0,1\\right] ^{N}\\text{\nand }t_{i}\\in \\lbrack \\gamma _{i-1},\\gamma _{i}),i=1,2,...,N,\n\\end{equation*\nthe\n\\begin{equation*}\n\\mathbf{E}\\left[ \\prod_{i=1}^{N}\\mathbf{S}_{i}^{\\mathbf{R}(t_{i})}\\right]\n=\\Omega _{N}(s_{11},s_{22},...,s_{NN}),\n\\end{equation*\nwhere $\\Omega _{1}(s)=s$ and\n\\begin{equation}\n\\Omega _{i+1}(s_{1},s_{2},...,s_{i+1})=s_{1}\\left( 1-\\sqrt{1-\\Omega\n_{i}(s_{2},...,s_{i+1})}\\right) ,\\,i=1,2,....  \\label{DefOmega}\n\\end{equation}\n\nIf now some intervals $[\\gamma _{i-1},\\gamma _{i})$ contain more than one\npoint of observation over the process $\\mathbf{R}(\\cdot )$, say, $\\gamma\n_{i-1}\\leq t_{i1}<t_{i2}<...<t_{ik_{i}}<\\gamma _{i},i=1,2,...,N,$ and $_{j\n\\mathbf{S}_{i}=\\left( _{j}s_{i1},_{j}s_{i2},...,_{j}s_{iN}\\right) \\in \\left[\n0,1\\right] ^{N}\\text{ }$ then, clearly\n\\begin{equation*}\n\\mathbf{E}\\left[ \\prod_{i=1}^{N}\\prod_{j=1}^{k_{i}}(_{j}\\mathbf{S}_{i})^\n\\mathbf{R}(t_{ij})}\\right] =\\Omega _{N}\\left(\n\\prod_{j=1}^{k_{1}}\\,_{j}s_{11},\\prod_{j=1}^{k_{2}}\\,_{j}s_{22},...\n\\prod_{j=1}^{k_{N}}\\,_{j}s_{NN}\\right) .\n\\end{equation*}\n\nTo describe the characteristics of the processes $\\mathbf{U}_{i}(\\cdot\n),i=1,...,N-1$, let, for $(s_{i},s_{i+1})\\in \\lbrack 0,1]^{2}\n\\begin{equation}\n\\varphi _{i}(y;s_{i},s_{i+1})=\\sqrt{1-s_{i+1}}\\frac{(1-s_{i})+\\sqrt{1-s_{i+1\n}\\tanh (b_{i}c_{iN}y\\sqrt{1-s_{i+1}})}{(1-s_{i})\\tanh (b_{i}c_{iN}y\\sqrt\n1-s_{i+1}})+\\sqrt{1-s_{i+1}}}  \\label{Deffi}\n\\end{equation\nwith the natural agreement $\\varphi _{i}(y;1,1)=0$ an\n\\begin{equation*}\n\\varphi _{i}(y;s_{i},1)=\\frac{1-s_{i}}{b_{i}c_{iN}y(1-s_{i})+1}.\n\\end{equation*\nDenot\n\\begin{eqnarray*}\nX_{i}(y;\\mathbf{s}) &=&X_{i}(y;s_{i},s_{i+1})=\\mathbf{E}\\left[ \\mathbf{s}^\n\\mathbf{U}_{i}(y)}|\\mathbf{U}_{i}(0)=\\mathbf{e}_{i}\\right] \\\\\n&=&\\mathbf{E}\\left[ s_{i}^{U_{ii}(y)}s_{i+1}^{U_{i,i+1}(y)}|\\mathbf{U\n_{i}(0)=\\mathbf{e}_{i}\\right]\n\\end{eqnarray*\nand se\n\\begin{equation*}\n\\bar{X}_{R_{i}}(y;\\mathbf{s})=\\bar{X}_{R_{i}}(y;s_{i},s_{i+1})=\\mathbf{E\n_{R_{i}}\\left[ \\mathbf{s}^{\\mathbf{U}_{i}(y)}\\right] =\\mathbf{E}_{R_{i}\n\\left[ s_{i}^{U_{ii}(y)}s_{i+1}^{U_{i,i+1}(y)}\\right] ,\n\\end{equation*\nwhere the symbol $\\mathbf{E}_{R_{i}}[\\cdot ]$ means that the process starts\nby a random number of type $i$ particles distributed as $R_{i}$ in (\\re\n{Dist_ro}).\n\nIt follows from the description of the branching mechanism for $\\mathbf{U\n_{i}(\\cdot )$ and the general theory of branching processes (see, for\ninstance, \\cite{AN72}, p. 201) that $X_{i}(y;s_{i},s_{i+1})$ solves the\ndifferential equatio\n\\begin{eqnarray*}\n\\frac{\\partial }{\\partial y}X_{i}(y;s_{i},s_{i+1}) &=&2b_{i}c_{iN}\\left(\n\\frac{1}{2}X_{i}^{2}(y;s_{i},s_{i+1})-X_{i}(y;s_{i},s_{i+1})+\\frac{1}{2\ns_{i+1}\\right) , \\\\\nX_{i}(0;s_{i},s_{i+1}) &=&s_{i}.\n\\end{eqnarray*\nDirect calculations show tha\n\\begin{equation}\nX_{i}(y;s_{i},s_{i+1})=1-\\varphi _{i}(y;s_{i},s_{i+1})  \\label{ExplX}\n\\end{equation\nand, as a result\n\\begin{equation}\n\\bar{X}_{R_{i}}(y;s_{i},s_{i+1})=1-(\\varphi\n_{i}(y;s_{i},s_{i+1}))^{1\/2^{i-1}}\\text{.}  \\label{ExplXstar}\n\\end{equation\nOne may check by (\\ref{Deffi}) and (\\ref{ExplXstar}) tha\n\\begin{equation}\n\\lim_{y\\downarrow 0}\\bar{X}_{R_{i}}(y;s_{i},s_{i+1})=1-(1-s_{i})^{1\/2^{i-1}\n\\text{ }  \\label{LxLeft}\n\\end{equation\nan\n\\begin{equation}\n\\lim_{y\\uparrow \\infty }\\bar{X\n_{R_{i}}(y;s_{i},s_{i+1})=1-(1-s_{i+1})^{1\/2^{i}}.  \\label{liminf}\n\\end{equation}\n\nFor $y_{k}\\in \\lbrack 0,\\infty ),\\,(s_{ki},s_{k,i+1})\\in \\left[ 0,1\\right]\n^{2},\\ k=1,2,...,p;\\ i=1,...,N-1$ denote $\\mathbf{y}_{l,p}=(y_{l},...,y_{p})$\nand $\\mathbf{S\n_{l,p}^{(i)}=(s_{li},s_{l,i+1},s_{l+1,i},s_{l+1,i+1},...,s_{pi},s_{p,i+1}).$\n\nUsing (\\ref{ExplX}) se\n\\begin{equation*}\nX_{i}^{(2)}\\left( \\mathbf{y}_{1,2};\\mathbf{S}_{1,2}^{(i)}\\right)\n=X_{i}\\left(\ny_{1};s_{1i}X_{i}(y_{2};s_{2i},s_{2,i+1}),s_{1,i+1}s_{2,i+1}\\right)\n\\end{equation*\nand, by induction\n\\begin{equation*}\nX_{i}^{(p)}\\left( \\mathbf{y}_{1,p};\\mathbf{S}_{1,p}^{(i)}\\right)\n=X_{i}\\left( y_{1};s_{1i}X_{i}^{(p-1)}\\left( \\mathbf{y}_{2,p};\\mathbf{S\n_{2,p}^{(i)}\\right) ,\\prod_{r=1}^{p}s_{r,i+1}\\right) .\n\\end{equation*\nFinally, recalling (\\ref{ExplXstar}) put\n\\begin{equation*}\n\\bar{X}_{R_{i}}\\left( \\mathbf{y}_{1,p};\\mathbf{S}_{1,p}^{(i)}\\right)\n=1-\\left( 1-X_{i}^{(p)}\\left( \\mathbf{y}_{1,p};\\mathbf{S}_{1,p}^{(i)}\\right)\n\\right) ^{1\/2^{i-1}}.\n\\end{equation*\nIt is not difficult to check tha\n\\begin{equation*}\n\\bar{X}_{R_{i}}\\left( \\mathbf{y}_{1,p};\\mathbf{S}_{1,p}^{(i)}\\right) \n\\mathbf{E}_{R_{i}}\\left[\ns_{1i}^{U_{ii}(y_{1})}s_{1,i+1}^{U_{i,i+1}(y_{1})}...s_{pi}^{U_{ii}(y_{p})}s_{p,i+1}^{U_{i,i+1}(y_{p})\n\\right] .\n\\end{equation*}\n\nTo complete the description of the limiting processes we are interesting in\nintroduce the function\n\\begin{equation*}\n\\psi (x;s)=\\frac{1}{x+(1-x)\/(1-s)},\\quad s\\in \\lbrack 0,1],\\,x\\in \\lbrack\n0,1],\n\\end{equation*\nand consider an $N-$dimensional process $\\mathbf{U}_{N}(\\cdot\n)=(U_{N1}(\\cdot ),...,U_{NN}(\\cdot ))$ in which the first $N-1$ components\nare equal to zero while $U_{NN}(\\cdot )$ may be obtained by a time-change\nfrom the following single-type continuous time Markov process $\\sigma\n(t),0\\leq t<\\infty $. The life-length distribution of particles in $\\sigma\n(\\cdot )$ is exponential with parameter 1. Dying each particle produces\nexactly two children. One may check (compare, for instance, with Example 3,\nSection 8, Chapter 1 in \\cite{Sev74}) that\n\\begin{equation*}\n\\mathbf{E}\\left[ s^{\\sigma (t)}|\\sigma (0)=1\\right] =1-\\psi (1-e^{-t};s).\n\\end{equation*\nAssuming that $\\sigma (0)\\overset{d}{=}R_{N}$ (recall (\\ref{DefroN})) and\nmaking the change of time $x=1-e^{-t},0\\leq t<\\infty ,$ we obtain an\ninhomogeneous single-type branching process, denoted by $U_{NN}(\\cdot )$\nsuch tha\n\\begin{equation*}\n\\bar{G}_{R_{N}}(x;s)=\\mathbf{E}_{R_{N}}\\left[ s^{U_{NN}(x)}\\right] =1-(\\psi\n(x;s))^{1\/2^{N-1}}\n\\end{equation*\nan\n\\begin{equation*}\n\\mathbf{E}\\left[ s^{U_{NN}(x+\\Delta )}|U_{NN}(x)=1\\right] =1-\\psi \\left(\n\\frac{\\Delta }{1-x};s\\right) ,\\ 0<x+\\Delta <1.\n\\end{equation*\nLet, further, for $x_{j}\\in \\lbrack 0,1)$ and $\\mathbf{S\n_{j,p}=(s_{j},...,s_{p}),\\,j=1,2,...,p\n\\begin{equation*}\nG^{(1)}(x_{1};s_{1})=G(x_{1};s_{1})=1-\\psi (x_{1};s_{1})\n\\end{equation*\nand, by inductio\n\\begin{equation*}\nG^{(p)}\\left( \\mathbf{x}_{1,p};\\mathbf{S}_{1,p}\\right) =G\\left(\nx_{1};s_{1N}G^{(p-1)}\\left( \\frac{\\mathbf{x}_{2,p}}{1-x_{1}};\\mathbf{S\n_{2,p}\\right) \\right) .\n\\end{equation*\nOne may check tha\n\\begin{eqnarray*}\n\\bar{G}_{R_{N}}(\\mathbf{x}_{1,p};\\mathbf{S}_{1,p}) &=&\\mathbf{E}_{R_{N}\n\\left[ s_{1}^{U_{NN}(x_{1})}s_{2}^{U_{NN}(x_{2})}...\\,s_{p}^{U_{NN}(x_{p})\n\\right] \\\\\n&=&1-(1-G^{(p)}(\\mathbf{x}_{1,p};\\mathbf{S}_{1,p}))^{1\/2^{N-1}}.\n\\end{eqnarray*}\n\n\\section{Convergence of one-dimensional distributions\\label{Sec4}}\n\nAs the first step in proving the main results of the paper we establish\nconvergence of one-dimensional distributions of $\\{ \\mathbf{Z}(m,n),0\\leq\nm\\leq n\\} $ given $\\mathbf{Z}(n)\\neq \\mathbf{0}$. Le\n\\begin{eqnarray*}\nH_{m,n}^{\\,(k,N)}(\\mathbf{s})=\\mathbf{E}\\left[ \\mathbf{s}^{\\mathbf{Z}(m,n)}\n\\mathbf{Z}(0)=\\mathbf{e}_{k}\\right], J_{m,n}^{\\,(k,N)}(\\mathbf{s})=\\mathbf{E\n\\left[ \\mathbf{s}^{\\mathbf{Z}(m,n)}|\\mathbf{Z}(n)\\neq \\mathbf{0},\\mathbf{Z\n(0)=\\mathbf{e}_{k}\\right],\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\mathbf{H}_{m,n}(\\mathbf{s}) =\\left( H_{m,n}^{\\,(1,N)}(\\mathbf{s\n),...,H_{m,n}^{\\,(N,N)}(\\mathbf{s})\\right),\\ \\mathbf{J}_{m,n}(\\mathbf{s\n)=\\left( J_{m,n}^{\\,(1,N)}(\\mathbf{s}),...,J_{m,n}^{\\,(N,N)}(\\mathbf{s\n)\\right).\n\\end{eqnarray*\nFor $\\mathbf{x}=(x_{1},...,x_{N})$ and $\\mathbf{y}=(y_{1},...,y_{N})$ put \n\\mathbf{x}\\otimes \\mathbf{y=}(x_{1}y_{1},x_{2}y_{2},...,x_{N}y_{N})$ and\ndenote\n\\begin{eqnarray}\ns_{k}^{\\prime }\n&=&s_{k}Q_{n-m}^{(k,N)}+(1-Q_{n-m}^{(k,N)})=1-(1-s_{k})Q_{n-m}^{(k,N)},\n\\notag \\\\\n&&  \\notag \\\\\n\\mathbf{s}^{\\prime } &=&(s_{1}^{\\prime },...,s_{N}^{\\prime })=\\mathbf{1}-\n\\mathbf{1}-\\mathbf{s})\\otimes \\mathbf{Q}_{n-m}.  \\label{sprimeVect}\n\\end{eqnarray\nIt is not difficult to understand that\n\\begin{equation*}\nH_{m,n}^{(k,N)}(\\mathbf{s})=H_{m}^{(k,N)}(\\mathbf{s}^{\\prime\n})=H_{m}^{(k,N)}(\\mathbf{1}-(\\mathbf{1}-\\mathbf{s})\\otimes \\mathbf{Q}_{n-m})\n\\end{equation*\nand that\n\\begin{equation}\nJ_{m,n}^{\\,(k,N)}(\\mathbf{s})=\\mathbf{E}\\left[ \\mathbf{s}^{\\mathbf{Z}(m,n)}\n\\mathbf{Z}(n)\\neq \\mathbf{0},\\mathbf{Z}(0)=\\mathbf{e}_{k}\\right] =1-\\frac\nQ_{m}^{\\,(k,N)}(\\mathbf{s}^{\\prime })}{Q_{n}^{\\,(k,N)}}.  \\label{BranProp2}\n\\end{equation}\n\n\\begin{theorem}\n\\label{T_familyMany}Let Hypothesis A be valid.\n\n1) If $m\\ll n^{\\gamma _{1}}$ then\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty }J_{m,n}^{\\,(1,N)}(\\mathbf{s})=\\lim_{n\\rightarrow\n\\infty }\\mathbf{E}\\left[ \\mathbf{s}^{\\mathbf{Z}(m,n)}|\\mathbf{Z}(n)\\neq\n\\mathbf{0},\\mathbf{Z}(0)=\\mathbf{e}_{1}\\right] =s_{1}.  \\label{LimSing}\n\\end{equation\n2) If $n^{\\gamma _{i}}\\ll m\\ll n^{\\gamma _{i+1}}$ for some $i\\in \\left\\{\n1,2,...,N-1\\right\\} $ then\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty }J_{m,n}^{\\,(1,N)}(\\mathbf{s\n)=1-(1-s_{i+1})^{1\/2^{i}}.  \\label{LimSIngLate}\n\\end{equation\n3) If $m=(y+l_{n})n^{\\gamma _{i}},\\,y\\in \\lbrack 0,\\infty )$ for some $i\\in\n\\left\\{ 1,2,...,N-1\\right\\} $ then\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty }J_{m,n}^{\\,(1,N)}(\\mathbf{s})=\\bar{X\n_{R_{i}}(y;s_{i},s_{i+1}).  \\label{LimTWo}\n\\end{equation\n4) If $m=(x+l_{n})n$, $x\\in \\lbrack 0,1)$ then\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty }J_{m,n}^{{}\\,(1,N)}(\\mathbf{s})=\\bar{G\n_{R_{N}}(x;s_{N}).  \\label{LimOLD}\n\\end{equation}\n\\end{theorem}\n\n\\textbf{Proof.} We start by observing that if $m\\ll n$ then\n\\begin{eqnarray*}\n1-s_{i}^{\\prime } &=&(1-s_{i})Q_{n-m}^{(i,N)}\\sim (1-s_{i})Q_{n}^{(i,N)} \\\\\n&\\sim &1-\\exp \\left\\{ -(1-s_{i})Q_{n}^{(i,N)}\\right\\} \\sim 1-\\exp \\left\\{\n-(1-s_{i})c_{iN}n^{-\\gamma _{i}}\\right\\} .\n\\end{eqnarray*\nThis representation allows us to use the previous results with $s_{i}$ and \n\\lambda _{i}$ replaced by $s_{i}^{\\prime }$ and $(1-s_{i})c_{iN}$,\nrespectively.\n\nRecalling (\\ref{SurvivSingle}) and applying Lemma \\ref{L_OneOnly} we ge\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }\\frac{Q_{m}^{(1,N)}(\\mathbf{s}^{\\prime}) }\nQ_{n}^{(1,N)}}=\\lim_{n\\rightarrow \\infty }n^{2^{-(N-1)}}\\frac\n1-H_{m}^{(1,N)}( \\mathbf{s}^{\\prime }) }{C_{N}}=1-s_{1}.\n\\end{equation*}\nHence (\\ref{LimSing}) follows.\n\nApplying Lemma \\ref{L_inbetween} with $n^{\\gamma _{i}}\\ll m\\ll n^{\\gamma\n_{i+1}}$ and recalling Lemma \\ref{L_DC} we conclud\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }\\frac{Q_{m}^{(1,N)}(\\mathbf{s}^{\\prime })}\nQ_{n}^{(1,N)}}=\\frac{D_{i}}{C_{N}\n((1-s_{i+1})c_{i+1,N})^{1\/2^{i}}=(1-s_{i+1})^{1\/2^{i}}\n\\end{equation*\nleading to (\\ref{LimSIngLate}).\n\n\\textbf{Proof of (\\ref{LimTWo}).} If $y=0$ then the needed statement follows\nfrom~(\\ref{LimSing}) and~(\\ref{LimSIngLate}). If $i\\in \\left\\{\n1,2,...,N-1\\right\\} $ is fixed and $m\\sim yn^{\\gamma _{i}},y>0,$ then for \nj\\geq i$\n\\begin{eqnarray*}\n1-s_{j}^{\\prime } &\\sim &1-\\exp \\left\\{ -(1-s_{j})c_{jN}n^{-\\gamma\n_{j}}\\right\\} \\\\\n&\\sim &1-\\exp \\left\\{ -(1-s_{j})c_{jN}y^{2^{j-i}}m^{-2^{j-i}}\\right\\} .\n\\end{eqnarray*\nHence, by (\\ref{SurvivSingle}) and Lemmas \\ref{L_twoOnly} and \\re\n{L_MultiSharp} we get\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }\\frac{Q_{m}^{(1,N)}(\\mathbf{s}^{\\prime })}\nQ_{n}^{(1,N)}}=\\frac{D_{i-1}}{C_{N}}\\left( \\frac{\\phi\n_{i}(c_{iN}(1-s_{i})y,c_{i+1,N}(1-s_{i+1})y^{2})}{y}\\right) ^{1\/2^{i-1}}\n\\end{equation*\nwhere we agree to write $D_{0}=1$. By~(\\ref{DefSimpl}) and~(\\ref{Const1})\n\\begin{eqnarray*}\n&&\\frac{\\phi _{i}(c_{iN}(1-s_{i})y,c_{i+1,N}(1-s_{i+1})y^{2})}{y} \\\\\n&&\\quad =\\sqrt{\\frac{m_{i,i+1}c_{i+1,N}(1-s_{i+1})}{b_{i}}}\\times \\\\\n&&\\qquad \\times \\frac{b_{i}c_{iN}(1-s_{i})+\\sqrt\nb_{i}m_{i,i+1}c_{i+1,N}(1-s_{i+1})}\\tanh y\\sqrt\nb_{i}m_{i,i+1}c_{i+1,N}(1-s_{i+1})}}{b_{i}c_{iN}(1-s_{i})\\tanh y\\sqrt\nb_{i}m_{i,i+1}c_{i+1,N}(1-s_{i+1})}+\\sqrt{b_{i}m_{i,i+1}c_{i+1,N}(1-s_{i+1})\n} \\\\\n&&\\quad =c_{iN}\\sqrt{1-s_{i+1}}\\times \\frac{b_{i}c_{iN}(1-s_{i})+b_{i}c_{iN\n\\sqrt{1-s_{i+1}}\\tanh (yb_{i}c_{iN}\\sqrt{1-s_{i+1}})}{b_{i}c_{iN}(1-s_{i}\n\\tanh (yb_{i}c_{iN}\\sqrt{1-s_{i+1}})+b_{i}c_{iN}\\sqrt{1-s_{i+1}}} \\\\\n&&\\quad =c_{iN}\\sqrt{1-s_{i+1}}\\times \\frac{1-s_{i}+\\sqrt{1-s_{i+1}}\\tanh\n(yb_{i}c_{iN}\\sqrt{1-s_{i+1}})}{(1-s_{i})\\tanh (yb_{i}c_{iN}\\sqrt{1-s_{i+1}\n)+\\sqrt{1-s_{i+1}}}.\n\\end{eqnarray*\nTo complete the proof of (\\ref{LimTWo}) it remains to recall (\\ref{CD}).\n\n\\textbf{Proof of (\\ref{LimOLD}).} If $x=0$ then (\\ref{LimOLD}) follows from \n\\ref{LimSIngLate}). Consider now the case $m\\sim xn,0<x<1$. Observe that for\n$\\mathbf{s}=(s_{1},s_{2},...,s_{N})\\in \\left[ 0,1\\right] ^{N}\n\\begin{eqnarray}\nH_{m}^{(1,N)}(\\mathbf{1}^{(N-1)},s_{N})-H_{m}^{(1,N)}(\\mathbf{s}) &=&\\mathbf\nE}\\left[ \\left( 1-s_{1}^{Z_{1}(m)}...\\,s_{N-1}^{Z_{N-1}(m)}\\right)\ns_{N}^{Z_{N}(m)}\\right]  \\notag \\\\\n\\quad \\leq \\mathbf{E}\\left[ 1-s_{1}^{Z_{1}(m)}...\\,s_{N-1}^{Z_{N-1}(m)\n\\right] &\\leq &\\mathbf{P}(T_{N-1}>m)\\leq cm^{-2^{-(N-2)}}.  \\label{Neglig}\n\\end{eqnarray\nThus\n\\begin{equation*}\n1-H_{m}^{(1,N)}(\\mathbf{s})=1-H_{m}^{(1,N)}\\left( \\mathbf{1\n^{(N-1)},s_{N}\\right) +\\varepsilon _{m,n}(\\mathbf{s})Q_{m}^{(1,N)}\n\\end{equation*\nwhere $\\varepsilon _{m,n}(\\mathbf{s})\\rightarrow 0$ as $n\\rightarrow \\infty\n, $ $m\\sim xn$ $\\ $uniformly in $\\mathbf{s}\\in \\left[ 0,1\\right] ^{N}.$\nTherefore,\n\\begin{equation*}\n1-H_{m}^{(1,N)}(\\mathbf{s}^{\\prime })=1-H_{m}^{(1,N)}\\left( \\hat{\\mathbf{s}\n,1-(1-s_{N})Q_{n-m}^{(N,N)}\\right) +\\varepsilon _{m,n}^{\\prime }(\\mathbf{s\n)Q_{n}^{(1,N)}\n\\end{equation*\nwhere $\\varepsilon _{m,n}^{\\prime }(\\mathbf{s})\\rightarrow 0$ as \nn\\rightarrow \\infty ,$ $m\\sim xn$ $\\ $uniformly in $\\hat{\\mathbf{s}}=\\left(\ns_{1}^{\\prime },...,s_{N-1}^{\\prime }\\right) \\in \\left[ 0,1\\right] ^{N-1}$.\n\nWe now select an integer $r=r(m,n)\\in \\mathbb{N}^{\\ast }=\\left\\{\n1,2,...,\\right\\} $ in such a way tha\n\\begin{equation*}\nH_{r-1}^{(N,N)}(0)\\leq 1-(1-s_{N})Q_{n-m}^{(N,N)}\\leq H_{r}^{(N,N)}(0)\n\\end{equation*\no\n\\begin{equation*}\nQ_{r}^{(N,N)}=1-H_{r}^{(N,N)}(0)\\leq (1-s_{N})Q_{n-m}^{(N,N)}\\leq\nQ_{r-1}^{(N,N)}=1-H_{r-1}^{(N,N)}(0).\n\\end{equation*\nThis is possible, since by (\\ref{SurvivSingle})\n\\begin{equation}\nQ_{n-m}^{(N,N)}\\sim \\frac{1}{(n-m)b_{N}}\\rightarrow 0,\\ n-m\\rightarrow\n\\infty .  \\label{ASRep}\n\\end{equation\nIn particular\n\\begin{equation}\nr\\sim \\frac{n-m}{1-s_{N}}.  \\label{ASSR}\n\\end{equation\nUnder our choice of $r$, for any $\\hat{\\mathbf{s}}\\in \\left[ 0,1\\right]\n^{N-1}$\n\\begin{equation*}\nH_{m}^{(1,N)}\\left( \\hat{\\mathbf{s}},H_{r-1}^{(N,N)}(0)\\right) \\leq\nH_{m}^{(1,N)}\\left( \\hat{\\mathbf{s}},1-(1-s_{N})Q_{n-m}^{(N,N)}\\right) \\leq\nH_{m}^{(1,N)}\\left( \\hat{\\mathbf{s}},H_{r}^{(N,N)}(0)\\right) .\n\\end{equation*\nLetting $\\hat{\\mathbf{s}}=\\left( H_{r}^{(1,N)}(\\mathbf{0\n),...,H_{r}^{(N-1,N)}(\\mathbf{0})\\right) $ we get by the branching property\nof generating functions the estimate\n\\begin{equation*}\nH_{m}^{(1,N)}\\left( \\hat{\\mathbf{s}},1-(1-s_{N})Q_{n-m}^{(N,N)}\\right) \\leq\nH_{m}^{(1,N)}(\\mathbf{H}_{r}(\\mathbf{0}))=H_{m+r}^{(1,N)}(\\mathbf{0})\n\\end{equation*\nimplying\n\\begin{equation*}\n1-H_{m}^{(1,N)}(\\mathbf{s}^{\\prime })\\geq 1-H_{m+r}^{(1,N)}(\\mathbf{0\n)+\\varepsilon _{m,n}^{\\prime }Q_{n}^{(1,N)}=Q_{m+r}^{(1,N)}+\\varepsilon\n_{m,n}^{\\prime }Q_{n}^{(1,N)},\n\\end{equation*\nwhere $\\varepsilon _{m,n}^{\\prime }\\rightarrow 0$ as $n\\rightarrow \\infty ,$\n$m\\sim xn,$ while $\\hat{\\mathbf{s}}=(H_{r-1}^{(1,N)}(\\mathbf{0\n),...,H_{r-1}^{(N-1,N)}(\\mathbf{0}))$ gives the inequality\n\\begin{equation*}\nH_{m}^{(1,N)}\\left( \\hat{\\mathbf{s}},1-(1-s_{N})Q_{n-m}^{(N,N)}\\right) \\geq\nH_{m}^{(1,N)}(\\mathbf{H}_{r}(\\mathbf{0}))=H_{m+r-1}^{(1,N)}(\\mathbf{0})\n\\end{equation*\nleading in the range under consideration to\n\\begin{equation*}\n1-H_{m}^{(1,N)}(\\mathbf{s}^{\\prime })\\leq Q_{m+r-1}^{(1,N)}+\\varepsilon\n_{m,n}^{\\prime }Q_{n}^{(1,N)}.\n\\end{equation*\nHenc\n\\begin{equation*}\n1-H_{m}^{(1,N)}(\\mathbf{s}^{\\prime })=Q_{m+r}^{(1,N)}+\\varepsilon\n_{m,n}^{\\prime \\prime }Q_{n}^{(1,N)}\n\\end{equation*\nwhere $\\varepsilon _{m,n}^{\\prime \\prime }\\rightarrow 0$ as $n\\rightarrow\n\\infty ,$ $m\\sim xn$. We now conclude by (\\ref{SurvivSingle}) tha\n\\begin{equation*}\n1-H_{m}^{(1,N)}(\\mathbf{s}^{\\prime })\\sim Q_{m+r}^{(1,N)}\\sim\nC_{N}(m+r)^{-2^{-(N-1)}}.\n\\end{equation*\nHence, on account of (\\ref{ASSR}) and $m\\sim xn,0<x<1,$ we get (recall (\\re\n{Const1})\n\\begin{eqnarray*}\n\\lim_{n\\rightarrow \\infty }\\frac{1-H_{m}^{(1,N)}(\\mathbf{s}^{\\prime })}\nQ_{n}^{(1,N)}} &=&\\lim_{n\\rightarrow \\infty }\\left( \\frac{n}\nnx+n(1-x)\/(1-s_{N})}\\right) ^{2^{-(N-1)}} \\\\\n&=&\\left( \\frac{1}{x+(1-x)\/(1-s_{N})}\\right) ^{2^{-(N-1)}}\n\\end{eqnarray*\ncompleting the proof of (\\ref{LimOLD}).\n\nTheorem \\ref{T_familyMany} is proved.\\\n\n\\textbf{Proof} \\textbf{of Theorem \\ref{T_Yaglom}}. Since our process is\ndecomposable and strongly critical, it is sufficient to check (\\ref{Yag})\nfor $i=1$ only. For $\\hat{s}_{N}=\\exp (-\\lambda \/(nb_{N}))$ we hav\n\\begin{equation*}\n\\mathbf{E}\\left[ \\exp \\left\\{ -\\lambda \\frac{Z_{N}(n)}{b_{N}n}\\right\\} \\Big\n\\mathbf{Z}(n)\\neq \\mathbf{0}\\right] =1-\\frac{1-H_{n}^{(1,N)}(\\mathbf{1\n^{(N-1)},\\hat{s}_{N})}{Q_{n}^{(1,N)}}.\n\\end{equation*\nWe now select an integer $r=r(\\lambda ,n)\\in \\mathbb{N}^{\\ast }=\\left\\{\n1,2,...,\\right\\} $ in such a way tha\n\\begin{equation*}\nH_{r-1}^{(N,N)}(0)\\leq \\hat{s}_{N}\\leq H_{r}^{(N,N)}(0).\n\\end{equation*\nIt follows from (\\ref{ASRep})\\ that $r\\sim n\\lambda ^{-1}$. Letting \ns_{i}=H_{r}^{(i,N)}(\\mathbf{0}),i=1,2,...,N,$ and setting $\\mathbf{s\n=(s_{1},...,s_{N})$ we get by (\\ref{Neglig}) after evident estimates tha\n\\begin{equation*}\n\\left\\vert H_{n}^{(1,N)}(\\mathbf{1}^{(N-1)},\\hat{s}_{N})-H_{n}^{(1,N)}\n\\mathbf{s})\\right\\vert \\leq cn^{1\/2^{N-2}}.\n\\end{equation*\nHence, using (\\ref{SurvivSingle}) with $i=1$ we obtain\n\\begin{eqnarray*}\n\\frac{1-H_{n}^{(1,N)}(\\mathbf{1}^{(N-1)},\\hat{s}_{N})}{Q_{n}^{(1,N)}} &\\sim \n\\frac{1-H_{n}^{(1,N)}(\\mathbf{s})}{Q_{n}^{(1,N)}}=\\frac{Q_{r+n}^{(1,N)}}\nQ_{n}^{(1,N)}} \\\\\n&\\sim &\\left( \\frac{n}{r+n}\\right) ^{1\/2^{N-1}}\\sim \\left( \\frac{\\lambda }\n1+\\lambda }\\right) ^{1\/2^{N-1}}\n\\end{eqnarray*\nas desired.\\\n\n\\section{Convergence of finite-dimensional distributions\\label{Sec5}}\n\nIn this section we study the limiting behavior of the finite-dimensional\ndistributions of the reduced process $\\left\\{ \\mathbf{Z}( m,n) ,0\\leq m\\leq\nn\\right\\} $. Our first theorem deals with the case $m\\ll n$.\n\n\\begin{theorem}\n\\label{T_findim} Let Hypothesis A be valid and $\\mathbf{S\n_{l}=(s_{l1},...,s_{lN}),l=1,2,...,p$.\n\n1) If, for a fixed $i\\in \\left\\{ 0,1,...,N-1\\right\\}$\n\\begin{equation*}\nn^{\\gamma _{i}}\\ll m_{l}\\ll n^{\\gamma _{i+1}},\\quad l=1,...,p\n\\end{equation*\nthe\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty }\\mathbf{E}\\left[ \\prod_{l=1}^{p}\\mathbf{S}_{l}^\n\\mathbf{Z}(m_{l},n)}\\,\\Big|\\,\\mathbf{Z}(n)\\neq 0\\right] =1-\\left(\n1-\\prod_{l=1}^{p}s_{l,i+1}\\right) ^{1\/2^{i}}.  \\label{Lim_diskr}\n\\end{equation}\n\n2) Let $0=Y_{1}<Y_{2}<...<Y_{p}<\\infty $ be a tuple of nonnegative numbers\nwith $y_{1}=0,$ $y_{l}=Y_{l}-Y_{l-1},\\,l=2,...,p.$ If, for a fixed $i\\in\n\\left\\{ 1,2,...,N-1\\right\\} $\n\\begin{equation*}\nm_{1}\\sim l_{n}n^{\\gamma _{i}},\\quad m_{l}\\sim Y_{l}n^{\\gamma\n_{i}},\\,l=2,...,p\n\\end{equation*\nthen\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty }\\mathbf{E}\\left[ \\prod_{l=1}^{p}\\mathbf{S}_{l}^\n\\mathbf{Z}(m_{l},n)}\\,\\Big|\\,\\mathbf{Z}(n)\\neq 0\\right] =\\bar{X}_{R_{i}}\\Big\n\\mathbf{y}_{1,p};\\mathbf{S}_{1,p}^{(i)}\\Big).  \\label{Lim_Multi}\n\\end{equation}\n\\end{theorem}\n\nThe second theorem is devoted to the finite-dimensional distributions of the\nreduced process when $m$ is of order $n$.\n\n\\begin{theorem}\n\\label{T_endpoint}Let\\textbf{\\ }Hypothesis A be valid and \n0=X_{1}<X_{2}<...<X_{p}<1$ be a tuple of nonnegative numbers with $x_{1}=0,$\n$x_{l}=X_{l}-X_{l-1},\\,l=2,...,p$. If\n\\begin{equation*}\nm_{1}\\sim l_{n}n,\\quad m_{l}\\sim X_{l}n,\\,l=2,...,p\n\\end{equation*\nthen\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }\\mathbf{E}\\left[ \\prod_{l=1}^{p}\\mathbf{S}_{l}^\n\\mathbf{Z}(m_{l},n)}\\,\\Big|\\,\\mathbf{Z}(n)\\neq 0\\right] =\\bar{G}_{R_{N}}\\Big\n\\mathbf{x}_{1,p};\\mathbf{S}_{1,p;N}\\Big),\n\\end{equation*\nwhere $\\mathbf{S}_{1,p;N}=(s_{1N},s_{2N},...,s_{pN})$.\n\\end{theorem}\n\nTo prove Theorems \\ref{T_findim} and \\ref{T_endpoint} we need additional\nnotation.\n\nFor $0\\leq m_{0}<m_{1}<...<m_{p}\\leq n$ set $\\mathbf{m\n=(m_{0},m_{1},...,m_{p}),$ put $\\Delta _{i}=m_{i}-m_{i-1}$, and denot\n\\begin{eqnarray*}\n\\hat{J}_{m_{0},m_{1},...,m_{p},n}^{\\,(i,N)}(\\mathbf{S}_{1},...,\\mathbf{S\n_{p}) &=&\\hat{J}_{\\mathbf{m},n}^{\\,(i,N)}(\\mathbf{S}_{1},...,\\mathbf{S}_{p})\n\\\\\n&=&\\mathbf{E}\\left[ \\prod_{l=1}^{p}\\mathbf{S}_{l}^{\\mathbf{Z}(m_{l},n)}\\\n\\Big|\\,\\mathbf{Z}(m_{0},n)=\\mathbf{e}_{i}\\right]\n\\end{eqnarray*\nan\n\\begin{equation*}\n\\mathbf{\\hat{J}}_{\\mathbf{m},n}(\\mathbf{S}_{1},...,\\mathbf{S}_{p})=\\left(\n\\hat{J}_{\\mathbf{m},n}^{\\,(1,N)}(\\mathbf{S}_{1},...,\\mathbf{S}_{p}),...,\\hat\nJ}_{\\mathbf{m},n}^{\\,(N,N)}(\\mathbf{S}_{1},...,\\mathbf{S}_{p})\\right) .\n\\end{equation*}\n\nThe next statement is a simple observation following from Corollary 2 in\n\\cite{VD06}.\n\n\\begin{lemma}\n\\label{L_convol}For any $0\\leq m_{0}<m_{1}<...<m_{p}\\leq n$ we hav\n\\begin{eqnarray*}\n&&\\hat{J}_{\\mathbf{m},n}^{\\,\\,(1,N)}(\\mathbf{S}_{1},...,\\mathbf{S}_{p})=\\hat\nJ}_{m_{0},m_{1},n}^{\\,(1,N)}\\left( \\mathbf{S}_{1}\\otimes \\mathbf{\\hat{J}\n_{m_{1},m_{2},...,m_{p},n}(\\mathbf{S}_{2},...,\\mathbf{S}_{p})\\right) \\\\\n&&\\quad=J_{\\Delta _{1},n-m_{0}}^{\\,(1,N)}\\left( \\mathbf{S}_{1}\\otimes\n\\mathbf{J}_{\\Delta _{2},n-m_{1}}\\left( \\mathbf{S}_{2}\\otimes ...(\\mathbf{S\n_{p-1}\\otimes \\mathbf{J}_{\\Delta _{p},n-m_{p-1}}(\\mathbf{S}_{p})\\right)\n....)\\right) .\n\\end{eqnarray*\nIn particular, if $\\mathbf{m}=(0,m_{1},m_{2})$ the\n\\begin{equation*}\n\\hat{J}_{\\mathbf{m},n}^{\\,(1,N)}(\\mathbf{S}_{1},\\mathbf{S\n_{2})=J_{m,n}^{\\,(1,N)}(\\mathbf{S}_{1}\\otimes \\mathbf{J}_{\\Delta\n_{2},n-m_{1}}(\\mathbf{S}_{2}))\n\\end{equation*\nand if $\\mathbf{m}=(m_{0},m_{1})$ then for $\\mathbf{s}=(s_{1},...,s_{N})\n\\begin{equation}\n\\hat{J}_{m_{0},m_{1},n}^{\\,(k,N)}(\\mathbf{s})=J_{\\Delta\n_{1},n-m_{0}}^{\\,(k,N)}(\\mathbf{s})=1-\\frac{1-H_{\\Delta _{1}}^{(k,N)}\\left(\n\\mathbf{1}-(\\mathbf{1}-\\mathbf{s})\\otimes \\mathbf{Q}_{n-m_{1}}\\right) }\nQ_{n-m_{0}}^{(k,N)}}.  \\label{Derivat}\n\\end{equation}\n\\end{lemma}\n\nUsing (\\ref{Derivat}) we prove the following statement.\n\n\\begin{lemma}\n\\label{L_derivative}If $m_{0}=(Y_{0}+l_{n})n^{\\gamma\n_{i}}<m_{1}=(Y_{1}+l_{n})n^{\\gamma _{i}}$ then for any $j\\geq i$ there\nexists a constant $\\chi \\in (0,\\infty )$ such that for all $n\\geq n_{0}$ \\\n\\begin{equation*}\n\\mathbf{P}(\\mathbf{Z}(m_{1},n)=\\mathbf{e}_{j}|\\mathbf{Z}(m_{0},n)=\\mathbf{e\n_{j})\\geq 1-\\chi (Y_{1}-Y_{0}).\n\\end{equation*}\n\\end{lemma}\n\n\\textbf{Proof.} By the decomposability assumption and the condition \nm_{jj}=1 $ implying\n\\begin{equation*}\nm_{jj}(\\Delta _{1})=\\frac{\\partial H_{\\Delta _{1}}^{(j,N)}(\\mathbf{s})}\n\\partial s_{j}}\\left\\vert _{\\mathbf{s}=\\mathbf{1}}\\right. =1\n\\end{equation*\nwe get\n\\begin{equation*}\n1-\\frac{\\partial H_{\\Delta _{1}}^{(j,N)}(\\mathbf{s})}{\\partial s_{j}\n\\left\\vert _{\\mathbf{s}=\\mathbf{H}_{n-m_{1}}(\\mathbf{0})}\\right. \\leq\n\\sum_{k=j}^{N}\\mathbf{E}Z_{j}(\\Delta _{1})(Z_{k}(\\Delta _{1})-\\delta\n_{kj})Q_{n-m_{1}}^{(k,N)}.\n\\end{equation*\nRecalling (\\ref{Momvariance}) and (\\ref{SurvivSingle}) and setting \nh=Y_{1}-Y_{0}$ we obtain\n\\begin{eqnarray*}\n\\mathbf{E}Z_{j}(\\Delta _{1})Z_{k}(\\Delta _{1})Q_{n-m_{1}}^{(k,N)} &\\leq\n&c_{0}(n-m_{1})^{-1\/2^{N-k}}(\\Delta _{1})^{k-j+1} \\\\\n&\\leq &c_{0}(n-m_{1})^{-1\/2^{N-k}}(hn^{1\/2^{N-i}})^{k-j+1} \\\\\n&\\leq &\\chi hn^{-1\/2^{N-k}}(n^{1\/2^{N-i}})^{k-j+1}\n\\end{eqnarray*\nfor some constants $0<c_{0}\\leq \\chi <\\infty $. On account of $k\\geq j\\geq i$\nwe hav\n\\begin{equation*}\n\\frac{k-j+1}{2^{N-i}}-\\frac{1}{2^{N-k}}=\\frac{1}{2^{N-i}}(k-j+1-2^{k-i})\\leq\n0.\n\\end{equation*\nThus\n\\begin{equation*}\n1-\\frac{\\partial H_{\\Delta _{1}}^{(j,N)}(\\mathbf{s})}{\\partial s_{j}\n\\left\\vert _{\\mathbf{s}=\\mathbf{H}_{n-m_{1}}(\\mathbf{0})}\\right. \\leq \\chi h.\n\\end{equation*\nHence, using the previous lemma and monotonicity of $Q_{r}^{(j,N)}$ in $r$\nwe ge\n\\begin{eqnarray}\n\\mathbf{P}(\\mathbf{Z}(m_{1},n)=\\mathbf{e}_{j}|\\mathbf{Z}(m_{0},n)=\\mathbf{e\n_{j}) &=&\\frac{Q_{n-m_{1}}^{(j,N)}}{Q_{n-m_{0}}^{(j,N)}}\\frac{\\partial\nH_{\\Delta _{1}}^{(j,N)}(\\mathbf{s})}{\\partial s_{j}}\\left\\vert _{\\mathbf{s}\n\\mathbf{H}_{n-m_{1}}(\\mathbf{0})}\\right.  \\label{ExplCoeff} \\\\\n&\\geq &\\frac{\\partial H_{\\Delta _{1}}^{(j,N)}(\\mathbf{s})}{\\partial s_{j}\n\\left\\vert _{\\mathbf{s}=\\mathbf{H}_{n-m_{1}}(\\mathbf{0})}\\right. \\geq 1-\\chi\nh.  \\notag\n\\end{eqnarray\nLemma \\ref{L_derivative} is proved. \\\n\n\\textbf{Proof} \\textbf{of Theorem \\ref{T_findim}}. Using (\\ref{BranProp2})\nand Theorem \\ref{T_familyMany} we see that\n\n1) if $m\\ll n^{\\gamma _{k}}$ the\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty }J_{m,n}^{\\,(k,N)}( \\mathbf{s}) =s_{k};\n\\label{LeftEnd}\n\\end{equation}\n\n2) if $m=(y+l_{n})n^{\\gamma _{k}}=(y+l_{n})n^{1\/2^{( N-k) }},\\,y\\in \\lbrack\n0,\\infty )$ then\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }J_{m,n}^{\\,(k,N)}( \\mathbf{s}) =X_{k}(\ny;s_{k},s_{k+1}) ;\n\\end{equation*}\n\n3) if $m=(x+l_{n})n,\\,x\\in \\left[ 0,1\\right] $ the\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty }J_{m,n}^{\\,(N,N)}(\\mathbf{s})=G\\left(\nx;s_{N}\\right) .  \\label{End1}\n\\end{equation}\n\n\\textbf{Proof of (\\ref{Lim_diskr})}. Consider first the case $p=2$ and take \n\\mathbf{m}=(0,m_{1},m_{2})$. By Lemma \\ref{L_convol\n\\begin{equation}\n\\hat{J}_{\\mathbf{m},n}^{\\,(1,N)}(\\mathbf{S}_{1},\\mathbf{S\n_{2})=J_{m_{1},n}^{\\,(1,N)}(\\mathbf{S}_{1}\\otimes \\mathbf{J}_{\\Delta\n_{2},n-m_{1}}(\\mathbf{S}_{2})).  \\label{JointDist}\n\\end{equation}\n\nIt follows from (\\ref{LimSIngLate}) that, given $n^{\\gamma _{i}}\\ll m_{1}\\ll\nn^{\\gamma _{i+1}}$\n\\begin{equation*}\nJ_{m_{1},n}^{\\,(1,N)}( \\mathbf{S}_{1}) \\rightarrow 1-( 1-s_{1,i+1})\n^{1\/2^{i}}\n\\end{equation*\nas $n\\rightarrow \\infty.$ Further, in view of $\\Delta _{2}=m_{2}-m_{1}\\ll\nn^{\\gamma _{i+1}}$ and~(\\ref{LeftEnd}) $J_{\\Delta _{2},n-m_{1}}^{\\,(i+1,N)}\n\\mathbf{S}_{2})\\rightarrow s_{2,i+1}$ as $n\\rightarrow \\infty.$ Hence, using\nthe continuity of the functions under consideration and (\\ref{JointDist}) we\nge\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }\\hat{J}_{\\mathbf{m},n}^{\\,(1,N)}(\\mathbf{S}_{1}\n\\mathbf{S}_{2})=1-( 1-s_{1,i+1}s_{2,i+1}) ^{1\/2^{i}}.\n\\end{equation*}\n\nThe validity of (\\ref{Lim_diskr}) for any $p>3$ may be checked by induction\nusing Lemma~\\ref{L_convol}.\n\n\\textbf{Proof of (\\ref{Lim_Multi})}. Consider again the case $p=2$ only. It\nfollows from (\\ref{LimTWo}) that, given $m_{l}\\sim Y_{l}n^{\\gamma\n_{i}},\\,l=1,2,$ with $Y_{1}=y_{1}$\n\\begin{equation*}\nJ_{m_{1},n}^{\\,(1,N)}(\\mathbf{s})\\rightarrow \\bar{X\n_{R_{i}}(y_{1};s_{i},s_{i+1})\n\\end{equation*\nas $n\\rightarrow \\infty $ an\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }J_{\\Delta _{2},n-m_{1}}^{\\,(i,N)}(\\mathbf{S\n_{2})=X_{i}(y_{2};s_{2i},s_{2,i+1}),\\quad \\lim_{n\\rightarrow \\infty\n}J_{\\Delta _{2},n-m_{1}}^{\\,(i+1,N)}(\\mathbf{S}_{2})=s_{2,i+1}.\n\\end{equation*\nHence, using the continuity of the functions involved and (\\ref{JointDist})\nwe ge\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }\\hat{J}_{\\mathbf{m},n}^{\\,(1,N)}(\\mathbf{S}_{1}\n\\mathbf{S}_{2})=\\bar{X}_{R_{i}}\\left(\ny_{1};s_{1i}X_{i}(y_{2};s_{2i},s_{2,i+1}),s_{1,i+1}s_{2,i+1}\\right)\n\\end{equation*\nproving (\\ref{Lim_Multi}) for $p=2$.\n\nTo justify (\\ref{Lim_Multi}) for $p>3$ it is necessary to use Lemma \\re\n{L_convol} and induction arguments. We omit the respective details. \\\n\n\\textbf{Proof} \\textbf{of Theorem \\ref{T_endpoint}}. We consider the case \np=2$ only and to this aim take $\\mathbf{m\n=(0,(x_{1}+l_{n})n,(x_{1}+x_{2}+l_{n})n)$. By (\\ref{JointDist}), (\\re\n{LimOLD}) and (\\ref{End1})\n\\begin{eqnarray*}\n\\lim_{n\\rightarrow \\infty }\\hat{J}_{\\mathbf{m},n}^{\\,(1,N)}(\\mathbf{S}_{1}\n\\mathbf{S}_{2}) &=&\\lim_{n\\rightarrow \\infty }J_{(x_{1}+l_{n})n,n}^{\\,(1,N)}\n\\mathbf{S}_{1}\\otimes \\mathbf{J}_{x_{2}n,n(1-x_{1}-l_{n})}(\\mathbf{S}_{2}))\n\\\\\n&=&\\bar{G}_{R_{N}}\\left( x_{1};s_{1N}G\\left( \\frac{x_{2}}{1-x_{1}\n;s_{2N}\\right) \\right) =\\bar{G}_{R_{N}}(\\mathbf{x}_{1,2};\\mathbf{S}_{1,2;N}).\n\\end{eqnarray*\nThe desired statement for $p>2$ follows by induction. \\\n\n\\textbf{Proof} \\textbf{of point 1) of Theorem \\ref{T_Skhod1}}. Let \n0=t_{0}<t_{1}<...<t_{p}<1$. If $\\gamma _{i-1}\\leq t_{1}<t_{p}<\\gamma _{i}$\nfor some $i\\in \\left\\{ 1,2,...,N\\right\\} $ then the needed convergence of\nfinite-dimensional distributions follows from (\\ref{Lim_diskr}). We now\nconsider another extreme case, namely, take a tuple \n0=t_{0}<t_{1}<...<t_{N}<1$ such that $\\gamma _{i-1}\\leq t_{i}<\\gamma _{i}$\nfor all $i=1,2,...,N$. Then for $m_{i}\\sim n^{t_{i}}g_{n}(t_{i})$ we have\n\\begin{equation*}\nn^{\\gamma _{i-1}}\\ll m_{i}\\ll n^{\\gamma _{i}},\\quad \\Delta\n_{i}=m_{i}-m_{i-1}\\sim m_{i},\\quad n-m_{i}\\sim n.\n\\end{equation*\nThese relations, (\\ref{LimSIngLate}), (\\ref{LeftEnd}), and the continuity of\nthe respective probability generating functions imply (recall (\\ref{DefOmega\n)\n\\begin{eqnarray*}\n&&\\lim_{n\\rightarrow \\infty }\\hat{J}_{\\mathbf{m},n}^{\\,(1,N)}(\\mathbf{S\n_{1},...,\\mathbf{S}_{N}) \\\\\n&&\\quad =\\lim_{n\\rightarrow \\infty }J_{m_{1},n}^{\\,(1,N)}(\\mathbf{S\n_{1}\\otimes \\mathbf{J}_{m_{2},n}(\\mathbf{S}_{2}\\otimes ...(\\mathbf{S\n_{N-1}\\otimes \\mathbf{J}_{m_{N},n}(\\mathbf{S}_{N}))...)) \\\\\n&&\\quad =s_{11}\\left( 1-\\sqrt{1-\\lim_{n\\rightarrow \\infty\n}J_{m_{2},n}^{\\,(2,N)}(\\mathbf{S}_{2}\\otimes ...(\\mathbf{S}_{N-1}\\otimes\n\\mathbf{J}_{m_{N},n}(\\mathbf{S}_{N}))...)}\\,\\right) \\\\\n&&\\quad =s_{11}\\left( 1-\\sqrt{1-s_{22}\\left( 1-\\sqrt{1-\\Omega\n_{N-2}(s_{33},...,s_{NN})}\\,\\right) }\\,\\,\\right) \\\\\n&&\\quad =...=\\Omega _{N}(s_{11},s_{22},...,s_{NN})\n\\end{eqnarray*\nas required.\n\nThe case when several values among $t_{j}$ are contained in a subinterval \n[\\gamma _{i-1},\\gamma _{i})$ may be considered by combining the previous\narguments. We omit the respective details. \\\n\n\\section{Tightness\\label{Sec6}}\n\nDenote by $\\mathbf{z}^{(i,i+1)},1\\leq i\\leq N-1,$ the $(N-2)$-dimensional\nvector obtained from $\\mathbf{z}=(z_{1},...,z_{N})\\in \\mathbb{Z}_{+}^{N}$ by\ndeleting the coordinates $i$ and $i+1$ and by $\\mathbf{z}^{(i)},1\\leq i\\leq\nN-1,$ the $(N-1)$-dimensional vector obtained from $\\mathbf{z}$ by deleting\nthe $i$-th coordinate. Let $\\left\\Vert \\mathbf{x}\\right\\Vert $ be the sum of\nabsolute values of all coordinates of the vector $\\mathbf{x}$.\n\nSet $\\mathcal{C}_{i}=\\left\\{ \\mathbf{z}\\in \\mathbb{Z}_{+}^{N}:\\left\\Vert\n\\mathbf{z}^{(i)}\\right\\Vert >0\\right\\} ,\\,\\mathcal{B}_{i}=\\mathbb{Z\n_{+}^{N}\\backslash \\mathcal{C}_{i}$ and\n\\begin{equation*}\n\\mathcal{C}_{i,i+1}=\\left\\{ \\mathbf{z}\\in \\mathbb{Z}_{+}^{N}:\\left\\Vert\n\\mathbf{z}^{(i,i+1)}\\right\\Vert >0\\right\\} .\n\\end{equation*}\n\nPut \\b{Z}$_{i}(m)=Z_{1}(m)+...+Z_{i}(m)$ and denot\n\\begin{equation*}\n\\text{\\b{Z}}_{i}(m,n)=\\sum_{k=1}^{i}Z_{k}(m,n),\\quad \\bar{Z\n_{i}(m,n)=\\sum_{k=i}^{N}Z_{k}(m,n).\n\\end{equation*}\n\nIn what follows it will be convenient to write $\\mathbf{P}_{n}(\\mathcal{B})$\nfor $\\mathbf{P}(\\mathcal{B}|\\mathbf{Z}(n)\\neq \\mathbf{0},\\mathbf{Z}(0)\n\\mathbf{e}_{1})$ for any admissible event~$\\mathcal{B}$.\n\nWe start checking the desired tightness of the prelimiting processes in\nTheorems \\ref{T_SkorohConst} and~\\ref{T_Skhod1} by proving two important\nlemmas.\n\nLet $A_{i}(n) =\\left\\{ m:n^{\\gamma _{i}}g_{n}(\\gamma _{i})\\leq m<n^{\\gamma\n_{i+1}-\\varepsilon }g_{n}(\\gamma _{i+1}-\\varepsilon\n)\\right\\},\\,\\varepsilon>0.$\n\n\\begin{lemma}\n\\label{L_Negli} For any $i=0,1,2,...,N-1$ and $\\varepsilon\\in (0,\\gamma_1)$\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }\\mathbf{P}_{n}( \\exists m\\in A_{i}( n) :\\mathbf{Z\n( m,n) \\in \\mathcal{C}_{i+1}) =0.\n\\end{equation*}\n\\end{lemma}\n\n\\textbf{Proof.} If $m\\in A_{i}(n)$ then $\\bar{Z}_{i+2}(m,n)\\leq \\bar{Z\n_{i+2}(n^{\\gamma _{i+1}-\\varepsilon }g_{n}(\\gamma _{i+1}-\\varepsilon ),n)$\nand\n\\begin{equation*}\n\\left\\{ \\text{\\b{Z}}_{i}(m,n)>0\\right\\} \\Rightarrow \\left\\{ \\text{\\b{Z}\n_{i}(m)>0\\right\\} \\Rightarrow \\left\\{ \\text{\\b{Z}}_{i}(n^{\\gamma\n_{i}}g_{n}(\\gamma _{i}))>0\\right\\} .\n\\end{equation*\nThus,\n\\begin{eqnarray*}\n\\mathbf{P}_{n}(\\exists m\\in A_{i}(n):\\mathbf{Z}(m,n)\\in \\mathcal{C}_{i+1})\n&\\leq &\\mathbf{P}_{n}(\\bar{Z}_{i+2}(n^{\\gamma _{i+1}-\\varepsilon\n}g_{n}(\\gamma _{i+1}-\\varepsilon ),n)>0) \\\\\n&&+\\mathbf{P}_{n}\\left( \\text{\\b{Z}}_{i}(n^{\\gamma _{i}}g_{n}(\\gamma\n_{i}))>0\\right) .\n\\end{eqnarray*\nLetting $n$ tend to infinity we see that the first summand at the right-hand\nside of the inequality vanishes by (\\ref{LimSIngLate}), while the second one\nis zero for $i=0$ and tends to zero for $1\\leq i\\leq N-1$ in view of\n\\begin{eqnarray*}\n\\mathbf{P}_{n}\\left( \\text{\\b{Z}}_{i-1}(n^{\\gamma _{i}}g_{n}(\\gamma\n_{i}))>0\\right) &=&\\frac{\\mathbf{P}(T_{i}>n^{\\gamma _{i}}g_{n}(\\gamma _{i})\n}{\\mathbf{P}(T_{N}>n)} \\\\\n&\\sim &\\frac{c_{1i}}{c_{1N}}\\frac{n^{1\/2^{N-1}}}{(n^{1\/2^{N-i}}g_{n}(\\gamma\n_{i}))^{1\/2^{i-1}}}=\\frac{c_{1i}}{c_{1N}}\\frac{1}{(g_{n}(\\gamma\n_{i}))^{1\/2^{i-1}}}.\n\\end{eqnarray*\nThe lemma is proved.\\\n\n\\begin{lemma}\n\\label{L_Skor1}If $N\\geq 3$ then for any $i=1,2,...,N-1\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }\\mathbf{P}_{n}( \\exists m\\in \\lbrack n^{3\\gamma\n_{i-1}},n^{3\\gamma _{i}}]:\\mathbf{Z}( m,n) \\in \\mathcal{C}_{i,i+1}) =0.\n\\end{equation*}\n\\end{lemma}\n\n\\textbf{Proof.} By the same arguments as in Lemma \\ref{L_Negli}, we conclude\n\\begin{eqnarray*}\n&&\\mathbf{P}_{n}(\\exists m\\in \\lbrack n^{3\\gamma _{i-1}},n^{3\\gamma _{i}}]\n\\mathbf{Z}(m,n)\\in \\mathcal{C}_{i,i+1}) \\\\\n&&\\qquad \\leq \\mathbf{P}_{n}(\\bar{Z}_{i+2}(n^{3\\gamma _{i}},n)>0)+\\mathbf{P\n_{n}(\\text{\\b{Z}}_{i-1}(n^{3\\gamma _{i-1}})>0).\n\\end{eqnarray*\nAccording to point 3) of Theorem \\ref{T_familyMany} the first summand tends\nto zero as $n\\rightarrow \\infty $ while the second is, by definition zero\nfor $i=1$ and is evaluated a\n\\begin{equation*}\n\\frac{\\mathbf{P}(T_{i-1}>n^{3\\gamma _{i-1}})}{\\mathbf{P}(T_{N}>n)}\\sim \\frac\nc_{1,i-1}}{c_{1N}}\\frac{n^{1\/2^{N-1}}}{(n^{3\/2^{N-i+1}})^{1\/2^{i-2}}}\\sim\n\\frac{c_{1,i-1}}{c_{1N}}\\frac{1}{n^{1\/2^{N-2}}}\n\\end{equation*\nfor $i\\geq 2$. \\ This completes the proof of the lemma.\n\n\\subsection{Macroscopic view\\label{Sec51}}\n\nIn this section we prove Theorem \\ref{T_SkorohConst}\\textbf{\\ }which\ndescribes the macroscopic structure of the family tree. Convergence of the\nfinite-dimensional distributions of $\\left\\{ \\mathbf{Z}(n^{t}g_{n}(t),n),\n\\leq t<1\\right\\} $ to the respective finite-dimensional distributions of \n\\left\\{ \\mathbf{R}(t),0\\leq t<1\\right\\} $ has been established in (\\re\n{Lim_diskr}). Thus, we concentrate on proving the tightness.\n\nSince $\\mathbf{Z}(n^{t}g_{n}(t),n)$ has integer-valued components we need to\ncheck\\ for each interval $A_{i}=\\left[ \\gamma _{i},\\gamma _{i+1}-\\varepsilon\n\\right] ,i=0,1,...,N-1,$ that (see \\cite{Bil68}, Theorem 15.3)\n\n1) for any positive $\\eta $ there exists $L$ such that\n\\begin{equation}\n\\mathbf{P}_{n}\\left( \\sup_{t\\in A_{i}}\\left\\Vert \\mathbf{Z\n(n^{t}g_{n}(t),n)\\right\\Vert >L\\right) \\leq \\eta ,~n\\geq 1;  \\label{Bi0}\n\\end{equation}\n\n2) for any positive $\\eta $ there exist $\\delta >0$ and $n_{0}$ such that,\nfor all $n\\geq n_{0}\n\\begin{equation}\n\\mathbf{P}_{n}\\left( \\max \\left( \\min_{k=1,2}\\left\\Vert \\mathbf{Z\n(n^{t}g_{n}(t),n)-\\mathbf{Z}(n^{t_{k}}g_{n}(t_{k}),n)\\right\\Vert \\right)\n\\neq 0\\right) \\leq \\eta ,  \\label{Bi01}\n\\end{equation\nwhere the $\\max $ is taken over all $\\gamma _{i}\\leq t_{1}\\leq t\\leq\nt_{2}\\leq \\gamma _{i+1}-\\varepsilon $ such that $t_{2}-t_{1}\\leq \\delta ;$\n\n\\begin{equation}\n\\mathbf{P}_{n}(\\exists t,s\\in \\left[ \\gamma _{i},\\gamma _{i}+\\delta \\right] \n\\mathbf{Z}(n^{t}g_{n}(t),n)\\neq \\mathbf{Z}(n^{s}g_{n}(s),n)\\,)\\leq \\eta ,\n\\label{Bi02}\n\\end{equation\nan\n\\begin{equation}\n\\mathbf{P}_{n}(\\exists t,s\\in \\lbrack \\gamma _{i+1}-\\delta -\\varepsilon\n,\\gamma _{i+1}-\\varepsilon ]:\\mathbf{Z}(n^{t}g_{n}(t),n)\\neq \\mathbf{Z\n(n^{s}g_{n}(s),n)\\,\\,)\\leq \\eta .  \\label{Bi03}\n\\end{equation}\n\nThe fact that the random variable $\\left\\Vert \\mathbf{Z}( n^{t}g_{n}(t),n)\n\\right\\Vert $ is monotone in $t$ for fixed $n$ essentially simplifies the\nproof.\n\nIndeed, in this case\n\\begin{equation*}\n\\mathbf{P}_{n}\\left( \\sup_{t\\in A_{i}}\\left\\Vert \\mathbf{Z\n(n^{t}g_{n}(t),n)\\right\\Vert >L\\right) \\leq \\mathbf{P}_{n}(\\left\\Vert\n\\mathbf{Z}(n^{1-\\varepsilon }g_{n}(1-\\varepsilon ),n)\\right\\Vert >L\\,)\n\\end{equation*\nand (\\ref{Bi0}) follows from the one-dimensional convergence established in \n\\ref{LimSIngLate}) for $i=N-1$.\n\nTo prove (\\ref{Bi01})-(\\ref{Bi03}) we introduce the event\n\\begin{eqnarray*}\n\\mathcal{D}_{i} &=&\\left\\{ \\forall t\\in A_{i}:\\mathbf{Z}(n^{t}g_{n}(t),n)\\in\n\\mathcal{B}_{i+1}\\right\\} , \\\\\n\\mathcal{F}_{i}(a,b) &=&\\left\\{ \\exists t,s\\in \\lbrack\na,b]:Z_{i+1}(n^{t}g_{n}(t),n)\\neq Z_{i+1}(n^{s}g_{n}(s),n)\\right\\} ,\n\\end{eqnarray*\ntake a sufficiently small $\\delta >0$ and observe that if $\\left[ a,b\\right]\n\\subset \\left[ \\gamma _{i},\\gamma _{i+1}-\\varepsilon \\right] $ then\n\\begin{eqnarray*}\n&&\\mathbf{P}_{n}(\\exists t,s\\in \\lbrack a,b]:\\mathbf{Z}(n^{t}g_{n}(t),n)\\neq\n\\mathbf{Z}(n^{s}g_{n}(s),n)\\,) \\\\\n&&\\qquad \\leq \\mathbf{P}_{n}(\\exists t\\in A_{i}:\\mathbf{Z\n(n^{t}g_{n}(t),n)\\in \\mathcal{C}_{i+1})+\\mathbf{P}_{n}(\\mathcal{D}_{i}\\cap\n\\mathcal{F}_{i}(\\gamma _{i},\\gamma _{i+1}-\\varepsilon )).\n\\end{eqnarray*\nBy Lemma \\ref{L_Negli} the first term at the right-hand side tends to zero\nas $n\\rightarrow \\infty $.\n\nFurther, for $i\\geq 1$\n\\begin{equation*}\n\\mathbf{P}_{n}( \\mathcal{D}_{i}\\cap \\mathcal{F}_{i}( \\gamma _{i},\\gamma\n_{i+1}-\\varepsilon ) ) \\leq \\mathbf{P}_{n}( Z_{i+1}( n^{\\gamma _{i}}g_{n},n)\n\\neq Z_{i+1}( n^{\\gamma _{i+1}-\\varepsilon }g_{n},n) ) \\rightarrow 0\n\\end{equation*\nby (\\ref{Lim_diskr}). This justifies (\\ref{Bi02})-(\\ref{Bi03}).\n\nTo check the validity of (\\ref{Bi01}) it remains to note that\n\n\\begin{eqnarray*}\n&&\\mathbf{P}_{n}\\left( \\max \\left( \\min_{k=1,2}\\left\\Vert \\mathbf{Z\n(n^{t}g_{n}(t),n)-\\mathbf{Z}(n^{t_{k}}g_{n}(t_{k}),n)\\right\\Vert \\right)\n\\neq 0\\right) \\\\\n&&\\qquad \\leq \\mathbf{P}_{n}(\\exists t,s\\in \\lbrack \\gamma _{i},\\gamma\n_{i+1}-\\varepsilon ]:\\mathbf{Z}(n^{t}g_{n}(t),n)\\neq \\mathbf{Z\n(n^{s}g_{n}(s),n)\\,)\n\\end{eqnarray*\nand to use the same arguments as before.\n\nTheorem \\ref{T_SkorohConst} is proved.\n\n\\subsection{Microscopic view\\label{Sec52}}\n\nWe follow in this section the ideas of paper \\cite{FZ} and to this aim\nformulate a particular and slightly modified case of Theorem 6.5.4 in \\cit\n{GS71} giving a convergence criterion in Skorokhod topology for a class of\nMarkov processes.\n\nLet $\\mathbf{K}_{n}(y),n=1,2,...$ be a sequence of Markov processes with\nvalues in $\\mathbb{Z}_{+}^{N}$ whose trajectories belong with probability 1\nto the space $D_{[a,b]}(\\mathbb{Z}_{+}^{N})$ of cadlag functions on $[a,b]$.\n\n\\begin{theorem}\n\\label{T_skoroh}If the finite-dimensional distributions of $\\left\\{ \\mathbf{\n}_{n}(y),a\\leq y\\leq b\\right\\} $ converge, as $n\\rightarrow \\infty ,$ to the\nrespective finite-dimensional distributions of a process $\\left\\{ \\mathbf{K\n(y),a\\leq y\\leq b\\right\\} $ and there exists a partition $\\mathbb{Z}_{+}^{N}\n\\mathcal{B}\\cup \\mathcal{C},\\mathcal{B}\\cap \\mathcal{C}=\\varnothing $ such\nthat\n\\begin{equation*}\n\\lim_{h\\downarrow 0}\\overline{\\lim_{n\\rightarrow \\infty }}\\sup_{0\\leq\ns-y\\leq h}\\sup_{\\mathbf{z}\\in \\mathcal{B}}\\mathbf{P}(\\mathbf{K}_{n}(s)\\neq\n\\mathbf{K}_{n}(y)|\\mathbf{K}_{n}(y)=\\mathbf{z})=0,\n\\end{equation*\nand\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }\\mathbf{P}(\\exists y\\in \\lbrack a,b]:\\mathbf{K\n_{n}(y)\\in \\mathcal{C})=0\n\\end{equation*\nthen, as $n\\rightarrow \\infty $\n\\begin{equation*}\n\\mathcal{L}\\left\\{ \\mathbf{K}_{n}(y),a\\leq y\\leq b\\right\\} \\Longrightarrow\n\\mathcal{L}\\left\\{ \\mathbf{K}(y),a\\leq y\\leq b\\right\\} .\n\\end{equation*}\n\\end{theorem}\n\nIn view of Lemma \\ref{L_convol} the law\\textbf{\\ }$\\mathbf{P}_{n}(\\left\\{\n\\mathbf{Z}(m,n),0\\leq m\\leq n\\right\\} \\in (\\cdot )|\\mathbf{Z}(n)\\neq \\mathbf\n0})$ specifies, for each fixed $n$ an inhomogeneous Markov branching\nprocess. We denote its transition probabilities by $\\mathbf{P}_{n}(m_{1}\n\\mathbf{z};m_{2},(\\cdot ))$.\n\nProving the tightness of $\\mathbf{U}_{i}\\left( \\cdot \\right) ,$ \ni=1,2,...,N, $ we need to construct an appropriate partition of $\\mathbb{Z\n_{+}^{N}$ and to use Theorem \\ref{T_skoroh} for each $\\left[ 0,b\\right]\n\\subset \\lbrack 0,\\infty )$.\n\nObserve that if $\\mathbf{w}=( w_{1},...,w_{N}) \\leq \\mathbf{z}=(\nz_{1},...,z_{N}) $ (where the inequality is understood componentwise) the\n\\begin{equation*}\n\\mathbf{P}_{n}( m_{0},\\mathbf{w};m_{1},\\left\\{ \\mathbf{w}\\right\\} ) \\geq\n\\mathbf{P}_{n}( m_{0},\\mathbf{z};m_{1},\\left\\{ \\mathbf{z}\\right\\} ) .\n\\end{equation*}\n\nLet $\\ \\mathcal{C}(k)=\\left\\{ \\mathbf{z}\\in \\mathbb{Z}_{+}^{N}:\\left\\Vert\n\\mathbf{z}\\right\\Vert \\leq k\\right\\} ,\n\\begin{equation*}\n\\mathcal{C}_{i}(k)=\\left\\{ \\mathbf{z}\\in \\mathbb{Z\n_{+}^{N}:z_{1}+...+z_{i-1}>0;\\left\\Vert \\mathbf{z}\\right\\Vert \\leq k\\right\\}\n,~\\mathcal{J}_{i}(k)=\\mathcal{C}(k)\\backslash \\mathcal{C}_{i}(k).\n\\end{equation*}\n\nFix $i\\in \\{1,...,N-1\\}$ and denote $m_{j}=( Y_{j}+l_{n}) n^{\\gamma\n_{i}},j=1,2$.\n\n\\begin{lemma}\n\\label{L_ceretain event.}Under Hypothesis A for any fixed $k$ and \n0<b<\\infty \n\\begin{equation*}\n\\lim_{h\\downarrow 0}\\overline{\\lim_{n\\rightarrow \\infty}}\\sup_{\\substack{\n0\\leq Y_{1}-Y_{0}\\leq h,  \\\\ Y_{1},Y_{0}\\in \\lbrack 0,b]}}\\sup_{\\mathbf{z\n\\in \\mathcal{J}_{i}(k)}\\mathbf{P}_{n}(\\mathbf{Z}(m_{1};n)\\neq \\mathbf{z}\n\\mathbf{Z}(m_{0};n)=\\mathbf{z})=0.\n\\end{equation*}\n\\end{lemma}\n\n\\textbf{Proof.} By the branching property, the decomposability assumption,\nand the positivity of the offspring number of each particle in the reduced\nprocess we have for all $m_{1}\\geq m_{0}$ and $\\mathbf{z}\\in \\mathcal{J\n_{i}(k)$\n\\begin{eqnarray*}\n\\mathbf{P}_{n}(m_{0},\\mathbf{z};m_{1},\\left\\{ \\mathbf{z}\\right\\} )\n&=&\\prod_{j=i}^{N}(\\mathbf{P}_{n}(m_{0},\\mathbf{e}_{j};m_{1},\\left\\{ \\mathbf\ne}_{j}\\right\\} ))^{z_{j}} \\\\\n&\\geq &\\prod_{j=i}^{N}(\\mathbf{P}_{n}(m_{0},\\mathbf{e}_{j};m_{1},\\left\\{\n\\mathbf{e}_{j}\\right\\} ))^{k}.\n\\end{eqnarray*\nUsing Lemma \\ref{L_derivative} we get for $m_{0}=(Y_{0}+l_{n})n^{\\gamma\n_{i}} $ and $m_{1}=(Y_{1}+l_{n})n^{\\gamma _{i}}$:\n\\begin{equation}\n\\inf_{\\substack{ 0\\leq Y_{1}-Y_{0}\\leq h,  \\\\ Y_{0},Y_{1}\\in \\left[ 0,\n\\right] }}\\mathbf{P}_{n}(m_{0},\\mathbf{z};m_{1},\\left\\{ \\mathbf{z}\\right\\}\n)\\geq (1-\\chi h)^{Nk}.  \\label{kapBel}\n\\end{equation\nThis implies the claim of the lemma. \\\n\n\\begin{lemma}\n\\label{L_ratio}If $m_{j}=(Y_{j}+l_{n})n^{\\gamma _{i}},~j=0,1,2,$ and $0\\leq\nY_{0}<Y_{1}<Y_{2}$ with $Y_{1}-Y_{0}\\leq h$, then for all $n\\geq n_{0}$\n\\begin{eqnarray*}\n&&\\mathbf{P}_{n}(\\mathbf{Z}(m_{1},n)=\\mathbf{z}|\\mathbf{Z}(m_{0},n)=\\mathbf{\n};\\left\\Vert \\mathbf{Z}(m_{2},n)\\right\\Vert \\leq k) \\\\\n&&\\qquad \\geq \\mathbf{P}_{n}(m_{0},\\mathbf{z};m_{1},\\left\\{ \\mathbf{z\n\\right\\} )\\frac{\\mathbf{P}_{n}(m_{1},\\mathbf{z};m_{2},\\mathcal{C}(k))}\n\\mathbf{P}_{n}(m_{1},\\mathbf{z};m_{2},\\mathcal{C}(k))\\mathbf{+}\\chi Nkh}.\n\\end{eqnarray*}\n\\end{lemma}\n\n\\textbf{Proof.} We hav\n\\begin{eqnarray*}\n&&\\mathbf{P}_{n}(\\mathbf{Z}(m_{1},n)=\\mathbf{z}|\\mathbf{Z}(m_{0},n)=\\mathbf{\n},\\left\\Vert \\mathbf{Z}(m_{2},n)\\right\\Vert \\leq k) \\\\\n&&\\qquad =\\mathbf{P}_{n}(m_{0},\\mathbf{z};m_{1},\\left\\{ \\mathbf{z}\\right\\} \n\\frac{\\mathbf{P}_{n}(m_{1},\\mathbf{z};m_{2},\\mathcal{C}(k))}{\\mathbf{P\n_{n}(m_{0},\\mathbf{z};m_{2},\\mathcal{C}(k))}.\n\\end{eqnarray*\nIn view of (\\ref{kapBel}\n\\begin{eqnarray*}\n\\mathbf{P}_{n}(m_{0},\\mathbf{z};m_{2},\\mathcal{C}(k)) &=&\\sum_{\\mathbf{z\n^{\\prime }}\\mathbf{P}_{n}(m_{0},\\mathbf{z};m_{1},\\left\\{ \\mathbf{z}^{\\prime\n}\\right\\} )\\mathbf{P}_{n}(m_{1},\\mathbf{z}^{\\prime };m_{2},\\mathcal{C}(k)) \\\\\n&\\leq &1-\\mathbf{P}_{n}(m_{0},\\mathbf{z};m_{1},\\left\\{ \\mathbf{z}\\right\\} )\n\\mathbf{P}_{n}(m_{1},\\mathbf{z};m_{2},\\mathcal{C}(k)) \\\\\n&\\leq &1-(1-\\chi h)^{Nk}+\\mathbf{P}_{n}(m_{1},\\mathbf{z};m_{2},\\mathcal{C\n(k)) \\\\\n&\\leq &\\chi Nkh+\\mathbf{P}_{n}(m_{1},\\mathbf{z};m_{2},\\mathcal{C}(k)).\n\\end{eqnarray*\nHence the needed statement follows.\n\n\\begin{lemma}\n\\label{L_Kskrokh}Under Hypothesis A for any fixed $k,$ $0<b<\\infty ,$ and \nm_{0}=( Y_{0}+l_{n}) n^{\\gamma _{i}},\\ m_{1}=( Y_{1}+l_{n}) n^{\\gamma\n_{i}},\\ m_{2}=2bn^{\\gamma _{i}}$ we hav\n\\begin{equation*}\n\\lim_{h\\downarrow 0}\\overline{\\lim_{n\\rightarrow \\infty }}\\sup_{\\substack{\n0\\leq Y_{1}-Y_{0}\\leq h  \\\\ Y_{1},Y_{0}\\in \\lbrack 0,b]}}\\sup_{\\mathbf{z}\\in\n\\mathcal{J}_{i}(k)}\\mathbf{P}_{n}( \\mathbf{Z}( m_{1};n) \\neq \\mathbf{z}\n\\mathbf{Z}( m_{0};n) =\\mathbf{z},\\left\\Vert \\mathbf{Z}( m_{2},n) \\right\\Vert\n\\leq k) =0.\n\\end{equation*}\n\\end{lemma}\n\n\\textbf{Proof.} By (\\ref{kapBel}) and Lemma \\ref{L_ratio} for \nm_{0}=(Y_{0}+l_{n})n^{\\gamma _{i}}$ and $m_{1}=(Y_{1}+l_{n})n^{\\gamma _{i}}$\n\\begin{eqnarray*}\n&&\\mathbf{P}_{n}(\\mathbf{Z}(m_{1},n)=\\mathbf{z}|\\mathbf{Z}(m_{0},n)=\\mathbf{\n};\\left\\Vert \\mathbf{Z}(m_{2},n)\\right\\Vert \\leq k) \\\\\n&&\\qquad \\geq (1-\\chi h)^{Nk}\\frac{\\mathbf{P}_{n}(m_{1},\\mathbf{z};m_{2}\n\\mathcal{C}(k))}{\\mathbf{P}_{n}(m_{1},\\mathbf{z};m_{2},\\mathcal{C}(k)\n\\mathbf{+}\\chi Nkh}.\n\\end{eqnarray*\nUsing the decomposability hypothesis and Lemma \\ref{L_derivative} we obtain\n\\begin{eqnarray*}\n&&\\mathbf{P}_{n}(m_{1},\\mathbf{z};m_{2},\\mathcal{C}(k))\\geq \\mathbf{P\n_{n}(m_{1},\\mathbf{z};m_{2},\\left\\{ \\mathbf{z}\\right\\} ) \\\\\n&&\\qquad =\\prod_{j=i}^{N}(\\mathbf{P}_{n}(m_{1},\\mathbf{e}_{j};m_{2},\\left\\{\n\\mathbf{e}_{j}\\right\\} ))^{z_{j}}\\geq \\prod_{j=i}^{N}\\mathbf{P\n_{n}^{k}(l_{n}n^{\\gamma _{i}},\\mathbf{e}_{j};2bn^{\\gamma _{i}},\\left\\{\n\\mathbf{e}_{j}\\right\\} ).\n\\end{eqnarray*\nIt follows from Theorem~\\ref{T_findim} that\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }\\prod_{j=i}^{N}\\mathbf{P}_{n}^{k}(l_{n}n^{\\gamma\n_{i}},\\mathbf{e}_{j};2bn^{\\gamma _{i}},\\left\\{ \\mathbf{e}_{j}\\right\\} )\n\\mathbf{P}^{k}(\\mathbf{U}_{i}(2b)=\\mathbf{e}_{i}|\\mathbf{U}_{i}(0)=\\mathbf{e\n_{i})=B>0.\n\\end{equation*\nHence we ge\n\\begin{eqnarray*}\n&&\\varliminf_{n\\rightarrow \\infty }\\inf_{\\substack{ 0\\leq Y_{1}-Y_{0}\\leq h\n\\\\ Y_{1},Y_{0}\\in \\lbrack 0,b]}}\\inf_{\\mathbf{z}\\in \\mathcal{J}_{i}(k)\n\\mathbf{P}_{n}(\\mathbf{Z}(m_{1};n)=\\mathbf{z}|\\mathbf{Z}(m_{0};n)=\\mathbf{z\n,\\left\\Vert \\mathbf{Z}(m_{2},n)\\right\\Vert \\leq k) \\\\\n&&\\qquad \\qquad \\geq (1-\\chi h)^{Nk}\\frac{B}{B\\mathbf{+}\\chi Nkh}.\n\\end{eqnarray*\nLetting $h\\downarrow 0$ completes the proof of the lemma. \\\n\n\\begin{corollary}\n\\label{C_skk}Under the conditions of Lemma \\ref{L_Kskrokh}\n\\begin{eqnarray*}\n&&\\mathcal{L}\\left\\{ \\mathbf{Z}((y+l_{n})n^{1\/2^{N-i}},n),0\\leq y\\leq b\\\n\\Big|\\,\\left\\Vert \\mathbf{Z}(m_{2},n)\\right\\Vert \\leq k,\\mathbf{Z}(n)\\neq\n\\mathbf{0}\\right\\} \\\\\n&&\\qquad\\qquad\\qquad\\qquad\\qquad\\quad\\,\\Longrightarrow \\mathcal{L\n_{R_{i}}\\left\\{ \\mathbf{U}_{i}(y),0\\leq y\\leq b|\\,\\left\\Vert \\mathbf{U\n_{i}(2b)\\right\\Vert \\leq k\\right\\} .\n\\end{eqnarray*}\n\\end{corollary}\n\n\\textbf{Proof.} Convergence of finite-dimensional distributions follows from\nthe respective results for the convergence of the processes established in\npoint 1) of Theorem \\ref{T_Skhod1}. Tightness follows from Lemma \\re\n{L_Kskrokh} and Theorem \\ref{T_skoroh} by taking $\\mathcal{B}=\\mathcal{J\n_{i}(k)$ and $\\mathcal{C=C}_{i}(k)$ and observing that\n\\begin{eqnarray*}\n&&\\lim_{n\\rightarrow \\infty }\\mathbf{P}_{n}(\\mathbf{Z}(l_{n}n^{\\gamma\n_{i}},n)\\in \\mathcal{C}_{i}(k)|\\left\\Vert \\mathbf{Z}(m_{2},n)\\right\\Vert\n\\leq k) \\\\\n&&\\qquad \\leq \\lim_{n\\rightarrow \\infty }\\mathbf{P}_{n}\\left( \\text{\\b{Z}\n_{i-1}(l_{n}n^{\\gamma _{i}})>0|\\left\\Vert \\mathbf{Z}(m_{2},n)\\right\\Vert\n\\leq k\\right) =0.\n\\end{eqnarray*}\n\n\\textbf{Proof} \\textbf{of Theorem \\ref{T_Skhod1}.} Let for $c>b$\n\\begin{eqnarray*}\n\\mathbf{P}_{n,i}(b;(\\cdot )) &=&\\mathbf{P}_{n}(\\left\\{ \\mathbf{Z\n((y+l_{n})n^{\\gamma _{i}},n),0\\leq y\\leq b\\right\\} \\in (\\cdot )), \\\\\n\\mathbf{P}_{n,i}^{(k)}(b,c;(\\cdot )) &=&\\mathbf{P}_{n}(\\left\\{ \\mathbf{Z\n((y+l_{n})n^{\\gamma _{i}},n),0\\leq y\\leq b\\right\\} \\in (\\cdot )|\\left\\Vert\n\\mathbf{Z}(cn^{\\gamma _{i}},n)\\right\\Vert \\leq k), \\\\\n\\mathbf{\\bar{P}}_{n,i}^{(k)}(b,c;(\\cdot )) &=&\\mathbf{P}_{n}(\\left\\{ \\mathbf\nZ}((y+l_{n})n^{\\gamma _{i}},n),0\\leq y\\leq b\\right\\} \\in (\\cdot )|\\left\\Vert\n\\mathbf{Z}(cn^{\\gamma _{i}},n)\\right\\Vert >k)\n\\end{eqnarray*\nan\n\\begin{eqnarray*}\n\\mathcal{P}_{i}(b;(\\cdot )) &=&\\mathbf{P}_{R_{i}}(\\left\\{ \\mathbf{U\n_{i}(y),0\\leq y\\leq b\\right\\} \\in (\\cdot )), \\\\\n\\mathcal{P}_{i}^{(k)}(b,c;(\\cdot )) &=&\\mathbf{P}_{R_{i}}(\\left\\{ \\mathbf{U\n_{i}(y),0\\leq y\\leq b\\right\\} \\in (\\cdot )|\\left\\Vert \\mathbf{U\n_{i}(c)\\right\\Vert \\leq k).\n\\end{eqnarray*\nThen for $0<b<\\infty $ and a continuous real function $\\psi $ on $D_{[0,b]}\n\\mathbb{Z}_{+}^{N})$ such that $\\left\\vert \\psi \\right\\vert \\leq q$ for a\npositive $q$ we have\n\\begin{eqnarray*}\n\\int \\psi (x)\\mathbf{P}_{n,i}(b;dx) &=&\\mathbf{P}_{n}(\\left\\Vert \\mathbf{Z\n(2bn^{\\gamma _{i}},n)\\right\\Vert >k)\\int \\psi (x)\\mathbf{\\bar{P}\n_{n,i}^{(k)}(b,2b;dx) \\\\\n&&+\\mathbf{P}_{n}(\\left\\Vert \\mathbf{Z}(2bn^{\\gamma _{i}},n)\\right\\Vert \\leq\nk)\\int \\psi (x)\\mathbf{P}_{n,i}^{(k)}(b,2b;dx).\n\\end{eqnarray*\nFor the first summand we ge\n\\begin{eqnarray*}\n&&\\lim \\sup_{n\\rightarrow \\infty }\\mathbf{P}_{n}(\\left\\Vert \\mathbf{Z\n(2bn^{\\gamma _{i}},n)\\right\\Vert >k)\\int \\psi (x)\\mathbf{\\bar{P}\n_{n,i}^{(k)}(b,2b;dx) \\\\\n&&\\quad \\leq q\\lim \\sup_{n\\rightarrow \\infty }\\mathbf{P}_{n}(\\left\\Vert\n\\mathbf{Z}(2bn^{\\gamma _{i}},n)\\right\\Vert >k)=q\\mathbf{P\n_{R_{i}}(\\left\\Vert \\mathbf{U}_{i}(2b)\\right\\Vert >k)=o(1)\n\\end{eqnarray*\nas $k\\rightarrow \\infty $ by the properties of $\\mathbf{U}_{i}(\\cdot )$. \\\n\nOn the other hand, letting first $n\\rightarrow \\infty $ and than \nk\\rightarrow \\infty $ we obtai\n\\begin{eqnarray*}\n&&\\lim_{k\\rightarrow \\infty }\\lim_{n\\rightarrow \\infty }\\mathbf{P\n_{n}(\\left\\Vert \\mathbf{Z}(2bn^{\\gamma _{i}},n)\\right\\Vert \\leq k)\\int \\psi\n(x)\\mathbf{P}_{n,i}^{(k)}(b,2b;dx) \\\\\n&&\\quad =\\lim_{k\\rightarrow \\infty }\\mathbf{P}_{R_{i}}(0<\\left\\Vert \\mathbf{\n}_{i}(2b)\\right\\Vert \\leq k)\\int \\psi (x)\\mathcal{P}_{i}^{(k)}(b,2b;dx) \\\\\n&&\\quad =\\lim_{k\\rightarrow \\infty }\\int_{\\left\\{ 0<\\left\\Vert \\mathbf{U\n_{i}(2b)\\right\\Vert \\leq k\\right\\} }\\psi (x)\\mathcal{P}_{i}(b,2b;dx)=\\int\n\\psi (x)\\mathcal{P}_{i}(b;dx).\n\\end{eqnarray*\nThus\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }\\int \\psi (x)\\mathbf{P}_{n}(\\mathbf{Z}((\\cdot\n+l_{n})n^{\\gamma _{i}},n)\\in dx)=\\int \\psi (x)\\mathcal{P}_{i}(b;dx)\n\\end{equation*\nfor any bounded continuous function on $D_{[0,b]}(\\mathbb{Z}_{+}^{N})$\nproving point 1) of Theorem~\\ref{T_Skhod1}.\n\nThe proof of point 2) of Theorem \\ref{T_Skhod1} needs only a few changes in\ncomparison with the proof of the respective theorem in \\cite{FZ} and we omit\nit. \\\n\n\\section{Proofs of Theorems \\protect\\ref{T_mrcaMany} and \\protect\\ref{T_type}\n\\label{Sec7}}\n\n\\textbf{Proof} \\textbf{of Theorem \\ref{T_mrcaMany}.} Our arguments are based\non the following simple observation\n\\begin{equation*}\n\\left\\{ \\bar{Z}_{1}(m,n)=1\\right\\} \\Leftrightarrow \\left\\{ \\beta _{n}\\geq\nm\\right\\} .\n\\end{equation*}\n\n\\textbf{Proof of 1).} According to (\\ref{LimSing}) for $m\\ll n^{\\gamma _{1}}$\n\\begin{eqnarray*}\n&&\\lim_{n\\rightarrow \\infty }\\mathbf{P}_{n}(\\bar{Z}_{1}(m,n)=1)=\\lim_{\n\\rightarrow \\infty }\\mathbf{P}_{n}(Z_{1}(m,n)=1) \\\\\n&&\\quad +\\lim_{n\\rightarrow \\infty }\\mathbf{P}_{n}(\\bar{Z}_{2}(m,n)=1)=1+0=1.\n\\end{eqnarray*}\n\n\\textbf{Proof of 2).} Observe that by point 2) of Theorem \\ref{T_familyMany\n\\begin{eqnarray*}\n&&\\lim_{n\\rightarrow \\infty }\\mathbf{P}_{n}(\\beta _{n}\\geq yn^{\\gamma\n_{i}})=\\lim_{n\\rightarrow \\infty }\\mathbf{P}_{n}(\\bar{Z}_{1}(yn^{\\gamma\n_{i}},n)=1) \\\\\n&&\\quad =\\lim_{n\\rightarrow \\infty }\\mathbf{P}_{n}(Z_{i}(yn^{\\gamma\n_{i}},n)+Z_{i+1}(yn^{\\gamma _{i}},n)=1) \\\\\n&&\\quad =\\lim_{n\\rightarrow \\infty }\\mathbf{P}_{n}(Z_{i}(yn^{\\gamma\n_{i}},n)=1)+\\lim_{n\\rightarrow \\infty }\\mathbf{P}_{n}(Z_{i+1}(yn^{\\gamma\n_{i}},n)=1).\n\\end{eqnarray*\nDirect calculations show tha\n\\begin{equation*}\n-\\frac{\\partial \\varphi _{i}(y;s_{i},s_{i+1})}{\\partial s_{i}}\\left\\vert\n_{s_{i}=s_{i+1}=0}\\right. =\\frac{1-\\tanh (yb_{i}c_{iN})}{1+\\tanh\n(yb_{i}c_{iN})}=e^{-2yb_{i}c_{iN}}\n\\end{equation*\nan\n\\begin{equation*}\n-\\frac{\\partial \\varphi _{i}(y;s_{i},s_{i+1})}{\\partial s_{i+1}}\\left\\vert\n_{s_{i}=s_{i+1}=0}\\right. =\\frac{\\tanh (yb_{i}c_{iN})}{1+\\tanh (yb_{i}c_{iN}\n}=\\frac{1}{2}-\\frac{1}{2}\\,e^{-2yb_{i}c_{iN}}.\n\\end{equation*\nThus,\n\\begin{eqnarray*}\n&&\\lim_{n\\rightarrow \\infty }\\mathbf{P}_{n}(Z_{i}(yn^{\\gamma\n_{i}},n)=1;\\beta _{n}\\geq yn^{\\gamma _{i}}) \\\\\n&&\\qquad \\qquad =-\\frac{\\partial (\\varphi _{i}(y;s_{i},s_{i+1}))^{1\/2^{i-1}\n}{\\partial s_{i}}\\left\\vert _{s_{i}=s_{i+1}=0}\\right. =\\frac{1}{2^{i-1}\n\\,e^{-2yb_{i}c_{iN}}\n\\end{eqnarray*\nan\n\\begin{eqnarray*}\n&&\\lim_{n\\rightarrow \\infty }\\mathbf{P}_{n}(Z_{i+1}(yn^{\\gamma\n_{i}},n)=1;\\beta _{n}\\geq yn^{\\gamma _{i}}) \\\\\n&&\\qquad \\quad =-\\frac{\\partial (\\varphi _{i}(y;s_{i},s_{i+1}))^{1\/2^{i-1}}}\n\\partial s_{i+1}}\\left\\vert _{s_{i}=s_{i+1}=0}\\right. =\\frac{1}{2^{i}\n(1-e^{-2yb_{i}c_{iN}}).\n\\end{eqnarray*\nCombining the previous estimates yields\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }\\mathbf{P}_{n}(\\beta _{n}\\leq yn^{\\gamma _{i}})=1\n\\frac{1}{2^{i}}-\\frac{1}{2^{i}}e^{-2yb_{i}c_{iN}}.\n\\end{equation*}\n\n\\textbf{Proof of 3).} This is evident.\n\n\\textbf{Proof of 4).} The needed statement follows from the equalit\n\\begin{equation*}\n-\\frac{\\partial }{\\partial s_{N}}\\left( \\frac{1}{x+(1-x)\/(1-s_{N})}\\right)\n^{2^{-(N-1)}}\\left\\vert _{s_{N}=0}\\right. =\\frac{1}{2^{N-1}}(1-x).\n\\end{equation*}\n\n\\textbf{Proof} \\textbf{of Theorem \\ref{T_type}.} Consider the case $N\\geq 4$\nand $i\\in \\left\\{ 2,3,..,N-2\\right\\} $ only. For $N=2,3$ or $N\\geq 4$ and \ni\\in \\left\\{ 1,N-1\\right\\} $ some of the random variables (events) below do\nnot exist (are empty) and the needed arguments become shorter.\n\nSince the total number of particles of all types in the reduced process does\nnot decrease with time, $\\mathbf{P}_{n}(\\beta _{n}<m)=\\mathbf{P}_{n}\\left(\n\\text{\\b{Z}}_{1}(m,n)\\geq 2\\right) $. We now tak\n\\begin{equation*}\nm_{i}=n^{\\gamma _{i}(1+\\gamma _{i})},\\ i=1,2,...,N-1,\n\\end{equation*\nand denote $\\mathcal{H}_{i}=\\left\\{ m:m_{i-1}\\leq m\\leq m_{i}\\right\\} $.\n\nSince $\\bar{Z}_{i}(k,n)$ is monotone increasing in $k$ for each fixed $n,$\nTheorem \\ref{T_familyMany} and~(\\ref{SurvivSingle}) imply, as $n\\rightarrow\n\\infty $\n\\begin{eqnarray*}\n&&\\mathbf{P}_{n}(\\zeta _{n}=i;\\beta _{n}\\notin \\mathcal{H}_{i})\\leq \\mathbf{\n}_{n}(\\exists k<m_{i-1}:Z_{i}(k,n)>0) \\\\\n&&\\qquad +\\mathbf{P}_{n}(\\exists k>m_{i}:Z_{i}(k,n)>0) \\\\\n&&\\quad \\leq \\mathbf{P}_{n}(\\bar{Z}_{i}(m_{i-1},n)>0)+\\mathbf{P\n_{n}(Z_{i}(m_{i})>0)=o(1).\n\\end{eqnarray*\nBy the same statements we conclude, as $n\\rightarrow \\infty \n\\begin{eqnarray*}\n&&\\mathbf{P}_{n}(\\zeta _{n}\\notin \\{i,i+1\\};\\beta _{n}\\in \\mathcal{H\n_{i})\\leq \\mathbf{P}_{n}\\left( \\exists k\\in \\mathcal{H}_{i}:\\text{\\b{Z}\n_{i-1}(k,n)+\\bar{Z}_{i+2}(k,n)>0\\right) \\\\\n&&\\quad \\leq \\mathbf{P}_{n}\\left( \\exists k\\in \\mathcal{H}_{i}:\\text{\\b{Z}\n_{i-1}(k)+\\bar{Z}_{i+2}(k,n)>0\\right) \\\\\n&&\\quad \\leq \\mathbf{P}_{n}(\\exists k\\in \\mathcal{H}_{i}:\\text{\\b{Z}\n_{i-1}(k)>0)+\\mathbf{P}_{n}\\left( \\exists k\\in \\mathcal{H}_{i}:\\bar{Z\n_{i+2}(k,n)>0\\right) \\\\\n&&\\quad \\leq \\mathbf{P}_{n}(\\text{\\b{Z}}_{i-1}(m_{i-1})>0)+\\mathbf{P}_{n}\n\\bar{Z}_{i+2}(m_{i},n)>0)=o(1).\n\\end{eqnarray*\nHence, as $n\\rightarrow \\infty \n\\begin{eqnarray}\n\\mathbf{P}_{n}(\\zeta _{n}=i) &=&\\mathbf{P}_{n}(\\zeta _{n}=i;\\beta _{n}\\in\n\\mathcal{H}_{i})+o(1)  \\notag \\\\\n&=&\\mathbf{P}_{n}(\\beta _{n}\\in \\mathcal{H}_{i})-\\mathbf{P}_{n}(\\zeta\n_{n}=i+1;\\beta _{n}\\in \\mathcal{H}_{i})+o(1).  \\label{Type}\n\\end{eqnarray\nIntroduce the event\n\\begin{equation*}\n\\mathcal{G}_{i}(j,n)=\\left\\{ \\text{\\b{Z}}_{i}(j;n)+\\bar{Z\n_{i+2}(j+1,n)=0;Z_{i+1}(j,n)=1\\right\\} .\n\\end{equation*\nClearly\n\\begin{eqnarray*}\n&&\\mathbf{P}_{n}(\\zeta _{n}=i+1;\\beta _{n}\\in \\mathcal{H}_{i})\n\\sum_{j=m_{i-1}}^{m_{i}}\\mathbf{P}_{n}(\\zeta _{n}=i+1;\\beta _{n}=j) \\\\\n&&\\,=\\sum_{j=m_{i-1}}^{m_{i}}\\mathbf{P}_{n}(\\mathcal{G}_{i}(j,n),\\bar{Z\n_{i+1}(j+1,n)\\geq 2) \\\\\n&&\\,=o(1)+\\sum_{j=m_{i-1}}^{m_{i}}\\mathbf{P}_{n}(\\mathcal{G}_{i}(j,n)\n\\mathbf{P}_{n}(Z_{i+1}(j+1,n)\\geq 2|\\mathbf{Z}(j,n)=\\mathbf{e}_{i+1}).\n\\end{eqnarray*\nIt is not difficult to check (recall (\\ref{DefNONimmigr}), (\\ref{Derivat})\nand (\\ref{ExplCoeff})) that\n\\begin{eqnarray*}\n\\mathbf{P}_{n}(Z_{i+1}(j+1,n)=1|\\mathbf{Z}(j,n)=\\mathbf{e}_{i+1}) &=&\\frac\nQ_{n-j-1}^{(i+1,N)}}{Q_{n-j}^{(i+1,N)}}\\frac{dh_{i+1}(s,\\mathbf{1}^{(N-i-1)}\n}{ds}\\left\\vert _{s=H_{n-j-1}^{(i+1,N)}(\\mathbf{0})}\\right. \\\\\n&\\geq &\\frac{dh_{i+1}(s,\\mathbf{1}^{(N-i-1)})}{ds}\\left\\vert\n_{s=H_{n-j-1}^{(i+1,N)}(\\mathbf{0})}\\right. \\\\\n&\\geq &1-2b_{i+1}Q_{n-j-1}^{(i+1,N)} \\\\\n&\\geq &1-2b_{i+1}Q_{n-m_{i}}^{(i+1,N)}.\n\\end{eqnarray*\nHence, using the estimat\n\\begin{eqnarray*}\n\\mathbf{P}_{n}(Z_{i+1}(j+1,n)\\geq 2|\\mathbf{Z}(j,n)=\\mathbf{e}_{i+1}) &=&1\n\\mathbf{P}_{n}(Z_{i+1}(j+1,n)=1|\\mathbf{Z}(j,n)=\\mathbf{e}_{i+1}) \\\\\n&\\leq &2b_{i}Q_{n-m_{i}}^{(i+1,N)}\n\\end{eqnarray*\nwe conclud\n\\begin{eqnarray*}\n\\mathbf{P}_{n}(\\zeta _{n}=i+1;\\beta _{n}\\in \\mathcal{H}_{i})\n&=&o(1)+O(m_{i}Q_{n-m_{i}}^{(i+1,N)}) \\\\\n&=&o(1)+O(n^{\\gamma _{i}(1+\\gamma _{i})}n^{-\\gamma _{i+1}})=o(1).\n\\end{eqnarray*\nThis, on account of (\\ref{recent_i}) and (\\ref{Type}) gives\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }\\mathbf{P}_{n}(\\zeta _{n}=i)=\\lim_{n\\rightarrow\n\\infty }\\mathbf{P}_{n}(\\beta _{n}\\in \\mathcal{H}_{i})=\\lim_{n\\rightarrow\n\\infty }\\mathbf{P}_{n}(n^{\\gamma _{i}}\\ll \\beta _{n}\\ll n^{\\gamma _{i+1}})\n\\frac{1}{2^{i}}\n\\end{equation*\nas desired.\n\nFinally\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }\\mathbf{P}_{n}( \\zeta _{n}=N) =1-\\sum_{i=1}^{N-1\n\\frac{1}{2^{i}}=\\frac{1}{2^{N-1}}.\n\\end{equation*}\n\nTheorem \\ref{T_type} is proved.\n\n\\textbf{Acknowledgement}. This work was partially supported by the Russian\nFoundation for Basic Research, project N14-01-00318. The author would also\nlike to thank prof. A.M.Zubkov for valuable remarks.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\nThe main goal of this\n paper is to give a\n fine analysis of  blow-up solutions of \nconformally invariant fully nonlinear second order elliptic equations.\n\n\n\nLet $n \\geq 3$ be an integer and\n\\begin{equation}\n\\Gamma\\subset {\\mathbb R}^n\\ \\mbox{be an open convex symmetric cone with vertex at the origin}\n\\label{01}\n\\end{equation}\nsatisfying\n\\begin{equation}\n\\Gamma_n\\subset\n\\Gamma\\subset\n\\Gamma_1,\n\\label{02}\n\\end{equation}\nwhere\n\\[\n\\Gamma_n \t:=\\{\\lambda\\in {\\mathbb R}^n\\ |\\\n\\lambda_i>0\\ \\forall\\ i\\}\n, \\qquad \\Gamma_1 :=\\Big\\{\\lambda\\in {\\mathbb R}^n\\ |\\\n\\sum_{i=1}^n\\lambda_i>0\\Big\\}.\n\\]\nWe assume that\n\\begin{equation}\nf\\in C^1(\\Gamma)\\cap C^0(\\overline \\Gamma)\\ \\mbox{is\nsymmetric in\n} \\ \\lambda_i,\n\\label{03}\n\\end{equation}\n\\begin{equation}\nf>0, \\ \\  \\frac{  \\partial f}{ \\partial \\lambda_i}>0\\ \n\\mbox{in}\\ \\Gamma\\ \\forall\\ i,\\ \\\nf=0\\ \\mbox{on}\\ \\partial \\Gamma,\n\\label{04}\n\\end{equation}\n\\begin{equation}\nf(\\lambda) < 1\\qquad\\forall\\\n\\lambda\\in \\Gamma \\text{ satisfying } \\ \\sum_{i=1}^n \\lambda_i <\\delta.\n\\label{FU-1a}\n\\end{equation}\n\nIn \\eqref{01} and \\eqref{03}, the symmetric property of $\\Gamma$ and $f$ is understood in the sense that if $\\lambda \\in \\Gamma$ and $\\tilde \\lambda$ is a permutation of $\\lambda$, then $\\tilde \\lambda \\in \\Gamma$ and $f(\\tilde\\lambda) = f(\\lambda)$. Also, throughout the paper, \n \\[\n \\parbox{.8\\textwidth}{whenever we write $f(\\lambda)$, we \n implicitly assume that $\\lambda\\in \\overline \\Gamma$.}\n \\]\n\nWhen $\\Gamma \\neq \\Gamma_1$, \\eqref{FU-1a} is a consequence of \\eqref{03} and \\eqref{04} (cf. \\cite[Proposition B.1]{LiNgBocher}). However, this does not have to be the case when $\\Gamma = \\Gamma_1$, for example when\n\\[\nf(\\lambda) = \\Big(\\sum_{i=1}^n \\lambda_i\\Big)^{\\frac{n-1}{n+3}} \\Big(\\sum_{i=1}^n \\lambda_i^2\\Big)^{\\frac{2}{n+3}}.\n\\]\n\nIlluminating examples of $(f, \\Gamma)$ are $(f, \\Gamma)=(\\sigma_k^{\\frac 1k}, \\Gamma_k)$ where $\\sigma_k(\\lambda) = \\sum \\lambda_{i_1} \\ldots \\lambda_{i_k}$ is the $k$-th elementary symmetric function and \n\\begin{align*}\n\\Gamma_k\n\t&= \\text{ the connected component of $\\{\\lambda \\in {\\mathbb R}^n: \\sigma_k(\\lambda) > 0\\}$ containing $\\Gamma_n$}\\\\\n\t&= \\{\\lambda \\in {\\mathbb R}^n: \\sigma_l(\\lambda) > 0 \\text{ for all } 1 \\leq l \\leq k\\}.\n\\end{align*}\nBesides \\eqref{01}-\\eqref{FU-1a}, $(\\sigma_k^{\\frac 1k}, \\Gamma_k)$ enjoys other nice and helpful properties, such as concavity and homogeneity properties of $\\sigma_k^{1\/k}$, Newton's inequalities, divergence and variational structures, etc., which we do not assume in this paper. In particular, we would like to note that no concavity or homogeneity assumption on $f$ is being made in the present paper.\n\n\n\nFor a positive $C^2$ function $u$, let $A^u$ be the $n\\times n$ matrix with entries\n\\[\n(A^u)_{ij} = - \\frac{2}{n-2} u^{-\\frac{n+2}{n-2}} \\nabla_i \\nabla_j u + \\frac{2n}{(n-2)^2} u^{-\\frac{2n}{n-2}}\\,\\nabla_i u \\,\\nabla_j u -  \\frac{2}{(n-2)^2} u^{-\\frac{2n}{n-2}}\\,|\\nabla u|^2\\,\\delta_{ij}.\n\\]\nThis is sometimes referred to as the conformal Hessian of $u$.\n\nThe conformal Hessian $A^u$ arises naturally in conformal geometry as follows. Recall that the Riemann curvature $Riem_g$ of a Riemannian metric $g$ can be decomposed into traced and traceless parts as\n\\[\nRiem_g = A_g \\owedge  g + W_g,\n\\]\nwhere $A_g = \\frac{1}{n-2}({\\rm Ric}_g - \\frac{1}{2(n-1)}R_g\\,g)$, ${\\rm Ric}_g$, $R_g$ and $W_g$ are the Schouten curvature, the Ricci curvature, the scalar curvature and the Weyl curvature of $g$ and $\\owedge$ denotes the Kulkarni-Nomizu product. While the $(1,3)$-valent Weyl curvature remains unchanged under a conformal change of the metric, the Schouten curvature is adjusted by a second order operator of the conformal factor. In particular, if we consider the metric $g_u :=  u^{\\frac{4}{n-2}}g_{flat}$ conformal to the flat metric $g_{flat}$ on ${\\mathbb R}^n$, then the Schouten curvature $A_{g_u}$ of $g_u$ is given by the conformal Hessian in the form\n\\[\nA_{g_u} = u^{\\frac{4}{n-2}}\\,(A^u)_{ij}\\,dx^i\\,dx^j.\n\\]\nConsequently, we have\n\\[\n\\lambda(A_{g_u}) = \\lambda(A^u)\n\\]\nwhere $\\lambda(A_{g_u})$ denotes the eigenvalues of $A_{g_u}$ with respect to the metric $g_u$ and $\\lambda(A^u)$ denotes those of the matrix $A^u$.\n\n\n$A^u$ enjoys a conformal invariance property, inherited from the conformal structure of ${\\mathbb R}^n$, which will be of special importance in our treatment. Recall that a map $\\varphi: {\\mathbb R}^n \\cup\\{\\infty\\} \\rightarrow {\\mathbb R}^n \\cup\\{\\infty\\}$ is called a M\\\"obius transformation if it is the composition of finitely many of the following types of transformations:\n\\begin{itemize}\n\\item a translation: $x \\mapsto x + \\bar x$ where $\\bar x$ is a given vector in ${\\mathbb R}^n$,\n\\item a dilation: $x \\mapsto a\\,x$ where $a$ is a given positive scalar,\n\\item a Kelvin transformation: $x \\mapsto \\frac{x}{|x|^2}$.\n\\end{itemize}\nFor a function $u$ and a M\\\"obius transformation $\\varphi$, let\n\\begin{equation}\nu_\\varphi = |J_\\varphi|^{\\frac{n-2}{2n}}u \\circ \\varphi,\n\t\\label{Eq:uvarphi}\n\\end{equation}\nwhere $J_\\varphi$ is the Jacobian of $\\varphi$. A calculation gives \n\\[\nA^{u_\\varphi}(x) = O_\\varphi(x)^t A^u(\\varphi(x)) O_\\varphi(x)\n\\]\nfor some orthogonal $n \\times n$ matrix $O_\\varphi(x)$. In particular,\n\\begin{equation}\n\\lambda(A^{u_\\varphi}(x)) = \\lambda(A^u( \\varphi(x))).\n\t\\label{Eq:CIProp}\n\\end{equation}\n\nThe main result of this paper concerns an analysis \non the behavior of a sequence  $\\{u_k\\}\\in C^2(B_3(0))$ satisfying\n\\begin{equation}\nf(\\lambda(A^{u_k})) =1, \\\nu_k>0,\\\n\\mbox{in}\\ B_3(0),\n\\label{eqB2}\n\\end{equation}\nand\n\\begin{equation}\n\\sup_{B_1(0)}u_k\\to \\infty,\n\\label{eqB1}\n\\end{equation}\nwhere\n$(f, \\Gamma)$ satisfies \\eqref{01}-\\eqref{FU-1a}. Note that no other assumptions on $u_k$ is made.\n\nAs is known, equation \\eqref{eqB2} is \na fully nonlinear\n elliptic equation. Fully nonlinear elliptic equations involving $f(\\lambda(\\nabla^2 u))$ were investigated in the classic paper of Caffarelli, Nirenberg and Spruck \\cite{C-N-S-Acta}. \n \n Our paper appears to be the first fine blow-up analysis in this fully nonlinear context. We expect this to serve as a crucial step in the study of the problem on Riemannian manifolds.\n \n\n\n\n\n\nTo obtain our result on fine analysis of  blow-up solutions,\nwe make use of the following Liouville theorems.\n\n\n\\begin{THM}[\\cite{LiLi05}]\\label{TheoremA}\n Let $(f, \\Gamma)$ satisfy \\eqref{01}-\\eqref{04}\nand\nlet $0<v\\in C^2({\\mathbb R}^n)$ satisfy \n\\begin{equation}\nf(\\lambda(A^v))=1\\qquad \\mbox{in}\\ {\\mathbb R}^n.\n\\label{eqB0}\n\\end{equation}\nThen \n\\begin{equation}\nv(x)\\equiv \\left( \\frac a{  1+b^2|x-\\bar x|^2}  \\right) ^{ \\frac {n-2}2 },\n\\qquad x\\in {\\mathbb R}^n\n\\label{vform}\n\\end{equation}\nfor some $\\bar x\\in {\\mathbb R}^n$ and some positive constants\n$a$ and $b$ satisfying \n\\[\nf(2b^2a^{-2}, \\cdots, 2b^2a^{-2})=1.\n\\]\n\\end{THM}\n\n\\begin{THM}[\\cite{Li09-CPAM}]\\label{TheoremADeg}\n Let $\\Gamma$ satisfy \\eqref{01} and \\eqref{02},\nand\nlet $0< v\\in C^{0,1}_{loc}({\\mathbb R}^n \\setminus \\{0\\})$ satisfy \n\\begin{equation}\n\\lambda(A^v) \\in \\partial \\Gamma \\mbox{ on }\\ {\\mathbb R}^n \\setminus \\{0\\}\n\\label{eqB0deg}\n\\end{equation}\nin the viscosity sense (see Definition \\ref{Def:DegVisSol} below). Then $v$ is radially symmetric about the origin and $v(r)$ is non-increasing in $r$.\n\\end{THM}\n\nFor $(f, \\Gamma)=(\\frac{1}{2n}\\sigma_1, \\Gamma_1)$, equation \\eqref{eqB2} is\nthe critical exponent equation $-\\Delta u= n(n-2)u^{ (n+2) \/(n-2) }$ and\nTheorem \\ref{TheoremA} was proved by Caffarelli, Gidas and Spruck \\cite{CGS}.\nSee also Gidas, Ni and Nirenberg \\cite{G-N-N-1981} under some decay assumption of $u$ at infinity.\nFor $(f, \\Gamma)=(\\sigma_2^{1\/2}, \\Gamma_2)$ in ${\\mathbb R}^4$ and $v \\in C^{1,1}_{loc}({\\mathbb R}^4)$, \nthe result was proved by Chang, Gursky and Yang \\cite{CGY03-IP}.\n \n\\medskip\nIn fact we need a stronger version of Theorem \\ref{TheoremA} (see Theorem \\ref{proposition1}) and a variant of Theorem \\ref{TheoremADeg} (see Theorem \\ref{proposition1-deg}). For simplicity, readers are advised that in the main body of the paper \n\\[\n\\parbox{.8\\textwidth}{all theorems, propositions and lemmas hold under \\eqref{01}-\\eqref{FU-1a}, instead of the stated weaker hypotheses on $(f,\\Gamma)$.}\n\\]\n\n\n\\begin{thm} Let $(f, \\Gamma)$ satisfy \n\\begin{align}\n&\\Gamma\\subset {\\mathbb R}^n\\ \\mbox{be an open symmetric set},\n\\label{01weak}\\\\\n&\\Gamma\\subset\n\\Gamma_1 \\text{ and } \\Gamma \\cap \\{\\lambda + t\\mu:  t > 0\\} \\text{ is convex for all $\\lambda \\in \\Gamma, \\mu \\in \\Gamma_n$},\n\\label{02weakX}\\\\\n&f \\in C^1(\\Gamma) \\text{ is symmetric in } \\lambda_i \\text{ and } \\frac{  \\partial f}{ \\partial \\lambda_i}>0\\ \n\\mbox{in}\\ \\Gamma\\ \\forall\\ i.\\label{0304weak}\n\\end{align}\n Assume that\n  $0<v\\in C^0({\\mathbb R}^n)$,\n$0<v_k\\in C^2(B_{R_k}(0))$,\n $R_k\\to\\infty$,\n\\begin{equation}\nf(\\lambda(A^{v_k}))=1\n\\quad \\ \\ \\mbox{in}\\ B_{R_k}(0),\n\\label{ve}\n\\end{equation}\nand \n\\begin{equation}\nv_k\\to v\\ \\mbox{in}\\ C^0_{loc}({\\mathbb R}^n).\n\\label{1new}\n\\end{equation}\nThen either $v$ is constant or $v$ is of the form\n\\eqref{vform}\nfor some $\\bar x\\in {\\mathbb R}^n$ and some positive constants\n$a$ and $b$.\n\n\n\nIf it holds in addition that\n\\begin{align}\n&\\text{there exists $t_0 > 0$ such that $f(t_0, \\ldots, t_0) < 1$,}\n\\label{Eq:ActaSIdiag}\n\\end{align}\nthen $v$ cannot be constant. If it holds further that\n\\begin{equation}\n\\Gamma + \\Gamma_n = \\{\\lambda + \\mu: \\lambda \\in \\Gamma, \\mu \\in \\Gamma_n\\} \\subset \\Gamma,\n\t\\label{02weakY}\n\\end{equation}\nthen the constants $a$ and $b$ in \\eqref{vform} satisfy $(2b^2a^{-2}, \\cdots, 2b^2a^{-2}) \\in \\Gamma$ and $$f(2b^2a^{-2}, \\cdots, 2b^2a^{-2})=1.$$\n\\label{proposition1}\n\\end{thm}\n\n\\begin{rem}\nIn Theorem \\ref{proposition1}, if condition \\eqref{Eq:ActaSIdiag} is dropped, the case that $v$ is constant can occur. See the counterexample in Remark \\ref{Rem:ASD}.\n\\end{rem}\n\n\\begin{thm} Let $(f, \\Gamma)$ satisfy \\eqref{01weak}-\\eqref{0304weak}. Assume that\n  $v_*\\in C^0({\\mathbb R}^n \\setminus \\{0\\})$,\n$0<v_k\\in C^2(B_{R_k}(0) \\setminus \\{0\\})$,\n $R_k\\to\\infty$,\n\\[\nf(\\lambda(A^{v_k}))=1\n\\quad \\ \\ \\mbox{in}\\ B_{R_k}(0) \\setminus \\{0\\},\n\\]\nand, for some $M_k \\rightarrow \\infty$,\n\\begin{equation}\nM_k\\,v_k\\to v_*\\ \\mbox{in}\\ C^0_{loc}({\\mathbb R}^n \\setminus \\{0\\}).\n\\label{1newdeg}\n\\end{equation}\nThen $v_*$ is radially symmetric about the origin, i.e. $v_*(x) = v_*(|x|)$. In particular, if $v_k \\in C^2(B_{R_k}(0))$, $f(\\lambda(A^{v_k}))=1$ in $B_{R_k}(0)$ and $M_kv_k$ converges to $v_*$ in $C^0_{loc}({\\mathbb R}^n)$, then $v_*$ is constant.\n\\label{proposition1-deg}\n\\end{thm}\n\n\n\n\n\n\n\n\n\\begin{rem}\nWhen $(f,\\Gamma)$ satisfies \\eqref{01}-\\eqref{04} and an additional hypothesis that $f$ is homogeneous of positive degree, the function $v_*$ in Theorem \\ref{proposition1-deg} is a viscosity solution of \\eqref{eqB0deg} and the conclusion follows from Theorem \\ref{TheoremADeg}. However, when $f$ is not homogeneous, $v_*$ is not necessarily a viscosity solution of \\eqref{eqB0deg}.\n\\end{rem}\n\n\n\n\nIt is not difficult to see that, under \\eqref{01}-\\eqref{04}, the function $v$ in Theorem \\ref{proposition1} is a viscosity solution of \\eqref{eqB0} (see Remark \\ref{Cor:VCConf}). We have the following conjecture.\n\n\n\\begin{conj}\nLet  $(f, \\Gamma)$ satisfy \\eqref{01}-\\eqref{04},\nand let  $0<v\\in C^0_{loc}({\\mathbb R}^n)$\nbe a viscosity solution of  \\eqref{eqB0}.\nThen $v$ is of the form \\eqref{vform}\nfor some $\\bar x\\in {\\mathbb R}^n$ and some positive constants\n$a$ and $b$.\n\\end{conj}\n\nThe notion of \nviscosity solutions given below is consistent with that in \\cite{Li09-CPAM}.\n\n\\begin{Def}\\label{Def:fVisSol} A positive continuous function\n $v$ in an open set $\\Omega\\subset {\\mathbb R}^n$\nis a viscosity supersolution (respectively, subsolution) of \n$$\nf(\\lambda(A^v))=1, \\ \\ \\mbox{in}\\ \\Omega,\n$$\nwhen the following holds: if $x_0\\in \\Omega$,\n$\\varphi\\in C^2(\\Omega)$, $(v-\\varphi)(x_0)=0$,\nand $v-\\varphi\\ge 0$ near $x_0$, then\n$$\nf(\\lambda(A^\\varphi(x_0))\\ge 1\n$$\n(respectively, if $(v-\\varphi)(x_0)=0$,\nand $v-\\varphi\\le 0$ near $x_0$, then either \n$\\lambda(A^\\varphi(x_0))  \\in {\\mathbb R}^n\\setminus \\overline \\Gamma$\nor\n$\nf(\\lambda(A^\\varphi(x_0)))\\le 1\n$).  We say that $v$ is a viscosity solution if it is both\na viscosity supersolution and a viscosity subsolution.\n\\end{Def}\n\n\\begin{Def}\\label{Def:DegVisSol} A positive continuous function\n $v$ in an open set $\\Omega\\subset {\\mathbb R}^n$\nis a viscosity supersolution (respectively, subsolution) of \n$$\n\\lambda(A^v) \\in \\partial \\Gamma, \\ \\ \\mbox{in}\\ \\Omega,\n$$\nwhen the following holds: if $x_0\\in \\Omega$,\n$\\varphi\\in C^2(\\Omega)$, $(v-\\varphi)(x_0)=0$,\nand $v-\\varphi\\ge 0$ near $x_0$, then\n$$\n\\lambda(A^\\varphi(x_0) \\in \\bar \\Gamma\n$$\n(respectively, if $(v-\\varphi)(x_0)=0$,\nand $v-\\varphi\\le 0$ near $x_0$, then either \n$\\lambda(A^\\varphi(x_0))  \\in {\\mathbb R}^n\\setminus \\overline \\Gamma$).  We say that $v$ is a viscosity solution if it is both\na viscosity supersolution and a viscosity subsolution.\n\\end{Def}\n\nIt is clear that for $C^2$ functions the notion of viscosity solutions and classical solutions coincide. Also, viscosity super- and sub-solutions are stable under uniform convergence, see Appendix \\ref{Sec:AppVS}.\n\n\n\n\n\n\n\n\nNote that for any $\\lambda = (\\lambda_1, \\cdots, \\lambda_n) \\in \\Gamma$, $t_1 = \\max\\lambda_i + 1 > 0$ and $(t_1, \\cdots, t_1) \\in \\lambda + \\Gamma_n$. In other words, the sets $\\lambda + \\Gamma_n$ have non-empty intersection with the ray $\\{(t, \\cdots, t): t > 0\\}$. Thus, if $f^{-1}(1) \\neq \\emptyset$, then, in view of \\eqref{04}, there exists some $c>0$ such that\n$f(c,\\cdots, c)=1$.  In such situation,  working with \n$\\tilde f(\\lambda):=f(\\frac c2 \\lambda)$ instead of $f$, \nwe\n may assume without loss of generality the following normalization\ncondition\n\\begin{equation}\nf(2,\\cdots, 2)=1.\n\\label{F1}\n\\end{equation}\n\n\n\n\n\n\n\n\nLet\n$$\nU(x):= \\left(\\frac 1{1+|x|^2}\\right)^{ \\frac {n-2}2 },\\qquad x\\in {\\mathbb R}^n.\n$$\nA calculation gives\n$$\nA^U\\equiv 2I.\n$$\n\nWith the normalization \\eqref{F1}, $U$ satisfies\n$$\nf(\\lambda(A^U))=1 \\quad\\mbox{on}\\ {\\mathbb R}^n.\n$$ \n\nFor $\\bar x\\in {\\mathbb R}^n$ and $\\mu>0$, let\n $$U^{\\bar x, \\mu}(x)= \\mu U(\\mu^{\\frac 2{n-2}}(x-\\bar x)) = \\Big(\\frac{\\mu^{\\frac 2{n-2}}}{1 + \\mu^{\\frac 4{n-2}} |x- \\bar x|^2}\\Big)^{\\frac{n-2}{2}}.\n$$\nNote that, in the sense of \\eqref{Eq:uvarphi}, $U^{\\bar x, \\mu} = U_\\varphi$ with $\\varphi(x) = \\mu^{\\frac{2}{n-2}}\\,(x - \\bar x)$. Hence, by the conformal invariance \\eqref{Eq:CIProp}, for any $\\bar x\\in {\\mathbb R}^n$ and $\\mu>0$, \n\\[\nf(\\lambda(A^{U_{\\bar x, \\mu}})) = 1 \\text{ on } {\\mathbb R}^n.\n\\]\n\n\\begin{thm}\\label{theorem4X}\n Let $(f, \\Gamma)$ satisfy \\eqref{01weak}-\\eqref{0304weak}, \\eqref{Eq:ActaSIdiag}-\\eqref{02weakY}, \\eqref{FU-1a}\n and the normalization condition \\eqref{F1}. Let $\\epsilon \\in (0,1\/2]$. There exist constants $\\bar m = \\bar m(f,\\Gamma) \\geq 1, {K} = {K}(f,\\Gamma) > 1$, ${\\delta_*} = {\\delta_*}(\\epsilon, f,\\Gamma) > 0, {C_*} = {C_*}(\\epsilon, f,\\Gamma) > 1$  such that for any positive $u \\in C^2(B_3(0))$ satisfying\\footnote{Note that, $\\bar m$ and ${K}$ are independent of $\\epsilon$.}\n %\n \\[\nf(\\lambda(A^u)) = 1 \\text{ in } B_{3}(0) \\text{ and } \\sup_{B_1(0)} u \\geq {C_*},\n\\]\nthere exists $\\{x^1, \\cdots,  x^m\\} \\subset B_{2}(0)$ with $1 \\leq m \\leq \\bar m$ satisfying\n\\begin{enumerate}[(i)]\n\\item $u(x^1) \\geq \\sup_{B_1(0)} u$,\n\\item $|x^i-x^j|\\ge \\frac{1}{{K}}$ for all $1 \\leq i\\ne j \\leq m$,\n\\item $\\frac 1{{K}} \\le\n\\frac {   u(x^i)  }{  u(x^j) }\\le {K}$\n for all $1 \\leq i, j \\leq m$,\n \\item $|u(x) - U^{x^i, u(x^i)}(x)| \\leq \\epsilon U^{x^i, u(x^i)}(x)$ for all $1 \\leq i \\leq m$, $x\\in B_{{\\delta_*}}(x^i)$,\n\\item $\\frac {1}{{K} \\delta_*^{n-2} u(x^1)}\\le u(x) \\le \\frac {{K}} { \\delta_*^{n-2}u(x^1)}$ for all $x \\in B_{\\frac{3}{2}}(0)\\setminus \\cup_{ i=1}^m\nB_{{\\delta_*} }(x^i)$,\n\\item $u(x^i) = \\sup_{B_{{\\delta_*}}(x^i)} u$.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{rem}\\label{Rem:Dist2B1}\nIf it holds further that $\\sup_{B_1(0)} u > \\frac{{K}^{1\/2}}{{\\delta_*}^{\\frac{n-2}{2}}}$, then\n\\[\n\\min_{1 \\leq i \\leq m} dist(x^i, B_1(0)) \\leq \\Big[\\sup_{B_1(0)} u\\Big]^{-\\frac{2}{n-2}}.\n\\]\nTo see this, let $x_*$ be a point in $\\bar B_1(0)$ such that $u(x_*) = \\sup_{B_1(0)} u$. In view of (v) and the stated condition on $\\sup_{B_1(0)} u$, $x_*$ belongs to some ball $B_{{\\delta_*}}(x^{i_0})$. By (iv), we then have\n\\[\nu(x_*) \\leq 2U^{x^{i_0},u(x^{i_0})}(x_*) \\leq \\frac{1}{|x_* - x^{i_0}|^{\\frac{n-2}{2}}},\n\\]\nwhich implies the assertion.\n\\end{rem}\n\nTheorem \\ref{theorem4X} can be stated equivalently as follows.\n\\begin{thm}\\label{theorem4}\n Let $(f, \\Gamma)$ satisfy \\eqref{01weak}-\\eqref{0304weak}, \\eqref{Eq:ActaSIdiag}-\\eqref{02weakY}, \\eqref{FU-1a}\n and the normalization condition \\eqref{F1}.\nAssume that  $0 < u_k\\in C^2(B_3(0))$ satisfy \\eqref{eqB2} and \\eqref{eqB1}. Let $\\epsilon \\in (0,1\/2]$. Then there exist $\\bar m = \\bar m(f,\\Gamma) \\geq 1, {K}(f,\\Gamma) > 1$ and ${\\delta_*} = {\\delta_*}(\\epsilon,f,\\Gamma) > 0$\nsuch that, \n after passing to a subsequence, still denoted by $u_k$, there exists $\\{x_k^1, \\cdots,  x_k^m\\}\\subset B_{2}(0)$ ($1 \\leq m \\leq \\bar m$) satisfying\\footnote{The constant $m$ is independent of $k$.}\n\\begin{enumerate}[(i)]\n\\item $u_k(x_k^1) \\geq \\sup_{B_1(0)} u_k$,\n\\item $|x_k^i-x_k^j|\\ge \\frac{1}{{K}}$ for all $k \\geq 1$, $1 \\leq i\\ne j \\leq m$, \n\\item $\\frac 1{{K}} \\le\n\\frac {   u_k(x_k^i)  }{  u_k(x_k^j) }\\le {K}$\n for all $k \\geq 1$, $1 \\leq i, j \\leq m$,\n \\item \n$|u_k(x) - U^{x_k^i, u_k(x_k^i)}(x)|\n\\le  \\epsilon U^{x_k^i, u_k(x_k^i)}(x)$ for all $k \\geq 1$, $1 \\leq i \\leq m$, $x\\in B_{{\\delta_*}}(x_k^i)$,\n\\item $\\frac {1}{{K} \\delta_*^{n-2}u_k(x_k^1)}\\le u_k(x) \\le \\frac {{K}} { \\delta_*^{n-2} u_k(x_k^1)}$ for all $k \\geq 1$, $x \\in B_{\\frac{3}{2}}(0)\\setminus \\cup_{ i=1}^m\nB_{{\\delta_*} }(x_k^i)$,\n\\item $u_k(x_k^i) = \\sup_{B_{{\\delta_*}}(x_k^i)} u_k$.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{rem}\nBy Remark \\ref{Rem:Dist2B1}, we have\n\\[\ndist(\\{x_k^1, \\ldots, x_k^m\\},B_1(0)) \\rightarrow 0 \\text{ as } k \\rightarrow \\infty.\n\\]\n\\end{rem}\n\nWhen $(f,\\Gamma)=(\\frac{1}{2n}\\sigma_1, \\Gamma_1)$, equation \\eqref{eqB2} is $-\\Delta u_k = n(n-2)\\,u_k^{\\frac{n+2}{n-2}}$ and Theorem \\ref{theorem4} in this case\nwas proved by  Schoen \\cite{SchoenNotes}. \n\nSee Li \\cite{Li95-JDE} and Chen and Lin \\cite{ChenLin} for analogous results for the equation $-\\Delta u_k = K(x)u_k^{\\frac{n+2}{n-2}}$.\n\n\nIn Theorems \\ref{theorem4X} and \\ref{theorem4}, $B_1(0)$, $B_2(0)$ and $B_3(0)$ can be replaced respectively by $B_{r_1}(0)$, $B_{r_2}(0)$ and $B_{r_3}(0)$, $0 < r_1 < r_2 < r_3$, and in this case the constants $\\bar m$, ${K}$, ${\\delta_*}$ and ${C_*}$ depend also on $r_1$, $r_2$ and $r_3$.\n\nThe following is a quantitative version of Theorem \\ref{TheoremA}, and is related to Theorem \\ref{proposition1} and Theorem \\ref{theorem4X}.\n\n\\begin{thm}[Quantitative Liouville Theorem]\nLet $(f, \\Gamma)$ satisfy \\eqref{01weak}-\\eqref{0304weak}, \\eqref{Eq:ActaSIdiag}-\\eqref{02weakY}, \\eqref{FU-1a} and the normalization\ncondition \\eqref{F1},\nand let $\\gamma, r_1 > 0$ be constants.\n  Then, for every $\\epsilon \\in (0,1\/2]$,\nthere exist some constants ${\\delta_*} > 0, R^* > 0$,  depending only on\n$(f, \\Gamma)$, $\\gamma, r_1$ and $\\epsilon$,\n such that if $0< v\\in C^2(B_R(0))$ for some $R\\ge R^*$,\n\\begin{equation}\nf(\\lambda(A^v)=1\\quad \n \\mbox{in}\\ B_R(0),\n\\label{AppB1}\n\\end{equation}\nand\n\\begin{equation}\nv\\ge \\gamma \\quad \\mbox{in}\\ \\ B_{r_1}(0),\n\\label{AppB2}\n\\end{equation}\nthen, for some $\\bar x\\in {\\mathbb R}^n$ satisfying\n\\begin{equation}\nv(\\bar x) = \\max_{B_{{\\delta_*} R}(0)} v \\leq \\frac{2^{n-1}}{\\gamma\\,r_1^{n-2}}, \\qquad |\\bar x| \\leq 2^{\\frac{1}{n-2}} \\gamma^{-\\frac{2}{n-2}},\n\\label{AppB3}\n\\end{equation}\nthere holds\n\\begin{equation}\n|v(y)- U^{\\bar x, v(\\bar x)}(y)|\n\\le \\epsilon\\,U^{\\bar x, v(\\bar x)}(y),\\qquad \\forall\\ |y - \\bar x|\\le {\\delta_*} R.\n\\label{AppB4}\n\\end{equation}\n\\label{quantitativeliouville}\n\\end{thm}\n\n\\begin{rem}\nThe constant ${\\delta_*}$ in Theorems \\ref{theorem4X} and \\ref{quantitativeliouville} can be chosen the same.\n\\end{rem}\n\n\n \n\\begin{rem}\nAn analogous result for the degenerate elliptic equation $\\lambda(A^v) \\in \\partial \\Gamma$ is a consequence of the local gradient estimate \\cite[Theorem 1.5]{LiNgBocher}.\n\\end{rem}\n\n\nAn ingredient in our proof of Theorems \\ref{theorem4X} and \\ref{theorem4} is the following local gradient estimate, which follows from Theorem \\ref{proposition1-deg} and the proof of \\cite[Theorem 1.10]{Li09-CPAM}.\n\n\\begin{thm}\\label{TheoremB}\n  Let $(f, \\Gamma)$ satisfy \\eqref{01weak}-\\eqref{0304weak} \n and let $v\\in C^2(B_2(0))$\nsatisfy, for some constant $b>0$,\n\\begin{equation}\n0<v\\le b \\qquad \\mbox{in}\\ B_2(0),\n\\label{36new}\n\\end{equation}\nand\n\\begin{equation}\nf(\\lambda(A^v))=1  \\qquad \\mbox{in}\\ B_2(0).\n\\label{35new}\n\\end{equation}\nThen, for some constant $C$ depending only on $(f, \\Gamma)$  and $b$,\n\\begin{equation}\n|\\nabla \\ln v|\\le C\\quad \\mbox{in}\\ B_1(0).\n\\label{abc}\n\\end{equation}\n\\end{thm}\n\nFor $(f,\\Gamma)=(\\sigma_k^{1\/k}, \\Gamma_k)$, the result was proved by Guan and Wang \\cite{GW03-IMRN}.\n\nWhen $(f,\\Gamma)$ satisfies \\eqref{01}-\\eqref{04} and is homogeneous of positive degree, Theorem \\ref{TheoremB} was proved in \\cite{Li09-CPAM}. \n\nThe rest of the paper is organized as follows. We start in Section \\ref{sec:Tp1} with the proof of Theorems \\ref{proposition1} and \\ref{proposition1-deg}. We then prove Theorem \\ref{TheoremB} in Section \\ref{sec:ThmB}. In Section \\ref{sec:T4X}, we first establish an intermediate quantitative Liouville result and then use it to prove Theorem \\ref{theorem4X}. In Section \\ref{sec:tQL}, we prove Theorem \\ref{quantitativeliouville} as an application of Theorem \\ref{theorem4X}. In Appendix \\ref{Sec:AppA}, we present a lemma about super-harmonic functions which is used in the body of the paper. In Appendix \\ref{Sec:AppVS}, we include a relevant remark on the limit of viscosity solutions of elliptic PDE. Finally we collect in Appendix \\ref{Sec:AppC} some relevant calculus lemmas.\n \n\n\n\n\\section{Non-quantitative Liouville theorems}\\label{sec:Tp1}\n\nIn this section, we prove Theorems \\ref{proposition1} and \\ref{proposition1-deg}. We use the method of moving spheres and establish along the way, as a tool, a gradient estimate which is in a sense weaker than that in Theorem \\ref{TheoremB} but suffices for the moment. (Note that the proof of Theorem \\ref{TheoremB} relies on Theorem \\ref{proposition1-deg}.)\n\n\\subsection{A gradient estimate}\n\n\n\\begin{thm}\\label{SoftGEst}\nLet $(f,\\Gamma)$ satisfy \\eqref{01weak}, \\eqref{0304weak} and \n\\[\n\\Gamma \\cap \\{\\lambda + t\\mu:  t > 0\\} \\text{ is convex for all $\\lambda \\in \\Gamma, \\mu \\in \\Gamma_n$}.\n\\]\nLet $0 < v\\in C^2(B_2(0))$\nsatisfy, for some constant $\\theta > 1$,\n\\begin{equation}\n\\sup_{B_2(0)} v \\leq \\theta \\inf_{B_2(0)} v,\n\\label{Eq:osclnv}\n\\end{equation}\nand\n\\[\nf(\\lambda(A^v))=1  \\qquad \\mbox{in}\\ B_2(0).\n\\]\nThen, for some constant $C$ depending only on $n$  and $\\theta$,\n\\[\n|\\nabla \\ln v|\\le C\\quad \\mbox{in}\\ B_1(0).\n\\]\n\\end{thm}\n\n\nThis type of gradient estimate was established and used in various work of the first named author and his collaborators under less general hypothesis on $(f,\\Gamma)$. It turns out that the same proof works in the current situation. We give a detailed sketch here for completeness.\n\nWe use the method of moving spheres as in \\cite{LiLi03, LiLi05, LiZhang03, LiZhu95-Duke}. For a function $w$ defined on a subset of ${\\mathbb R}^n$, we define\n\\[\nw_{x,\\lambda}(y) = \\frac{\\lambda^{n-2}}{|y - x|^{n-2}}w\\Big(x + \\frac{\\lambda^2(y -x )}{|y -x|^2}\\Big)\n\\]\nwherever the expression makes sense. We will use $w_\\lambda$ to denote $w_{0,\\lambda}$. We start with a simple result.\n\n\n\\begin{lem} \\label{lem-1}\nLet $R > 0$ and $w$ be a positive Lipschitz function in $\\bar B_R(0)$ such that, for some $L > 0$,\n\\[\n|\\ln w(y) - \\ln w(z)| \\leq L|y - z| \\text{ for all } y, z \\in \\bar B_R(0). \n\\]\nThen for $\\underline{\\lambda} = \\min(\\frac{n-2}{2L},\\frac{R}{2})$ we have \n\\begin{equation}\nw_\\lambda \\le  w\\\n\\mbox{in}\\ B_{\\underline{\\lambda}}(0)\\setminus B_\\lambda, \n\\forall 0<\\lambda<\n\\underline{\\lambda}.\n\\label{aa-1}\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\nWrite  $w$\nin polar coordinates $w(r, \\theta)$.\nIt is easy to see that \\eqref{aa-1}\nis equivalent to\n\\begin{equation}\nr^{\\frac {n-2}2} w(r, \\theta))\\le \\ \ns^{\\frac {n-2}2} w(s, \\theta),\n\\ \\ \\forall\\ 0<r<s<\\underline{\\lambda},\\ \\forall\\ \\theta.\n\\label{aa-2}\n\\end{equation}\nEstimate \\eqref{aa-2} is readily seen from the estimates\n\\begin{align*}\n\\ln w(s, \\theta) -\\ln  w(r, \\theta)\n\t&\\geq  - L|s - r|\n\t\t\\geq - \\frac{n-2}{2\\underline{\\lambda}} (s - r)\\\\\n\t&\\geq - \\frac{n-2}{2}[\\ln s - \\ln r].\n\\end{align*}\nLemma \\ref{lem-1} is established.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem \\ref{SoftGEst}]\nBy Lemma \\ref{lem-1}, there exists some $r_0 \\in (0,1\/3)$ such that\n\\[\nv_{x,\\lambda} \\leq v \\text{ in } B_{r_0}(x) \\setminus B_{\\lambda}(x) \\text{ for all } \\lambda \\in (0,r_0) \\text{ and } x \\in B_{4\/3}(0).\n\\]\nIt is easy to see that, for some $r_1 \\in (0,r_0)$, \n\\[\nv_{x,\\lambda} \\leq v \\text{ in } B_{5\/3}(0) \\setminus B_{r_0}(x) \\text{ for all } \\lambda \\in (0,r_1) \\text{ and } x \\in B_{4\/3}(0).\n\\]\nWe then define, for $x \\in B_{4\/3}(0)$,\n\\[\n\\bar\\lambda(x) = \\sup\\Big\\{\\lambda \\in (0, 5\/3 - |x|): v_{x,\\lambda} \\leq v \\text{ in } B_{5\/3}(0) \\setminus B_{\\lambda}(x)\\Big\\}.\n\\]\n\nWe have\n\\[\nv_{x,\\bar \\lambda(x)} \\le v\\\n\\mbox{in}\\ B_{5\/3}(0)\\setminus B_{\\bar \\lambda(x)}(x), \n\\]\nBy the conformal invariance \\eqref{Eq:CIProp}, \n$v_{x,\\bar \\lambda(x)}$ satisfies\n\\[\nf(\\lambda(A^{v_{x,\\bar \\lambda(x)}}))=1,\n\\quad \\ \\ \\mbox{in}\\ B_{5\/3}(0)\\setminus \\overline{  B_{\\bar \\lambda(x)}(x) }.\n\\]\nUsing the above two displayed equations, the definition\nof $\\bar \\lambda(x)$,\nand using the ellipticity of the\nequation satisfied by $v$ and $v_{x,\\bar \\lambda(x)}$,\nwe can apply \nthe strong maximum principle and Hopf Lemma to infer that either $\\bar\\lambda(x) = 5\/3 - |x|$ or there exists \nsome $y \\in \\partial B_{5\/3}(0)$ such that\n$$\nv_{x,\\bar \\lambda(x)}(y)=v(y)\n$$\n--- see the proof of \\cite[Lemma 4.5]{LiLi05}.\n\nIn the latter case, \\eqref{Eq:osclnv} implies that\n\\[\n\\theta \\geq \\frac{v\\big(x + \\frac{\\bar\\lambda(x)^2(y-x)}{|y - x|^2}\\big)}{v(y)} = \\frac{|y - x|^{n-2}}{\\bar\\lambda(x)^{n-2}} \\geq \\frac{(5\/3 - |x|)^{n-2}}{\\bar\\lambda(x)^{n-2}}.\n\\]\nIn either case, we obtain that\n\\[\n\\bar \\lambda(x) \\geq c(n,\\theta) > 0 \\text{ for all } x \\in B_{4\/3}(0).\n\\]\nThe conclusion then follows from \\cite[Lemma A.2]{LiLi05}.\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{proposition1}}\n\n\n\n\\begin{rem} \\label{Rem:ASD}\nIf we drop condition \\eqref{Eq:ActaSIdiag}, the case that $v$ is constant in Theorem \\ref{proposition1} can occur. For example, consider $n \\geq 3$ and \n\\begin{align*}\nf(\\lambda) &= \\sigma_2(\\lambda) + 1, \\qquad \\Gamma = \\Big\\{\\lambda \\in {\\mathbb R}^n\\Big| \\sum_{j=1}^n \\lambda_j - \\lambda_i > 0  \\text{ for all } i = 1, \\ldots, n\\Big\\},\\\\\nv_k(x) &= \\Big(\\frac{1}{R_k^{n+4}}|x - x_k|^{-\\frac{n-4}{2}} + 1\\Big)^{\\frac{2(n-2)}{n-4}} \\text{ for } |x| < \\frac{|x_k|}{2} = R_k \\rightarrow \\infty.\n\\end{align*}\nIt is readily seen that $f(t, \\ldots, t) > 1$ for all $(t, \\ldots, t) \\in \\Gamma$, $v_k$ satisfies \\eqref{ve} (cf. \\cite[Theorem 1.6]{LiNgBocher}) and $v_k \\rightarrow 1$ in $C^{0}_{loc}({\\mathbb R}^n)$.\n\\end{rem}\n\n\n\n\\medskip\n\\begin{proof}[Proof of Theorem \\ref{proposition1}]\nWe may assume that $R_k\\ge 5$ for all $k$.\n\nClearly, for every $\\beta>1$, there exists some positive constant \n$C(\\beta)$, independent of $k$, such that \n$1\/C(\\beta)\\le \nv_k\\le C(\\beta)$ in $B_\\beta(0)$. \n It follows from Theorem \\ref{SoftGEst}\nthat $|\\nabla \\ln v_k|\n\\le C'(\\beta)$ in $B_{\\beta\/2}(0)$.\nIt follows, after passing to a subsequence, that for every $0<\\alpha<1$,\n$v_k\\to v$ in $C^{\\alpha}_{loc}({\\mathbb R}^n)$, $v\\in C^{0,1}_{loc}({\\mathbb R}^n)$ and $v$ is super-harmonic on ${\\mathbb R}^n$.\n\n\nUsing the positivity, the superharmonic of $v$, and the\nmaximum principle, we can find $c_0 > 0$ such that  \n\\begin{equation}\nv(y)\\ge\n2c_0 |y|^{2-n} ,\\ \\ \\forall\\ |y|\\ge 1.\n\t\\label{Eq:vSuperhar}\n\\end{equation}\nPassing to a subsequence and shrinking $R_k$ and $c_0 > 0$, if necessary, we may assume that\n\\begin{equation}\n|v_k(y)-v(y)|\\le R_k^{-n},\\quad\n\\ \\ \\forall\\ |y|\\le R_k,\n\\label{aa-0}\n\\end{equation}\nand\n\\begin{equation}\nv_k(y) \\geq c_0(1+|y|)^{2-n}, \\ \\ \\forall\\   |y| < R_k.\n\\label{aa-0Ex}\n\\end{equation}\n\n\n\n\\begin{lem} \\label{lem-1new}\nUnder the hypotheses of Theorem \\ref{proposition1},\nthere \n exists a function $\\lambda^{(0)}: \n{\\mathbb R}^n \\rightarrow (0,\\infty)$ such that, for all $k$,\n$$\n(v_k)_{x,\\lambda} \\le v_k\\\n\\mbox{in}\\ B_{R_k}(0)\\setminus B_\\lambda(x), \\forall \\ 0<\\lambda<\n\\lambda^{(0)}(x)\\\n\\mbox{and}\\  |x|\\le \\frac {R_k}5.\n$$\n\\end{lem}\n\n\\begin{proof}\nFor $|x| \\leq \\frac{R_k}{5}$, we have, by \\eqref{aa-0} and \\eqref{aa-0Ex}, for all $k$ that\n\\begin{equation}\n\\frac{1}{c_1(x)}\\le  v_k\\le c_1(x)\\ \\mbox{in}\\ B_{4r_1(x)}(x) \\subset B_{R_k}(0), \\label{bound1}\n\\end{equation}\nwhere \n\\begin{align*}\nr_1(x) = \\frac{1}{4} + |x|, \\text{ and }\nc_1(x) = \\max \\Big\\{1+ \\sup_{B_{4r_1(x)}(x)} v, \\frac{1}{c_0} (1 + |x| + 4r_1(x))^{n-2}\\Big\\}.\n\\end{align*}\n\n\n\nBy \\eqref{bound1} and Theorem \\ref{SoftGEst},\nthere exists $c_2(x)>0$, independent of $k$,  such that\n$$\n|\\nabla \\ln v_k|\\le c_2(x)\\ \\mbox{in}\\ B_{2r_1(x)}(x).\n$$\nThus, by Lemma \\ref{lem-1}, we can find $0 <  \\lambda_1(x) \n< r_1(x)$ independent of $k$ such that\n\\begin{equation}\n(v_k)_{x,\\lambda} \\le v_k\\\n\\mbox{in}\\ B_{\\lambda_1(x)}(x)\\setminus B_\\lambda(x) \\ \\ \\forall\\ \n 0 < \\lambda < \\lambda_1(x).\n\\label{bound2}\n\\end{equation}\n\nFor $0 < \\lambda < \\lambda_1(x)$,  we have, using \\eqref{bound1}, that\n\\begin{equation}\n(v_k)_{x,\\lambda}(y) \\leq \n \\frac{\\lambda^{n-2}c_1(x)}{|y - x|^{n-2}}  ,\n\\quad \\forall \\ y \\in  B_{R_k}(0)\\setminus B_{\\lambda}(x).\n\t\\label{Eq:B2-1}\n\\end{equation}\nWhen $y \\in B_{R_k}(0) \\setminus B_{4r_1(x)}(x)$, $\\frac{1}{2}(1 + |y|) < |y - x|$ and we obtain, using \\eqref{Eq:B2-1} and \\eqref{aa-0Ex}, that\n\\begin{equation}\n(v_k)_{x,\\lambda}(y) \\leq \\frac{(2\\lambda)^{n-2}c_1(x)}{(1 + |y|)^{n-2}} \\leq \\frac{(2\\lambda)^{n-2}c_1(x)}{c_0}\\,v_k(y).\n\t\\label{Eq:B3-1}\n\\end{equation}\nWhen $y \\in B_{4r_1(x)}(x) \\setminus B_{\\lambda_1(x)}(x)$, $1 + |y| \\leq 2(1 + 3|x|)$, $|y - x| \\geq \\lambda_1(x)$ and we obtain, using \\eqref{Eq:B2-1} and \\eqref{aa-0Ex}, that\n\\begin{equation}\n(v_k)_{x,\\lambda}(y) \\leq \\frac{\\lambda^{n-2} c_1(x)}{\\lambda_1(x)^{n-2}} \\leq \\frac{(2\\lambda)^{n-2} c_1(x) (1 + 3|x|)^{n-2}}{\\lambda_1(x)^{n-2} c_0} v_k(y).\n\t\\label{Eq:B4-1}\n\\end{equation}\nLetting\n\\[\n\\lambda^{(0)}(x) = \\min \\Big\\{ \\lambda_1(x), \\frac{\\lambda_1(x)}{2(1 + 3|x|)}\\Big[\\frac{c_1(x)}{c_0}\\Big]^{\\frac{1}{n-2}}\\Big\\} \\leq \\lambda_1(x),\n\\]\nwe derive from \\eqref{Eq:B3-1} and \\eqref{Eq:B4-1} that\n\\begin{equation}\n(v_k)_{x,\\lambda} \\leq \n v_k \\ \\ \\ \\mbox{in}\\\n B_{R_k}(0) \\setminus B_{\\lambda_1(x)}(x) \n\\text{ and } 0 < \\lambda < \\lambda^{(0)}(x).\n\\label{bound3}\n\\end{equation}\nLemma \\ref{lem-1new} follows from\n\\eqref{bound2} and \\eqref{bound3}.\n\\end{proof}\n\n\n\\bigskip\n\nDefine, \n for $x\\in {\\mathbb R}^n$ and $|x|\\le  R_k\/5$,\nthat \n\\[\n\\bar \\lambda_k(x)\n=\\sup\\Big\\{0<\\mu\\le \\frac {R_k}5\\  |\\\n(v_k)_{x,\\lambda} \\le v_k\\\n\\mbox{in}\\ B_{R_k}(0)\\setminus B_\\lambda(x), \\forall 0<\\lambda<\\mu\\Big\\}.\n\\]\n\n\n\\bigskip\n\nBy Lemma \\ref{lem-1new},\n$$\\bar\\lambda(x):=\n\\displaystyle{\n\\liminf_{k\\to\\infty}\\bar\\lambda_k(x)\n}\n\\in [\\lambda^{(0)}(x), \\infty], \\qquad x\\in {\\mathbb R}^n.\n$$\n\nBy \\eqref{Eq:vSuperhar},\n $$\n\\alpha:= \\liminf_{|y|\\to \\infty}\n|y|^{n-2}v(y)\\in (0, \\infty].\n$$\n\n\n\\begin{lem}\\  Assume \\eqref{01weak}-\\eqref{0304weak}. Then either $v$ is constant or\n$$\n\\alpha=\\lim_{|y|\\to \\infty}\n|y|^{n-2}v_{x,\\bar\\lambda(x)}(y)=\n\\bar \\lambda(x)^{n-2}v(x)<\\infty,\\qquad\n\\forall\\ x\\in {\\mathbb R}^n.\n$$\n\\label{lem-0.3}\n\\end{lem}\n\n\n\\begin{proof}\n\n\\noindent{\\bf Step 1.}\\ If $\\bar \\lambda(x)<\\infty$ for some $x\\in {\\mathbb R}^n$,\nthen \n$$\n\\alpha= \\lim_{|y|\\to \\infty}\n|y|^{n-2}v_{x,\\bar\\lambda(x)}(y)\n=\\bar \\lambda(x)^{n-2}v(x)\n<\\infty.\n$$\n\n\nSince\n $\\bar \\lambda(x)<\\infty$,\nwe have,   along a subsequence, $\\bar \\lambda_k(x)\\to \\bar \\lambda(x)$ ---\nbut for simplicity, we still use $\\{\\bar \\lambda_k(x)\\}$,\n $\\{v_k\\}$, etc to denote the\nsubsequence.\nBy the definition of $\\bar\\lambda_k(x)$, we have\n\\begin{equation}\n(v_k)_{x,\\bar \\lambda_k(x)} \\le v_k\\\n\\mbox{in}\\ B_{R_k}(0)\\setminus B_{\\bar \\lambda_k(x)}(x), \n\\label{vkeq}\n\\end{equation}\nBy the conformal invariance \\eqref{Eq:CIProp}, \n$(v_k)_{x,\\bar \\lambda_k(x)}$ satisfies\n\\begin{equation}\nf(\\lambda(A^{(v_k)_{x,\\bar \\lambda_k(x)}}))=1,\n\\quad \\ \\ \\mbox{in}\\ B_{R_k}(0)\\setminus \\overline{  B_{\\bar \\lambda_k(x)}(x) }.\n\\label{vkeqnew}\n\\end{equation}\nUsing  \\eqref{ve}, \\eqref{vkeq}, \\eqref{vkeqnew}, the definition\nof $\\bar \\lambda_k(x)$,\nand using the ellipticity of the\nequation satisfied by $v_k$ and $(v_k)_{x,\\bar \\lambda_k(x)}$,\nwe can apply \nthe strong maximum principle and Hopf Lemma to infer the existence of \nsome $y_k\\in \\partial B_{R_k}(0)$ such that\n$$\n(v_k)_{x,\\bar \\lambda_k(x)}(y_k)=v_k(y_k)\n$$\n--- see the proof of \\cite[Lemma 4.5]{LiLi05}.\n\nIt follows \nthat\n$$\n\\lim_{k\\to\\infty} |y_k|^{n-2} v_k(y_k)\n= \\lim_{k\\to\\infty} |y_k|^{n-2} (v_k)_{x,\\bar \\lambda_k(x)}(y_k)\n=(\\bar \\lambda(x))^{n-2}v(x).\n$$\nThis implies, in view of \\eqref{aa-0}, that\n$$\n\\alpha\\le \\lim_{k\\to\\infty} |y_k|^{n-2} v(y_k)=\\bar \\lambda(x)^{n-2}v(x)\n=\\lim_{|y|\\to \\infty}\n|y|^{n-2}v_{x,\\bar\\lambda(x)}(y)<\\infty.\n$$\n\nOn the other hand, if $\\hat y_i$ is such that\n$|\\hat y_i|\\to \\infty$ and\n$$\n\\alpha= \\lim_{i\\to\\infty} |y_i|^{n-2}v(y_i),\n$$\nthen, since \n$v_{ x, \\bar \\lambda(x) }\\le v$\nin ${\\mathbb R}^n\\setminus B_{\\bar \\lambda(x) }(x)$, we have\n$$\nv(y_i)\\ge \\frac {   \\bar \\lambda(x) ^{n-2} }\n{  |y_i-x|^{n-2}  }\nv\\left(x+\\frac {\\bar\\lambda(x)^2(y_i-x) }{   |y_i-x|^2  }\\right).\n$$\nThis gives\n$$\n\\alpha= \\lim_{i\\to\\infty} |y_i|^{n-2}v(y_i)\n\\ge \\bar \\lambda(x)^{n-2}v(x).\n$$\n\nStep 1 is established.\n\n\n\\medskip\n\n\\noindent{\\bf Step 2.}\\  It remains to show that either $v$ is constant or, for every $x\\in {\\mathbb R}^n$,\n$\\bar \\lambda(x)<\\infty$.\n\n\\medskip\n\nTo this end, we show that if $\\bar \\lambda(x)=\\infty$ for some $x \\in {\\mathbb R}^n$, then $v$ is constant. Indeed, assume that  $\\bar \\lambda_k(x)\n\\to \\infty$ as $k\\to\\infty$.\nWe easily derive from this and the convergence of $v_k$ to $v$ that\n\\begin{equation}\nv_{x,\\lambda}\\le v\\quad \\mbox{in}\\\n{\\mathbb R}^n\\setminus B_\\lambda(x)\\ \\ \\forall\\ \\lambda>0.\n\\label{vx}\n\\end{equation}\nThe above is equivalent to the property that for every\nfixed unit vector $e$, \n$r^{ \\frac{n-2}2 }v(x+ re)$ is non-decreasing\nin $r$.  Thus\n$$\nr^{n-2} \\min_{ \\partial B_r(x) }v\\ge \nr^{ \\frac {n-2}2 }\n\\min_{ \\partial B_1(x) }v\\quad \\forall\\ r\\ge 1.\n$$\nIn particular,\n$\\alpha=\\liminf_{ |y|\\to \\infty} |y|^{n-2}v(y)=\\infty$.\nThis implies, by Step 1, that \n$\\bar\\lambda(x)=\\infty$ for {\\it every} $x\\in {\\mathbb R}^n$, and therefore\n\\eqref{vx} holds for {\\it every} $x\\in {\\mathbb R}^n$.\nThis implies that\n$v$ is a constant, see Corollary \\ref{lem-app1}.\n\\end{proof}\n\n\n\\begin{lem}\nAssume \\eqref{01weak}-\\eqref{0304weak} and \\eqref{Eq:ActaSIdiag}. Then the function $v$ in Theorem \\ref{proposition1} cannot be constant.\n\\end{lem}\n\n\\begin{proof}\nFix some $t > 0$ for the moment. Set $\\varphi(x) = v(0) - t\\,|x|^2$ and fix some $r > 0$ such that $\\varphi > 0$ in $B_{r}(0)$ and $\\varphi < v_k$ on $\\partial B_r(0)$ for all sufficiently large $k$. Let\n\\[\n\\gamma_k = \\sup_{B_r(0)} (\\varphi - v_k) \\text{ and } \\varphi_k = \\varphi - \\gamma_k.\n\\]\nThen $\\varphi_k \\leq v_k$ in $B_r(0)$ and $\\varphi_k(x_k) = v_k(x_k)$ for some $x_k \\in \\bar B_r(0)$. Noting that\n\\[\n\\gamma_k = (\\varphi - v)(x_k) - (v_k - v)(x_k)    = -t|x_k|^2 - (v_k - v)(x_k) \n\\]\nand\n\\[\n\\lim_{k \\rightarrow \\infty} \\gamma_k = \\sup_{B_r(0)} (\\varphi - v) = 0\n\\]\nwe deduce that $x_k \\rightarrow 0$. This leads to\n\\[\n\\varphi_k(x_k) = v_k(x_k), \\nabla \\varphi_k(x_k) = \\nabla v_k(x_k), \\nabla^2\\varphi_k(x_k) \\leq \\nabla^2 v_k(x_k)\n\\]\nand\n\\[\nA^{\\varphi_k}(x_k) \\geq A^{v_k}(x_k).\n\\]\nNoting that there is some $C > 0$ independent of $\\delta$ and $k$ such that, for large $k$, \n\\[\nA^{\\varphi_k}(x_k) \\leq \\Big(\\frac{4}{n-2}v(0)^{-\\frac{n+2}{n-2}}t + C\\delta\\Big)I.\n\\]\nThus, we can select $t$ and $\\delta$ such that\n\\[\nA^{v_k}(x_k) \\leq A^{\\varphi_k}(x_k) \\leq t_0\\,I.\n\\]\nwhere $t_0$ is the constant in \\eqref{Eq:ActaSIdiag}. Since $f(\\lambda(A^{v_k}(x_k))) = 1$, this contradicts \\eqref{02weakX}, \\eqref{0304weak} and \\eqref{Eq:ActaSIdiag}.\n\\end{proof}\n\nRecall that\n$0<v\\in C^{0,1}_{loc}(\\mathbb R^n)$,\n$\\Delta v\\le 0\\ \\mbox{in}\\ \\mathbb R^n$,\nand it remains to consider the case that, for every $x\\in\\mathbb R^n$, there exists $0<\\bar\\lambda(x)<\\infty$ such that\n\\[\nv_{x,\\bar\\lambda(x)}(y)\\le v(y),\\ \\forall\\ |y-x|\\ge \\bar\\lambda(x),\n\\]\nand\n\\[\n\\lim_{|y|\\to \\infty}\n|y|^{n-2}v_{x,\\bar\\lambda(x)}(y)=\n\\alpha:=\\liminf_{|y|\\to\\infty}|y|^{n-2}v(y)<\\infty.\n\\]\n\n\nIf $v$ is in $C^2(\\mathbb R^n)$, the conclusion of Theorem \\ref{proposition1}\nfollows from the proof of theorem 1.3 in \\cite{LiLi05}.\nAn observation made in \\cite{Li07-ARMA} easily\nallows the proof to hold for $v\\in C^{0,1}_{loc}(\\mathbb R^n)$.\nFor readers' convenience, we outline the proof below.\n\n\n\nLet\n$\\psi(y)=\\frac y{|y|^2}$. We denote\n$$\nv_\\psi := |J_\\psi|^{ \\frac {n-2}{2n}} v\\circ \\psi.\n$$\nWe know that\n$\\lambda(A^{ v_\\psi }(y))=\\lambda(A^v (\\psi (y))$. Namely, $A^{ v_\\psi }(y)$ and $ A^v (\\psi (y))$ differ only\nby an orthogonal conjugation.\n\nIntroduce\n\\[\nw^{(x)}:=(v_{x,\\bar\\lambda(x)})_\\psi,\\ x\\in {\\mathbb R}^n.\n\\]\nWe deduce from the above properties of $v$ that\nfor every $x\\in \\mathbb R^n$, there exists some $\\delta(x)>0$ such that\n\\[\nv_\\psi\\ge w^{(x)}\\ \\mbox{in}\\ B_{\\delta(x)}(0)\\setminus\\{0\\},\n\\]\n\\[\nw^{(x)}(0)=\\alpha=\\liminf_{y\\to 0}v_\\psi(y),\n\\]\n\\[\n\\Delta v_\\psi\\le 0,\\ \\mbox{in}\\ B_{\\delta(x)}(0)\\setminus\\{0\\},\n\\]\n\n Let\n$D=\\{x\\in {\\mathbb R}^n\\ |\\ v\\ \\mbox{is differentiable at}\n \\ x\\}$.  Since $v\\in C^{0,1}_{loc}({\\mathbb R}^n)$, the  Lebesgue measure\nof $\\mathbb R^n\\setminus D$ is $0$.\nIt is clear that \n $w^{(x)}(y)$ is differentiable at $y=0$\nif $v$ is differentiable at $x$.\n\nBy \\cite[Lemma 4.1]{LiLi05}, \n\\[\n\\nabla w^{(x)}(0)=\\nabla w^{(\\tilde x)}(0),\\ \\ \\forall\\ x, \\tilde x\\in D.\n\\]\nNamely, for some $V\\in {\\mathbb R}^n$,\n\\[\n\\nabla w^{(x)}(0)=V,\\ \\forall\\ x\\in D.\n\\]\n\nA calculation yields\n\\[\n\\nabla w^{(x)}(0)=(n-2)\\alpha x+\\alpha^{ \\frac n{n-2} }\nv(x)^{ \\frac n{n-2} }\n\\nabla v(x).\n\\]\n Thus\n\\[\n\\nabla_x\\big( \\frac {n-2}2 \\alpha^{ \\frac n{n-2} }\nv(x) ^{ -\\frac 2{n-2} }\n- \\frac {n-2}2 \\alpha |x|^2 +V\\cdot x \\big)=0,\\ \\forall\\ x\\in D.\n\\]\n\n Consequently, for some $\\bar x\\in {\\mathbb R}^n$ and $d\\in {\\mathbb R}$,\n\\[\nv(x) ^{ -\\frac 2{n-2} }\n\\equiv\n\\alpha^{ -\\frac 2{n-2} }\n|x-\\bar x|^2+d \\alpha^{ -\\frac 2{n-2} }.\n\\]\n\n Since $v>0$, we must have $d>0$, so\n\\[\nv(x)\\equiv \\big( \\frac {  \\alpha^{ \\frac 2{n-2} }  }\n{d+|x-\\bar x|^2}\\big) ^{ \\frac {n-2}2 }.\n\\]\nWe have proved that $v$ is of the form\n\\eqref{vform}\nfor some $\\bar x\\in {\\mathbb R}^n$ and some positive constants\n$a$ and $b$. \n\nTo finish the proof, we show that $f(2b^2a^{-2}, \\ldots, 2b^2a^{-2}) = 1$ when \\eqref{Eq:ActaSIdiag} and \\eqref{02weakY} are in effect. For $\\delta > 0$, let\n\\[\nv_\\delta(x) = v(x) - \\delta|x|^2.\n\\]\nSince $v_k \\rightarrow v$ in $C^0(\\bar B_\\delta(0))$, there exists $\\beta_k \\rightarrow 0$ and $x_k \\rightarrow 0$ such that $\\hat v_k := v_\\delta + \\beta_k$ satisfies\n\\[\n(v_k - \\hat v_k)(x_k) = 0 \\text{ and } v_k - \\hat v_k \\geq 0 \\text{ near } x_k.\n\\]\nWe have $A^{\\hat v_k}(x_k) \\geq A^{v_k}(x_k)$. Therefore, by \\eqref{02weakY}, $\\lambda(A^{\\hat v_k}(x_k)) \\in \\Gamma$, and by \\eqref{0304weak}, \n\\begin{equation}\nf(\\lambda(A^{\\hat v_k}(x_k))) \\geq f(\\lambda(A^{v_k}(x_k))) = 1.\n\t\\label{Eq:13IV2016}\n\\end{equation}\nNoting that $A^{\\hat v_k}(x_k) \\rightarrow 2b^2a^{-2}I$ as $\\delta \\rightarrow 0, k \\rightarrow \\infty$, we infer that $2b^2a^{-2} > t_0$. (Indeed, if $2b^2a^{-2} \\leq t_0$, then, for small $\\rho > 0$, we have $A^{\\hat v_k}(x_k) < (t_0 + \\rho)I$ for small $\\delta$ and large $k$, which implies, by \\eqref{Eq:13IV2016}, \\eqref{02weakX} and \\eqref{0304weak}, that $1 \\leq f(\\lambda(A^{\\hat v_k}(x_k)) < f(t_0 + \\rho, \\ldots, t_0 + \\rho)$, which contradicts \\eqref{Eq:ActaSIdiag}.) In view of \\eqref{02weakY}, this implies that $(2b^2a^{-2}, \\ldots, 2b^2a^{-2}) \\in \\Gamma$. We can now send $k \\rightarrow \\infty$ and then $\\delta \\rightarrow 0$ in \\eqref{Eq:13IV2016} to obtain\n\\[\nf(\\lambda(A^v(0))) \\geq 1, \\text{ i.e. } f(2b^2a^{-2}, \\ldots, 2b^2a^{-2}) \\geq 1.\n\\]\n\nUsing $v^\\delta(x) = v(x) + \\delta|x|^2$ instead of $v_\\delta$ and the fact that $(2b^2a^{-2}, \\ldots, 2b^2a^{-2}) \\in \\Gamma$, one can easily derive\n\\[\nf(2b^2a^{-2}, \\ldots, 2b^2a^{-2}) \\leq 1.\n\\]\nTheorem \\ref{proposition1} is established.\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{proposition1-deg}}\\label{sec:Tp1deg}\n\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{proposition1-deg}]\nWe start with some preparation as in the proof of Theorem \\ref{proposition1}. We may assume that $R_k \\geq 5$ for all $k$.\n\nBy hypotheses, $v_k$ is super-harmonic and positive on ${\\mathbb R}^n \\setminus \\{0\\}$. Therefore, $v_*$ is super-harmonic and non-negative on ${\\mathbb R}^n \\setminus \\{0\\}$. Hence either $v_* \\equiv 0$ or $v_* > 0$ in ${\\mathbb R}^n \\setminus \\{0\\}$. In the former case we are done. We assume henceforth that the latter holds.\n\n\nNow, for every $\\beta>2$, there exists some positive constant \n$C(\\beta)$, independent of $k$, such that \n$C(\\beta)^{-1} \\leq M_k v_k\\le C(\\beta)$ in $B_\\beta(0) \\setminus B_{1\/\\beta}(0)$. It follows from Theorem \\ref{SoftGEst}\nthat $|\\nabla \\ln v_k|\n\\le C'(\\beta)$ in $B_{\\beta\/2}(0) \\setminus B_{2\/\\beta}(0)$.\nIt follows, after passing to a subsequence, that for every $0<\\alpha<1$,\n$M_k v_k\\to v_*$ in $C^{\\alpha}_{loc}({\\mathbb R}^n \\setminus \\{0\\})$, $v_*\\in C^{0,1}_{loc}({\\mathbb R}^n \\setminus \\{0\\})$ and\n\\begin{equation}\n|\\nabla \\ln v_*|\n\\le C'(\\beta) \\text{ in } B_{\\beta\/2}(0) \\setminus B_{2\/\\beta}(0).\n\t\\label{Eq:v*Harnack}\n\\end{equation}\n\nBy the super-harmonicity and the positivity of $v_*$, we can find $c_0 > 0$ such that \n\\begin{equation}\nv_*(y)\\ge\n2c_0 |y|^{2-n} ,\\ \\ \\forall\\ |y|\\ge 1.\n\t\\label{Eq:vSuperhardeg}\n\\end{equation}\nHence, passing to a subsequence and shrinking $R_k$ and $c_0 > 0$ if necessary, we can assume without loss of generality that, for all $k$,\n\\begin{equation}\n|M_kv_k(y)-v_*(y)|\\le (R_k)^{-n},\\quad\n\\ \\ \\forall\\ R_k^{-1}\\le |y|\\le R_k\n\\label{aadeg-0}\n\\end{equation}\nand\n\\begin{equation}\nM_kv_k(y) \\geq c_0(1+|y|)^{2-n} \\ \\ \\forall 0< |y| < R_k.\n\\label{aadeg-0Ex}\n\\end{equation}\n\n\nDenote\n$$\n(v_k)_{x,\\lambda}(y):=(\\frac \\lambda {|y-x|})^{n-2}v_k(x+\n\\frac {\\lambda^2(y-x)} { |y-x|^2}),\n$$\n the Kelvin transformation of $v_k$.\nWe use $(v_k)_\\lambda$ to denote $(v_k)_{0, \\lambda}$.\n\n\\begin{lem} \\label{lem-1deg}\nUnder the hypotheses of Theorem \\ref{proposition1-deg},\nthere \n exists a function $\\lambda^{(0)}: \n{\\mathbb R}^n \\setminus \\{0\\} \\rightarrow (0,\\infty)$ such that $\\lambda^{(0)}(x) \\leq |x|$ and, for all $k$,\n$$\n(v_k)_{x,\\lambda} \\le v_k\\\n\\mbox{in}\\ B_{R_k}(0)\\setminus (B_\\lambda(x) \\cup \\{0\\}), \\forall \\ 0<\\lambda<\n\\lambda^{(0)}(x)\\\n\\mbox{and}\\  |x|\\le \\frac {R_k}5.\n$$\n\\end{lem}\n\n\\begin{proof}\nWe adapt the proof of Lemma \\ref{lem-1new}. For $0 < |x| < \\frac{R_k}{5}$, we have, by \\eqref{aadeg-0} and \\eqref{aadeg-0Ex},  for all $k$ that\n\\begin{equation}\n\\frac{1}{c_1(x)}\\le  M_k v_k\\le c_1(x)\\ \\mbox{in}\\ B_{4r_1(x)}(x) \\subset B_{R_k}(0),\n\\label{bound1-deg}\n\\end{equation}\nwhere\n\\begin{align*}\nr_1(x) = \\frac{1}{8}|x| \\text{ and } c_1(x) = \\max\\Big\\{1 + \\sup_{B_{4r_1(x)}(x)} v_*, \\frac{1}{c_0}(1 + 2|x|)^{n-2}\\Big\\}.\n\\end{align*}\n\nBy Theorem \\ref{SoftGEst} and \\eqref{bound1-deg}, there exists $c_2(x)>0$, independent of $k$,  such that\n$$\n|\\nabla \\ln v_k|\\le c_2(x)\\ \\mbox{in}\\ B_{2r_1(x)}(x).\n$$\nThus, by Lemma \\ref{lem-1}, we can find $0 < \\lambda_1(x) < r_1(x)$ independent of $k$ such that\n\\begin{equation}\n(v_k)_{x,\\lambda} \\leq v_k \\text{ in } B_{\\lambda_1(x)}(x) \\setminus (B_\\lambda(x) \\cup \\{0\\}) \\text{ for all } 0 < \\lambda < \\lambda_1(x).\n\t\\label{Eq:E2-1}\n\\end{equation}\nFor $0 < \\lambda < \\lambda_1(x)$, we have, using \\eqref{bound1-deg}, that\n\\begin{equation}\n(v_k)_{x,\\lambda}(y) \\leq \\frac{\\lambda^{n-2}c_1(x)}{|y - x|^{n-2}} \\text{ for } y \\in B_{R_k}(0) \\setminus (B_{\\lambda}(x) \\cup\\{0\\}).\n\t\\label{Eq:E3-1}\n\\end{equation}\nFor $y \\in B_{R_k}(0) \\setminus (B_{1 + 4|x|}(x) \\cup \\{0\\})$, we have $\\frac{1}{2}(1 + |y|) \\leq |y - x|$ and we obtain, using \\eqref{Eq:E3-1} and \\eqref{aadeg-0Ex}, that\n\\begin{equation}\n(v_k)_{x,\\lambda}(y) \\leq \\frac{(2\\lambda)^{n-2} c_1(x)}{c_0} v_k(y).\n\t\\label{Eq:E3-2}\n\\end{equation}\nFor $y \\in B_{1 + 4|x|}(x) \\setminus  (B_{\\lambda_1(x)}(x) \\cup \\{0\\})$, we have $1 + |y| \\leq 2(1+3|x|)$, $|y - x| \\geq \\lambda_1(x)$ and we obtain, using \\eqref{Eq:E3-1} and \\eqref{aadeg-0Ex}, that\n\\begin{equation}\n(v_k)_{x,\\lambda}(y) \\leq \\frac{(2\\lambda)^{n-2} c_1(x) (1 + 3|x|)^{n-2}}{c_0 \\lambda_1(x)^{n-2}} v_k(y).\n\t\\label{Eq:E4-1}\n\\end{equation}\n\nLetting\n\\[\n\\lambda^{(0)}(x) = \\min \\Big\\{ \\lambda_1(x), \\frac{\\lambda_1(x)}{2(1+3|x|)} \\Big[\\frac{c_1(x)}{c_0}\\Big]^{\\frac{1}{n-2}}\\Big\\} \\leq \\lambda_1(x),\n\\]\nwe see that the conclusion of Lemma \\ref{lem-1deg} follows from \\eqref{Eq:E2-1}, \\eqref{Eq:E3-2} and \\eqref{Eq:E4-1}.\n\\end{proof}\n\nDefine, \n for $0 < |x|\\le  R_k\/5$,\nthat \n\\[\n\\bar \\lambda_k(x)\n=\\sup\\Big\\{0<\\mu\\le \\min(|x|,\\frac {R_k}5)\\  |\\\n(v_k)_{x,\\lambda} \\le v_k\\\n\\mbox{in}\\ B_{R_k}(0)\\setminus (B_\\lambda(x) \\cup \\{0\\}), \\forall 0<\\lambda<\\mu\\Big\\}.\n\\]\n\n\n\\bigskip\n\nBy Lemma \\ref{lem-1deg},\n$$\\bar\\lambda(x):=\n\\displaystyle{\n\\liminf_{k\\to\\infty}\\bar\\lambda_k(x)\n}\n\\in [\\lambda^{(0)}(x), |x|], \\qquad x\\in {\\mathbb R}^n \\setminus \\{0\\}.\n$$\nClearly,\n\\[\n(v_*)_{x,\\bar\\lambda(x)} \\leq v_* \\text{ in } {\\mathbb R}^n \\setminus (B_{\\bar\\lambda(x)}(x) \\cup \\{0\\}) \\text{ for all } x \\in {\\mathbb R}^n \\setminus \\{0\\}.\n\\]\n\nWe have a dichotomy:\n\\begin{align}\n\\text{ either } &\\bar \\lambda(x) = |x| \\text{ for all }x \\in {\\mathbb R}^n \\setminus \\{0\\},\n\t\\label{Eq:Casei}\\\\\n\\text{ or } &\\bar\\lambda(x_0) < |x_0| \\text{ for some }x_0 \\in {\\mathbb R}^n \\setminus \\{0\\}.\n\t\\label{Eq:Caseii}\n\\end{align}\nIn case \\eqref{Eq:Casei}, we obtain that $v_*$ is radially symmetric about the origin thanks to Lemma \\ref{lem-app1Sing}. To finish the proof, we assume in the rest of the argument that \\eqref{Eq:Caseii} holds and derive a contradiction.\n\n\nWe first collect some properties of $\\bar\\lambda(x)$. We start with an analogue of Lemma \\ref{lem-0.3}. By \\eqref{Eq:vSuperhardeg}, let\n $$\n\\alpha:= \\liminf_{|y|\\to \\infty}\n|y|^{n-2}v_*(y)\\in (0, \\infty].\n$$\n\n\n\\begin{lem}\\  Under the hypotheses of Theorem \\ref{proposition1-deg}, if $\\bar \\lambda(x)< |x|$ for some $x\\in {\\mathbb R}^n \\setminus \\{0\\}$,\nthen \n$$\n\\alpha= \\lim_{|y|\\to \\infty}\n|y|^{n-2}(v_*)_{x,\\bar\\lambda(x)}(y)\n=\\bar \\lambda(x)^{n-2}v_*(x)\n<\\infty.\n$$\n\\label{lem-0.3deg}\n\\end{lem}\n\n\\begin{proof} We adapt Step 1 in the proof of Lemma \\ref{lem-0.3}. Assume that $\\bar \\lambda(x)<|x|$ and (without loss of generality) that $\\bar \\lambda_k(x)\\to \\bar \\lambda(x)$. Arguing as before but using the strong maximum principle for solutions with isolated singularities \\cite[Theorem 1.6]{Li06-JFA} instead of the standard strong maximum principle, this leads to the existence of \nsome $y_k\\in \\partial B_{R_k}(0)$ such that\n$$\n(v_k)_{x,\\bar \\lambda_k(x)}(y_k)=v_k(y_k).\n$$\n\nIt follows \nthat\n$$\n\\lim_{k\\to\\infty} |y_k|^{n-2} M_k v_k(y_k)\n= \\lim_{k\\to\\infty} |y_k|^{n-2} (M_k v_k)_{x,\\bar \\lambda_k(x)}(y_k)\n=(\\bar \\lambda(x))^{n-2}v_*(x).\n$$\nThis implies, in view of \\eqref{aadeg-0}, that\n$$\n\\alpha\\le \\lim_{k\\to\\infty} |y_k|^{n-2} v_*(y_k)=\\bar \\lambda(x)^{n-2}v_*(x)\n=\\lim_{|y|\\to \\infty}\n|y|^{n-2}(v_*)_{x,\\bar\\lambda(x)}(y)<\\infty.\n$$\nOn the other hand, as in the proof of Lemma \\ref{lem-0.3}, we can use $(v_*)_{ x, \\bar \\lambda(x) }\\le v_*$\nin ${\\mathbb R}^n\\setminus (B_{\\bar \\lambda(x) }(x) \\cup \\{0\\})$ to show that \n$$\n\\alpha\n\\ge \\bar \\lambda(x)^{n-2}v_*(x).\n$$\nThe conclusion is readily seen.\n\\end{proof}\n\n\n\\begin{lem}\\label{Lem:lamCont}\nUnder the hypotheses of Theorem \\ref{proposition1-deg}, if $\\bar\\lambda(x_0) < |x_0|$ for some $x_0 \\in {\\mathbb R}^n \\setminus \\{0\\}$, then\n\\[\n\\limsup_{x \\rightarrow x_0} \\bar\\lambda(x) \\leq \\bar\\lambda(x_0).\n\\]\n\\end{lem}\n\n\\begin{proof} Along a subsequence, we have $\\bar\\lambda_k(x_0) \\rightarrow \\bar\\lambda(x_0)$.\n\nAs in the proof of Lemma \\ref{lem-0.3deg}, there exists $y_k\\in \\partial B_{R_k}(0)$ such that\n\\begin{equation}\n(v_k)_{x_0,\\bar \\lambda_k(x_0)}(y_k)=v_k(y_k).\n\t\\label{Eq:F2-1}\n\\end{equation}\n\nWe know that\n\\[\n\\eta_k := \\sup_{B_{|x_0|\/2}(x_0)} |M_k v_k - v_*| \\rightarrow 0 \\text{ as } k \\rightarrow \\infty.\n\\]\nLet $m$ denote the modulus of continuity of $v_*$ in $B_{|x_0|\/2}(x_0)$, i.e.\n\\[\nm(r) = \\sup \\Big\\{ |v_*(x) - v_*(y)|: x, y \\in B_{|x_0|\/2}(x_0), |x - y| < r\\Big\\}.\n\\]\n\nIn the computation below, we use $o(1)$ to denote quantities such that $$\\lim_{k \\rightarrow \\infty} o(1) = 0.$$\n\n\n\nFix some $\\delta > 0$ and consider $|x - x_0| < |x_0|\/2$. We note that\n\\[\n\\Big|\\Big(x + \\frac{(\\bar\\lambda_k(x_0) + \\delta)^2(y_k - x)}{|y_k - x|^2}\\Big) - \\Big(x_0 + \\frac{\\bar\\lambda_k(x_0)^2(y_k - x_0)}{|y_k - x_0|^2}\\Big)\\Big| = |x - x_0| + o(1).\n\\]\nThus, \n\\[\n\\Big|v_*\\Big(x + \\frac{(\\bar\\lambda_k(x_0) + \\delta)^2(y_k - x)}{|y_k - x|^2}\\Big) - v_*\\Big(x_0 + \\frac{\\bar\\lambda_k(x_0)^2(y_k - x_0)}{|y_k - x_0|^2}\\Big)\\Big| \\leq m(|x - x_0| + o(1)).\n\\]\nIt follows that\n\\begin{align*}\nM_k (v_k)_{x, \\bar\\lambda_k(x_0) + \\delta}(y_k)\n\t&= \\Big(\\frac{\\bar\\lambda_k(x_0) + \\delta}{|y_k - x|}\\Big)^{n-2} (M_kv_k)\\Big(x + \\frac{(\\bar\\lambda_k(x_0) + \\delta)^2(y_k - x)}{|y_k - x|^2}\\Big)\\\\\n\t&\\geq \\Big(\\frac{\\bar\\lambda_k(x_0) + \\delta}{|y_k - x|}\\Big)^{n-2} \\Big[(M_kv_k)\\Big(x_0 + \\frac{\\bar\\lambda_k(x_0) ^2(y_k - x_0)}{|y_k - x_0|^2}\\Big)\\\\\n\t\t&\\qquad\\qquad\\qquad -2\\eta_k - m(|x - x_0| + o(1))\\Big]\\\\\n\t&= \\Big(1 + \\frac{\\delta}{\\bar\\lambda_k(x_0)}\\Big)^{n-2} M_k (v_k)_{x_0, \\bar\\lambda_k(x_0)}(y_k)\\\\\n\t\t&\\qquad\\qquad - \\Big(\\frac{\\bar\\lambda_k(x_0) + \\delta}{|y_k - x|}\\Big)^{n-2}[2\\eta_k + m(|x - x_0| + o(1))].\n\\end{align*}\nRecalling \\eqref{Eq:F2-1}, we arrive at\n\\begin{align*}\nM_k (v_k)_{x, \\bar\\lambda_k(x_0) + \\delta}(y_k)\n\t&\\geq \\Big(1 + \\frac{\\delta}{\\bar\\lambda_k(x_0)}\\Big)^{n-2} M_k v_k(y_k)\\\\\n\t\t&\\qquad\\qquad - \\Big(\\frac{\\bar\\lambda_k(x_0) + \\delta}{|y_k|}\\Big)^{n-2}[o(1)+ m(|x - x_0| + o(1))].\n\\end{align*}\nThus, in view of \\eqref{aadeg-0Ex}, we can find small $\\bar\\epsilon > 0$ depending only on $\\delta$, $c$, $\\bar\\lambda(x_0)$ and the function $m(\\cdot)$ such that, for all $|x - x_0| < \\bar\\epsilon$ and for large $k$,\n\\begin{align*}\nM_k (v_k)_{x, \\bar\\lambda_k(x_0) + \\delta}(y_k)\n\t&\\geq \\Big(1 + \\frac{\\delta}{4\\bar\\lambda_k(x_0)}\\Big)^{n-2} M_k v_k(y_k).\n\\end{align*}\nThis implies that (cf. \\eqref{Eq:F2-1}), that\n\\[\n\\bar\\lambda_k(x) \\leq \\bar \\lambda_k(x_0) + \\delta \\text{ for all } |x - x_0| < \\bar\\epsilon \\text{ and large } k.\n\\]\nThe conclusion follows.\n\\end{proof}\n\nWe now return to drawing a contradiction from \\eqref{Eq:Caseii}. By Lemma \\ref{Lem:lamCont}, we infer from \\eqref{Eq:Caseii} that there exists some $r_0 > 0$ such that $\\bar\\lambda(x) < |x|$ for all $x \\in B_{r_0}(x_0)$. We can then argue as in the proof of Theorem \\ref{proposition1}, using Lemma \\ref{lem-0.3deg} instead of Lemma \\ref{lem-0.3} to obtain\n\\[\nv_*(x) = \\Big(\\frac{a}{1 +  b^2|x - \\bar x|^2}\\Big)^{\\frac{n-2}{2}} \\qquad x \\in B_{r_0}(x_0).\n\\]\nfor some $ \\bar x \\in {\\mathbb R}^n$ and some $a, b > 0$. For small $\\delta > 0$, let\n\\[\nv_*^\\delta(x) := v_*(x) + \\delta |x - x_0|^2.\n\\]\nSince $M_k v_* \\rightarrow v_*$ in $C^{0}(\\bar B_\\delta(x_0))$, there exists $\\beta_k \\rightarrow 0$ and $x_k \\rightarrow x_0$ such that the function $\\xi_{k,\\delta} := v_*^\\delta + \\beta_k$ satisfies\n\\[\n(M_k v_k - \\xi_{k,\\delta})(x_k) = 0 \\text{ and } M_k v_k - \\xi_{k,\\delta} \\leq 0 \\text{ near } x_k.\n\\]\nIt follows that\n\\[\nA^{v_k}(x_k) \\geq A^{\\frac{1}{M_k} \\xi_{k,\\delta}}(x_k) = M_k^{\\frac{4}{n-2}} A^{\\xi_{k,\\delta}}(x_k) .\n\\]\nOn the other hand, by hypothesis, there is some $\\lambda_* \\in \\Gamma$ such that $f(\\lambda_*) = 1$ (e.g. $\\lambda_* = \\lambda(A^{v_1}(0))$). By \\eqref{0304weak}, we can find $\\hat\\lambda_* \\in \\Gamma$ such that $f(\\hat \\lambda_*) > 1$. As $M_k \\rightarrow \\infty$ and $A^{\\xi_{k,\\delta}}(x_k) = 2b^{2}a^{-2}I + O(\\delta)$, we can find $k$ sufficiently large such that $M_k^{\\frac{4}{n-2}} A^{\\xi_{k,\\delta}}(x_k) > {\\rm diag}(\\hat\\lambda_*)$. We are thus led to\n\\[\nA^{v_k}(x_k) > {\\rm diag}(\\hat\\lambda_*).\n\\]\nAs $f(\\lambda(A^{v_k})) = 1$ and $f(\\hat \\lambda_*) > 1$, the above contradicts \\eqref{02weakX} and \\eqref{0304weak}.\n\\end{proof}\n\n\\section{Local gradient estimates} \\label{sec:ThmB}\n\nIn this section, we adapt the argument in \\cite{Li09-CPAM} to prove Theorem \\ref{TheoremB}.\n\nFor a locally Lipschitz function $w$ in $B_2(0)$, $0 < \\alpha < 1$, $x \\in B_2(0)$  and $0 < \\delta < 2 - |x|$, define\n\\begin{align*}\n[w]_{\\alpha,\\delta}(x) \n\t&= \\sup_{0 < |y - x| < \\delta} \\frac{|w(y) - w(x)|}{|y - x|^\\alpha},\\\\\n\\delta(w,x,\\alpha) \n\t&= \\left\\{\\begin{array}{ll}\n\t\\infty & \\text{ if } (2 - |x|)^\\alpha\\,[w]_{\\alpha,2 - |x|}(x) < 1,\\\\\n\t\\mu & \\text{ where } 0 < \\mu \\leq 2 - |x| \\text{ and } \\mu^\\alpha\\,[w]_{\\alpha,\\mu}(x) = 1\\\\\n\t\t& \\text{ if } (2 - |x|)^\\alpha\\,[w]_{\\alpha,2 - |x|}(x) \\geq 1.\n\\end{array}\\right.\n\\end{align*}\nNote that $\\delta(w,x,\\alpha)$ is well defined as $[w]_{\\alpha,\\delta}(x)$ is continuous and non-decreasing in $\\delta$. The object $\\delta(w,x,\\alpha)$ was introduced in \\cite{Li09-CPAM}. Its reciprocal $\\delta(w,x,\\alpha)^{-1}$ plays a role similar to that of $|\\nabla w(x)|$ in performing a rescaling argument for a sequence of functions blowing up in $C^\\alpha$-norms. For example, when $\\delta = \\delta(w,x,\\alpha) < \\infty$, the rescaled function $\\hat w(y) := w(x + \\delta y) - w(x)$ satisfies\n\\[\n\\hat w(0) = 0 \\text{ and } [\\hat w]_{\\alpha,1}(0) = \\delta^\\alpha[\\hat w]_{\\alpha,\\delta}(x) = 1.\n\\]\n\n\\begin{thm}\nTheorem \\ref{TheoremB} holds if we relax \\eqref{01}-\\eqref{04} to \\eqref{01weak}-\\eqref{Eq:ActaSIdiag}.\n\\end{thm}\n\n\\begin{proof} By the conformal invariance \\eqref{Eq:CIProp}, it suffices to show bound $|\\nabla \\ln v|$ in $B_{1\/4}(0)$. \n\nWe first claim that\n\\begin{equation}\n\\sup_{x \\neq y \\in B_{1\/2}(0)} \\frac{|\\ln v(x) - \\ln v(y)|}{|x - y|^\\alpha} \\leq \nC(\\Gamma, \\alpha) \\text{ for any } 0 < \\alpha < 1\n\t.\\label{11M16-1}\n\\end{equation}\n\nAssume otherwise that \\eqref{11M16-1} fails for some $0 < \\alpha < 1$. Then there exist $0 < v_i \\in C^2(B_2(0))$ such that $f(\\lambda(A^{v_i})) = 1$ and $v_i \\leq b$ in $B_2(0)$ but\n\\[\n\\sup_{x \\neq y \\in B_{1\/2}(0)} \\frac{|\\ln v_i(x) - \\ln v_i(y)|}{|x - y|^\\alpha} \\rightarrow \\infty.\n\\]\nThis implies that, for any fixed $0 < r < 1\/2$, \n\\[\n\\sup_{x \\in B_{1\/2}(0)} [\\ln v_i]_{\\alpha,r}(x) \\rightarrow \\infty \\text{ and } \\inf_{x \\in B_{1\/2}(0)} \\delta(\\ln v_i, x, \\alpha) \\rightarrow 0.\n\\]\nTherefore, there exists $x_i \\in B_{1}(0)$, \n\\[\n\\frac{1 - |x_i|}{\\delta(\\ln v_i, x_i, \\alpha)} = \\sup_{x \\in B_{1}(0)} \\frac{1 - |x|}{\\delta(\\ln v_i, x,\\alpha)} \\rightarrow \\infty.\n\\]\nLet $\\sigma_i = \\frac{1 - |x_i|}{2}$ and $\\epsilon_i = \\delta(\\ln v_i, x_i, \\alpha)$. Then\n\\begin{equation}\n\\frac{\\sigma_i}{\\epsilon_i} \\rightarrow \\infty, \\epsilon_i \\rightarrow 0,  \\text{ and } \\epsilon_i \\leq 2\\,\\delta(\\ln v_i,z,\\alpha) \\text{ for any } |z - x_i| \\leq \\sigma_i\n\t.\\label{11M16-1x}\n\\end{equation}\n\nWe now define\n\\[\n\\hat v_i(y) = \\frac{1}{v_i(x_i)}\\,v_i(x_i + \\epsilon_i\\,y) \\text{ for } |y| \\leq \\frac{\\sigma_i}{\\epsilon_i}.\n\\]\nThen\n\\begin{equation}\n[\\ln \\hat v_i]_{\\alpha,1}(0) = \\epsilon_i^\\alpha\\,[\\ln v_i]_{\\alpha,\\epsilon_i}(x_i) = 1\n\t.\\label{11M16-2}\n\\end{equation}\nAlso, by \\eqref{11M16-1x}, for any fixed $\\beta > 1$ and $|y| < \\beta$, there holds\n\\begin{align}\n[\\ln \\hat v_i]_{\\alpha,1}(y) \n\t&= \\epsilon_i^\\alpha\\,[\\ln v_i]_{\\alpha,\\epsilon_i}(x_i + \\epsilon_i\\,y)\\nonumber\\\\\n\t&\\leq 2^{-\\alpha}\\,\\epsilon_i^\\alpha \\Big\\{ \\sup_{|z - (x_i + \\epsilon_i y)| \\leq \\epsilon_i} [\\ln v_i]_{\\alpha,\\epsilon_i\/4}(z) + [\\ln v_i]_{\\alpha,\\epsilon_i\/4}(x_i + \\epsilon_i\\,y)\\Big\\}\\nonumber\\\\\n\t&\\leq  \\sup_{|z - (x_i + \\epsilon_i y)| \\leq \\epsilon_i} \\delta(\\ln v_i,z,\\alpha)^\\alpha\\,[\\ln v_i]_{\\alpha,\\delta(\\ln v_i,z,\\alpha)}(z)\\nonumber\\\\\n\t\t&\\qquad\\qquad + \\delta(\\ln v_i,x_i + \\epsilon_i\\,y,\\alpha)^\\alpha\\,[\\ln v_i]_{\\alpha,\\delta(\\ln v_i,x_i + \\epsilon_i\\,y,\\alpha)}(x_i + \\epsilon_i\\,y)\\nonumber\\\\\n\t&= 2\n\t\\label{11M16-3}\n\\end{align}\nfor all sufficiently large $i$. Since $\\hat v_i(0) = 1$ by definition, we deduce from \\eqref{11M16-2} and \\eqref{11M16-3} that\n\\begin{equation}\n\\frac{1}{C(\\beta)} \\leq \\hat v_i(y) \\leq C(\\beta) \\text{ for } |y| \\leq \\beta \\text{ and all sufficiently large $i$}\n\t.\\label{21F11-4}\n\\end{equation}\nWe can now apply Theorem \\ref{SoftGEst} to obtain\n\\begin{equation}\n|\\nabla \\ln \\hat v_i| \\leq C(\\beta) \\text{ in } B_{\\beta\/2}(0) \\text{ for all sufficiently large $i$}.\n\t\\label{21F11-7}\n\\end{equation}\nPassing to a subsequence and recalling \\eqref{11M16-1x} and \\eqref{21F11-4}, we see that $\\hat v_i$ converges in $C^{0,\\alpha'}$ ($\\alpha < \\alpha' < 1$) on compact subsets of ${\\mathbb R}^n$ to some positive, locally Lipschitz function $v_*$.\n\nOn the other hand, if we define\n\\[\n\\bar v_i(y) = \\epsilon_i^{\\frac{n-2}{2}}\\,v_i(x_i + \\epsilon_i\\,y) \\text{ for } |y| \\leq \\frac{\\sigma_i}{\\epsilon_i},\n\\]\nthen by the conformal invariance \\eqref{Eq:CIProp}, we have\n\\[\nf(\\lambda(A^{\\bar v_i})) = 1 \\text{ in } B_{\\sigma_i\/\\epsilon_i}(0).\n\\]\nSince $\\frac{\\sigma_i}{\\epsilon_i} \\rightarrow \\infty$, $\\hat v_i = M_i\\,\\bar v_i$ where $M_i = v_i(x_i)^{-1} \\epsilon_i^{-\\frac{n-2}{2}} \\rightarrow \\infty$ (thanks to the bound $v_i \\leq b$), we then conclude from Theorem \\ref{proposition1-deg} that $v_*$ is constant, namely\n\\[\nv_* \\equiv v_*(0) = \\lim_{i \\rightarrow \\infty} \\hat v_i(0) = 1.\n\\]\nThis contradicts \\eqref{11M16-2}, in view of \\eqref{21F11-7} and the convergence of $\\hat v_i$ to $v_*$. We have proved \\eqref{11M16-1}.\n\nFrom \\eqref{11M16-1}, we can find some universal constant $C > 1$ such that\n\\[\n\\frac{u(0)}{C} \\leq u \\leq C\\,u(0) \\text{ in } B_{1\/2}(0).\n\\]\nApplying Theorem \\ref{SoftGEst} again we obtain the required gradient estimate in $B_{1\/4}(0)$.\n\\end{proof}\n\n\\section{Fine blow-up analysis} \\label{sec:T4X}\n\n\\subsection{A quantitative centered Liouville-type result}\n\nIn this subsection, we establish: \n\\begin{prop}\\label{prop-C16-1new}\nLet $(f, \\Gamma)$  satisfy \n\\eqref{01weak}-\\eqref{0304weak}, \\eqref{Eq:ActaSIdiag}-\\eqref{02weakY}, \\eqref{FU-1a} and the normalization condition \\eqref{F1}.\nAssume that  for a sequence $R_k\\to\\infty$,  \n $0 < v_k \\in C^2(B_{R_k})$ satisfy\n\\begin{equation}\nf(\\lambda(A^{v_k}))(y)=1,\\ \\ 0<v_k(y)\\le v_k(0)=1,\n\\quad |y|\\le R_k.\n\\label{ab1new}\n\\end{equation}\n  Then for every \n$\\epsilon>0$, there exists a constant\n${\\delta_{0}} > 0$, depending only on $(f,\\Gamma)$ and $\\epsilon$, \nsuch that, for all sufficiently large $k$,\n\\begin{equation}\n|v_k(y)-U(y)|\\le 2 \\epsilon\nU(y),\\qquad\\forall\\\n|y|\\le {\\delta_{0}} R_k.\n\\end{equation}\n\\end{prop}\n\nRecall that $U = (1 + |x|^2)^{-\\frac{n-2}{2}}$, $A^U \\equiv 2I$ and $f(\\lambda(A^U)) = 1$ on ${\\mathbb R}^n$.\n\nProposition \\ref{prop-C16-1new} is equivalent to the following proposition.\n\n\\begin{prop}\\label{prop:BbUp1}\nLet $(f,\\Gamma)$ satisfy \\eqref{01weak}-\\eqref{0304weak}, \\eqref{Eq:ActaSIdiag}-\\eqref{02weakY}, \\eqref{FU-1a} and the normalization condition \\eqref{F1}. For any $\\epsilon > 0$ there exist ${\\delta_{0}}, {C_{0}} > 0$ depending only on $(f,\\Gamma)$ and $\\epsilon$  such that if $0 < u \\in C^2(B_{R}(0))$, $R > 0$, satisfies\n\\[\nf(\\lambda(A^u)) = 1 \\text{ in } B_{R}(0) \\text{ and } u(0) = \\sup_{B_R(0)} u \\geq {C_{0}}\\, R^{-\\frac{n-2}{2}},\n\\]\nthen\n\\[\n|u(x) - U^{0,u(0)}(x)| \\leq 2\\epsilon U^{0,u(0)}(x) \\text{ for all } x \\in B_{{\\delta_{0}} R}(0).\n\\]\n\\end{prop}\n\n\\begin{proof}[Proof of the equivalence between Proposition \\ref{prop-C16-1new} and Proposition \\ref{prop:BbUp1}] It is clear that Proposition \\ref{prop:BbUp1} implies Proposition \\ref{prop-C16-1new}.\n\nConsider the converse. Let ${\\delta_{0}} = {\\delta_{0}}(\\epsilon)$ be as in Proposition \\ref{prop-C16-1new}. Arguing by contradiction, we assume that there are some $\\epsilon > 0$ and a sequence of $R_k$ and $u_k \\in C^2(B_{R_k}(0))$ such that \n\\[\nf(\\lambda(A^{u_k})) = 1 \\text{ in } B_{R_k}(0) \\text{ and } u_k(0) = \\sup_{B_{R_k}(0)} u_k \\geq k\\,R_k^{-\\frac{n-2}{2}}\n\\]\nbut the last estimate in Proposition \\ref{prop:BbUp1} fails for each $k$.\n\n\nDefine\n\\[\n\\bar u_k(y) = \\frac{1}{u_k(0)} u_k\\Big(\\frac{y}{u_k(0)^{\\frac{2}{n-2}}}\\Big) \\text{ for } |y| \\leq R_k\\,u_k(0)^{\\frac{2}{n-2}} =: \\bar R_k.\n\\]\nThen $f(\\lambda(A^{\\bar u_k})) = 1$ in $B_{\\bar R_k}(0)$, $\\sup_{B_{\\bar R_k}(0)} \\bar u_k = \\bar u_k(0) = 1$, and $\\bar R_k \\geq k^{\\frac{2}{n-2}} \\rightarrow \\infty$. By Proposition \\ref{prop-C16-1new},\n\\[\n|\\bar u_k(y) - U(y)| \\leq 2\\epsilon U(y) \\text{ in } B_{{\\delta_{0}}\\,\\bar R_k}(0) \\text{ for all sufficiently large }k.\n\\]\nReturning to the original sequence $u_k$ we arrive at a contradiction.\n\\end{proof}\n\n\n\n\n\\begin{lem}\\label{lem0.1}\nUnder the hypotheses of Proposition \\ref{prop-C16-1new}\nexcept for \\eqref{FU-1a}, we have\n\\begin{equation}\nv_k\\to U,\\qquad \\mbox{in}\\ C^\\beta_{loc}({\\mathbb R}^n),\\ \\ \\forall\\ 0<\\beta<1.\n\\label{limit}\n\\end{equation}\nMoreover,  \n for  every $\\epsilon>0$, there exists   $k_0\\ge 1$\nsuch that\n\\begin{equation}\n\\min_{|y|=r}v_k(y)\\le (1+\\epsilon)U(r),\\qquad\\forall\\  0<r<\n R_k\/5, \\ k\\ge k_0.\n\\label{eq35}\n\\end{equation}\n\\end{lem}\n\n\n\\begin{proof}  We first prove \n\\eqref{limit}.  Since $v_k$ satisfies \\eqref{ab1new},\n  we deduce from \nTheorem \\ref{TheoremB} that\n$$\n|\\nabla \\ln v_k|\\le C\\quad\\mbox{in}\\ B_{ R_k-1},\\ \\ \\forall\\ k,\n$$\nwhere $C$ is independent of $k$.\nThis yields \\eqref{limit} in view of Theorem \\ref{proposition1}.\n\nWe now prove \\eqref{eq35}.\nSuppose the contrary, then there exists some $\\epsilon>0$ and  sequences of\n$k_i\\to \\infty$, $0<r_i<R_{k_i}\/5$ such that\n\\begin{equation}\nv_{k_i}> (1+\\epsilon)U\\ \\ \\ \\mbox{on}\\ \\partial B_{ r_i}.\n\\label{ri}\n\\end{equation}\nBecause of \\eqref{limit},\n $r_i\\to\\infty$.\n\nAs in the proof of Lemma \\ref{lem-1new},\nthere exists $\\lambda^{(0)}_i>0$ such that\n\\begin{equation}\n(v_{k_i})_\\lambda \\le v_{k_i}\\\n\\mbox{in}\\ B_{r_i}\\setminus B_\\lambda, \\forall 0<\\lambda<\n\\lambda^{(0)}_i\\\n\\mbox{and}\\  |x|\\le r_i.\n\\label{26}\n\\end{equation}\nBy the explicit expression of $U$, there exists some small $\\delta>0$ independent of $i$\nsuch that, for large $i$, \n$$\nU_\\lambda(y)\\le (1+\\frac \\epsilon 4)U(y),\\quad\n\\forall\\ y\\in \\partial B_{ r_i},\\ \n\\lambda^{(0)}_i \\le \\lambda\\le 1+\\delta,\n$$\nBy the uniform convergence of $v_{k_i}$ to $U$ on compact\nsubsets of ${\\mathbb R}^n$, we have, for large $i$,\n$$\n(v_{k_i})_\\lambda \\le (1+\\frac \\epsilon 2) \nU(y),\\quad\n\\forall\\ y\\in \\partial B_{ r_i},\\\n\\lambda^{(0)}_i \\le \\lambda\\le 1+\\delta,\n$$\nAs in the proof of Lemma \\ref{lem-1new}, the moving sphere \nprocedure does not stop before reaching $\\lambda=1+\\delta$, namely\nwe have, for large $i$, \n$$\n(v_{k_i})_\\lambda \\le v_{k_i}\\\n\\mbox{in}\\ B_{r_i}\\setminus B_\\lambda, \\forall 0<\\lambda<\n1+\\delta\\\n\\mbox{and}\\  |x|\\le r_i.\n$$\nSending $i$ to $\\infty$ leads to\n$$\nU_{1+\\delta}(y)\\le U(y),\\ \\ \\forall\\ 1+\\delta\\le |y|\\le 2.\n$$\nA contradiction --- since we see from the explicit expression of $U$ that\n$U_{1+\\delta}(y)>U(y)$\nfor all $1<1+\\delta< |y|\\le 2$.\n\\end{proof}\n\n\n\\begin{lem}\\label{lem-energy1}\nUnder the hypotheses of Proposition \\ref{prop-C16-1new}, \n for  any $\\epsilon > 0$, there exist a small $\\delta_1>0$ and a large  $r_1 > 1$,\ndepending only on $(f, \\Gamma)$ and $\\epsilon$,\nsuch that, for all sufficiently large $k$,\n\\begin{align} \n&v_k(y) \\ge (1-\\epsilon)U(y),\n\\qquad\\forall\\ |y|\\le \\delta_1 R_k,\n\\label{U00}\\\\\n\\text{ and }\\qquad &\\int_{ r_1\\le |y|\\le \\delta_1 R_k }\nv_k^{\\frac {n+2}{n-2}}\n\\le \\epsilon.\n\\label{energy1}\n\\end{align}\n\\end{lem}\n\n\\begin{proof} Assume without loss of generality that $\\epsilon \\in (0,1\/2)$. Since\n$v_k\\to U$ in $C^0_{loc}({\\mathbb R}^n)$, \nthere exist $r_2>1$ and $k_1$,  depending on $\\epsilon$, \n such that for all $k\\ge k_1$\n\\begin{align}\nv_k(y)&\\ge (1-\\epsilon^2)U(y),\\qquad\n\\forall\\ |y|\\le r_2,\n\\label{U10}\\\\\nv_k(y)&\\ge (1-\\epsilon^2)U(r_2)\\ge (1-2\\epsilon^2) r_2^{2-n},\n\\qquad\\forall\\ |y|=r_2.\n\\label{U0}\n\\end{align}\n\n\n\n\n \nBy \\eqref{FU-1a},\n$$\nTrace\\ (A^{v_k})\\ge \\delta > 0,\n$$\nand therefore\n\\begin{equation}\n-\\Delta v_k(y)\\ge \\frac {n-2}2 \\delta v_k(y) ^{ \\frac {n+2}{n-2} }\n\\qquad\\mbox{in}\\  r_2\\le \n |y| \\le R_k.\n\\label{U4}\n\\end{equation}\n\nUsing the superharmonicity of $v_k$\nand the maximum principle, we obtain\n$$\nv_k(y)\\ge \n(1-\\epsilon^2)\n\\left( |y|^{2-n}-\nR_k^{2-n}\\right),\\qquad r_2\\le |y|\\le R_k.\n$$\nThus, for any $\\delta_2 \\in (0,\\epsilon^{\\frac{2}{n-2}})$, we have for all sufficiently large $k$ that\n\\begin{equation}\nv_k(y)\\ge \n(1-\\epsilon^2)(1 - \\delta_2^{n-2}) \n|y|^{2-n} \\geq (1-2\\epsilon^2)\n|y|^{2-n},\\qquad\nr_2\\le |y|\\le \\delta_2 R_k.\n\\label{U5}\n\\end{equation}\nNow if $\\delta_1 < \\delta_2$, \\eqref{U00} is readily seen from \\eqref{U10} and \\eqref{U5}.\n\nLet\n$$\n\\hat v_k(y):= v_k(y)- (1-2\\epsilon^2)\n|y|^{2-n}.\n$$\nThen\n$$\n-\\Delta \n\\hat v_k(y)\\geq \\hat f(y):=\n\\frac {n-2}2 \\delta v_k(y) ^{ \\frac {n+2}{n-2} }\n\\qquad\\mbox{in}\\  r_2\\le \n |y| \\le \\delta_2 R_k,\n$$\nand\n$$\n\\hat v_k(y)\\ge 0,\\quad \\mbox{for}\\ y\\in \\partial (B_{\\delta_2 R_k}\\setminus \nB_{ r_2}).\n$$\nLet $R_k' = \\frac{\\delta_2R_k}{2}$. Enlarging $k_1$ if necessary, we can apply Corollary \\ref{cor-App1-4} \nin Appendix \\ref{Sec:AppA} to get\n\\begin{equation}\n\\min_{ |x|= R_k'}\n \\hat v_k(x)\n\\ge \nC^{-1} (\\delta_2 R_k)^{2-n}\n\\int_{ 2r_2\\le |y|\\le \\delta_2 R_k\/8} \\delta  v_k(y) ^{ \\frac {n+2}{n-2} }\ndy, \\qquad\n\\forall\\ k \\geq k_1,\n\\label{U6}\n\\end{equation}\nwhere here and below $C$ is some positive constant depending only on $n$. On the other hand, by Lemma \\ref{lem0.1}, we have (after enlarging $k_1$ if necessary) \n\\[\n\\min_{ |x|= R_k'} v_k(x) \\le(1+\\epsilon^2) U(R_k')\n\\le   (1+2\\epsilon^2)(R_k')^{2-n}, \\qquad  \\forall\\ k \\geq k_1,\n\\label{U7}\n\\]\nwhich implies that\n\\begin{equation}\n\\min_{ |x|= R_k'} \\hat v_k(x) \\le C\\,\\epsilon^2 (\\delta_2 R_k)^{2-n}, \\qquad  \\forall\\ k \\geq k_1.\n\\label{U7}\n\\end{equation} \nIt now follows from \\eqref{U6} and \\eqref{U7} that\n$$\n\\int_{ 2r_2\\le |y|\\le \\delta_2 R_k\/8} \\delta  v_k(y) ^{ \\frac {n+2}{n-2} }\ndy\n\\le \nc_1\\,\\epsilon^2\n$$\nwhere $c_1$ depends only on $n$. \\eqref{energy1} is then established for $\\epsilon \\leq \\frac{1}{c_1}$ with $r_1 = 2r_2$ and $\\delta_1 = \\delta_2\/8$. The conclusion for $\\epsilon > 1\/c_1$ also follows.\n\\end{proof}\n\n\n\n\\begin{lem}  Let $(f, \\Gamma)$ satisfy \\eqref{01weak}-\\eqref{0304weak}. \nThen there  exist $\\delta_3>0$ and $C_3>1$, depending only on $(f, \\Gamma)$, such that if $u\\in\nC^2(B_2(0))$ satisfies\n$$\nf(\\lambda(A^u))=1,   u>0,  \\qquad \\mbox{in}\\ B_2(0),\n$$\nand\n$$\n\\int_{ B_2(0)} u^{ \\frac {2n}{n-2} }\\le \\delta_3,\n$$\nthen \n$$\nu\\le C_3\\quad\\mbox{in}\\ B_1(0).\n$$\n\\label{lem-smallenergy}\n\nIf $(f,\\Gamma)$ satisfies in addition the conditions \\eqref{Eq:ActaSIdiag}, \\eqref{02weakY} and the normalization condition \\eqref{F1}, then $\\delta_3$ can be chosen to be any constant smaller than $\\int_{{\\mathbb R}^n} U^{\\frac{2n}{n-2}}\\,dx$.\n\\end{lem}\n\n\\begin{proof}\nWe adapt the proof of \\cite[Lemma 6.4]{LiLi03}. Arguing by contradiction, we can find a sequence of $0 < u_j \\in C^2(B_2)$ such that $f(\\lambda(A^{u_j})) = 1$ in $B_2(0)$,\n\\[\n\\int_{B_2(0)} u_j ^{ \\frac {2n}{n-2} } \\rightarrow 0\n\\]\nbut\n\\[\nd(y_j)^{\\frac{n-2}{2}}u_j(y_j) = \\max_{\\bar B_{3\/2}(0)} d(y)^{\\frac{n-2}{2}}u_j(y) \\rightarrow \\infty,\n\\]\nwhere $y_j \\in B_{3\/2}(0)$ and $d(y) = 3\/2 - |y|$.\n\nLet $\\sigma_j = \\frac{1}{2}d(y_j) > 0$,\n\\[\nv_j(z) = \\frac{1}{u_j(y_j)}u_j\\Big(y_j + \\frac{1}{u_j(y_j)^{\\frac{2}{n-2}}} z\\Big) \\text{ for } |z| < r_j := u_j(y_j)^{\\frac{2}{n-2}} \\sigma_j \\rightarrow \\infty.\n\\]\nThen by the conformal invariance property \\eqref{Eq:CIProp}, $f(\\lambda(A^{v_j})) = 1$ in $B_{r_j}(0)$, $v_j(0) = 1$, $v_j \\leq 2^{\\frac{n-2}{2}}$ in $B_{r_j}(0)$ and\n\\begin{equation}\n\\int_{B_{r_j}(0)} v_j^{\\frac{2n}{n-2}} \\rightarrow 0.\n\t\\label{Eq:SLcrit}\n\\end{equation}\n\nBy Theorem \\ref{TheoremB}, there is a constant $C$ independent of $j$ such that\n\\[\n|\\nabla \\ln v_j| \\leq C \\text{ in } B_{r_j\/2}(0).\n\\]\nThus, after passing to a subsequence, we can assume that $v_j$ converges in $C^0_{loc}({\\mathbb R}^n)$ to some positive function $v$ (as $v_j(0) = 1$). This contradicts \\eqref{Eq:SLcrit}.\n\nThe above argument can be adapted to prove the last assertion of the lemma: Equation \\eqref{Eq:SLcrit} is replaced by\n\\[\n\\int_{B_{r_j}(0)} v_j^{\\frac{2n}{n-2}} \\leq \\delta_3 <  \\int_{{\\mathbb R}^n} U^{\\frac{2n}{n-2}}\\,dx.\n\\]\nOn the other hand, by Theorem \\ref{proposition1}, we have $v_j \\rightarrow U$ in $C^0_{loc}({\\mathbb R}^n)$. This gives a contradiction.\n\\end{proof}\n\n\\begin{lem}\\label{lem:Ressmallenergy}\n\\  Let $(f, \\Gamma)$ satisfy \\eqref{01weak}-\\eqref{0304weak} and let $\\delta_3$, $C_3$ be as in Lemma \\ref{lem-smallenergy}. If $u\\in\nC^2(B_{2R}(0))$ satisfies\n$$\nf(\\lambda(A^u))=1,   u>0,  \\qquad \\mbox{in}\\ B_{2R}(0),\n$$\nand\n$$\n\\int_{ B_{2R(0)}} u^{ \\frac {2n}{n-2} }\\le \\delta_3,\n$$\nthen \n$$\nu\\le C_3\\,R^{-\\frac{n-2}{2}}\\quad\\mbox{in}\\ B_{R}(0).\n$$\n\\end{lem}\n\n\\begin{proof} This follows from Lemma \\ref{lem-smallenergy} and a change of variables, $\\tilde u(y) = R^{\\frac{n-2}{2}}u(Ry)$ for $|y| \\leq 2$.\n\\end{proof}\n\n\\bigskip\n\n\n\n\n\n\n\n\n\n\n\\begin{lem}\\label{lem-upperbound}\nUnder the hypotheses of Proposition \\ref{prop-C16-1new}, there exist positive constants $\\delta_4 > 0$ and $C_4 > 1$, depending only on $(f,\\Gamma)$,  such that, for all sufficiently large $k$,\n\\begin{equation}\nv_k(y)\\le C_4 U(y),\n\\qquad\\forall\\\n|y|\\le \\delta_4 R_k.\n\\end{equation}\n\\end{lem}\n\n\\begin{proof} Let $\\delta_3$ be as in Lemma \\ref{lem-smallenergy}. Since $v_k \\leq 1$, we deduce from Lemma \\ref{lem-energy1}, there is $r_1 > 1$ and $\\delta_1 > 0$ such that\n\\begin{equation}\n\\int_{ r_1\\le |y|\\le \\delta_1 R_k} v_k^{ \\frac {2n}{n-2} }\n\\le  \\epsilon.\n\\label{energy2}\n\\end{equation}\n\nFor any $2r_1<r< \\delta_1 R_k\/2$, consider\n$$\n\\tilde v_k(z)= r^{ \\frac {n-2}2 } v_k(rz),\\quad\n\\frac 12<|z|<2.\n$$\nBy \\eqref{energy2}, we have, for large $k$,   \n$$\n\\int_{ \\frac 12<|z|<2 } \\tilde v_k(z)^{ \\frac {2n}{n-2} }\n=\\int_{  \\frac r2 <|\\eta|< 2r}\nv_k(\\eta)^{ \\frac {2n}{n-2} }\n\\le  \\delta_3.\n$$\nIt follows from Lemma \\ref{lem:Ressmallenergy} that\n$$\n\\tilde v_k(z)\\le C \\qquad \\forall\\ \\frac 23 <|z|<  \\frac 54,\n$$\nfor some universal constant $C$. Since $\\tilde v_k$ also satisfies $f(\\lambda(A^{\\tilde v_k})) = 1$, we can apply Theorem \\ref{TheoremB} to obtain\n\\[\n|\\nabla \\ln \\tilde v_k(z)| \\leq C \\qquad \\forall\\  |z| = 1,\n\\]\nwhich implies that $\\max_{|z| = 1} \\tilde v_k \\leq C\\,\\min_{\\partial B_1}\\tilde v_k$. Returning to $v_k$, we obtain\n\\[\n\\max_{\\partial B_r} v_k \\leq C\\,\\min_{\\partial B_r} v_k \n\\]\nwhere $C$ is universal. The conclusion then follows from Lemma \\ref{lem0.1}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop-C16-1new}]\nFix $\\epsilon > 0$. In view of Lemma \\ref{lem-energy1} (cf. \\eqref{U00}), we only need to prove that there exist ${\\delta_{0}} > 0$ such that, for all sufficiently large $k$,\n\\begin{equation}\nv_k(y)\\le (1+2\\epsilon)U(y), \\qquad \\forall\\ |y|\\le {\\delta_{0}} R_k.\n\\label{C17-1}\n\\end{equation}\nSuppose the contrary of the above, then, after passing to a subsequence \nand renaming the subsequence still as $\\{v_k\\}$ and $\\{R_k\\}$,\nthere exist $|y_k|=\\delta_k R_k$, $\\delta_k\\to 0^+$,\nsuch that\n\\begin{equation}\nv_k(y_k)=\\max_{ |y|=\\delta_k R_k}\nv_k(y)> (1+2\\epsilon) U(y_k).\n\\label{C17-3}\n\\end{equation}\nIn view of the convergence of $v_k$ to $U$,\n $|y_k|\\to \\infty$ as $k\\to \\infty$.\n\nConsider the following two rescalings of $v_k$:\n\\begin{equation}\n\\hat v_k(z):= |y_k|^{n-2} \nv_k(|y_k|z) \\text{ and } \\bar v_k(z) = |y_k|^{\\frac{n-2}{2}}v_k(|y_k|z),\\qquad\n|z|<\\frac {  R_k}{ |y_k|}\\to \\infty.\n\\label{C18-2}\n\\end{equation}\nBy Lemma \\ref{lem-upperbound}, we have\n\\begin{equation}\n\\hat v_k(z)\\le C|z|^{2-n} \\text{ and } \\bar v_k(z) \\leq C\\,|y_k|^{-\\frac{n-2}{2}} |z|^{2-n}\n\t\\label{Eq:hatbarvkUB}\n\\end{equation}\nfor some constant $C$ independent of $k$.\n\nIn view of the conformal invariance \\eqref{Eq:CIProp} and \\eqref{ab1new}, \n\\[\nf(\\lambda(A^{\\bar v_k}(z))) = 1 \\text{ for } |z|< \\frac {  R_k}{ |y_k|}.\n\\]\nRecalling \\eqref{Eq:hatbarvkUB}, we can apply Theorem \\ref{TheoremB} to obtain that\nfor all $0<\\alpha<\\beta <\\infty$, there exists\npositive constant $C(\\alpha, \\beta)$ such that\nfor large $k$,\n\\begin{equation}\n|\\nabla \\ln \\bar v_k(z)|\\le C(\\alpha, \\beta),\\qquad\n\\forall \\ \\alpha<|z|<\\beta,\n\\label{C22-2X}\n\\end{equation}\nwhich implies that\n\\begin{equation}\n|\\nabla \\ln \\hat v_k(z)|\\le C(\\alpha, \\beta),\\qquad\n\\forall \\ \\alpha<|z|<\\beta.\n\\label{C22-2}\n\\end{equation}\n\n\n\n\nWe know from \\eqref{C18-2},  \\eqref{C17-3} and\nLemma \\ref{lem0.1} that\n\\begin{equation}\n\\min_{ |z|=1} \\hat v_k(z)\\le \n(1+ \\epsilon)\\frac{|y_k|^{n-2}}{U(y_k)},\n\\label{C19-2}\n\\end{equation}\nand\n\\begin{equation}\n\\max_{ |z|=1} \\hat v_k(z)\\ge (1+ 2\\epsilon)\\frac{|y_k|^{n-2}}{U(y_k)}.\n\\label{C20-1}\n\\end{equation}\n\nWe deduce from \\eqref{C22-2}, \\eqref{C19-2} and \\eqref{C20-1}, after passing to a subsequence,\nthat for some positive function\n$\\hat v^*$ in $C^{0,1}_{loc}({\\mathbb R}^n\\setminus\\{0\\})$,\n\\begin{equation}\n\\hat v_k\\to \\hat v^*\\qquad \\mbox{in}\\ C^{\\alpha}_{loc}({\\mathbb R}^n\\setminus\n\\{0\\}),\\ \\forall\\ 0<\\alpha<1.\n\\label{C23-1}\n\\end{equation}\nBy Theorem \\ref{proposition1-deg}, $\\hat v^*$ is radially symmetric.\nOn the other hand, we  deduce from \\eqref{C19-2} and\n\\eqref{C20-1} after passing to limit that\n\\begin{equation}\n\\min_{ |z|=1} \\hat v^*(z)\\le \n1+\\epsilon,\n\\quad\\mbox{and}\\quad\n\\max_{ |z|=1} \\hat v^*(z)\\ge 1+2\\epsilon.\n\\label{C20-1new}\n\\end{equation}\nThe above violates the radial symmetry of $\\hat v^*$.\n Proposition \\ref{prop-C16-1new} is established.\n\\end{proof}\n\n\n\n\\subsection{Detailed blow-up landscape}\n\n\nThe proof of Theorem \\ref{theorem4X} uses the following consequence of the Harnack-type inequality for conformally invariant equations, see \\cite{SchoenNotes, ChenLin, LiLi03}. \n\n\\begin{lem}\\label{Lem:EBnd}\nLet $(f,\\Gamma)$ satisfy \\eqref{01weak}-\\eqref{0304weak} and \\eqref{FU-1a}. There exists a constant $C_6$, depending only on $(f,\\Gamma)$, such that if $u \\in C^2(B_{3}(0))$ is a positive solution of\n\\[\nf(\\lambda(A^u)) = 1 \\text{ in } B_{3}(0)\n\\]\nthen \n\\[\n\\int_{B_1(0)} |u|^{\\frac{2n}{n-2}}\\,dx \\leq C_6.\n\\]\n\\end{lem}\n\n\\begin{proof} We give the proof here for completeness. By \\eqref{FU-1a}, \n\\[\n-\\Delta u \\geq \\frac{n-2}{2}\\delta\\,u^{\\frac{n+2}{n-2}} > 0 \\text{ in } B_2(0).\n\\]\nThus, by Corollary \\ref{cor-App1-3X} in Appendix \\ref{Sec:AppA} as well as the maximum principle,\n\\[\n\\inf_{B_{2}(0)} u = \\inf_{B_{2}(0) \\setminus B_{3\/2}(0)} u \\geq \\frac{1}{C} \\int_{B_1(0)} u^{\\frac{n+2}{n-2}}\\,dx.\n\\]\nIt follows that\n\\[\n\\int_{B_1(0)} u^{\\frac{2n}{n-2}}\\,dx \\leq C\\sup_{B_1(0)} u \\,\\inf_{B_{2}(0)} u.\n\\]\nThe conclusion follows from the above estimate and the Harnack-type inequality \\cite[Theorem 1.2]{LiLi05}. (Note that \\eqref{FU-1a} is used again here.)\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{theorem4X}]\nIn view of Proposition \\ref{prop:BbUp1} and (vi), it suffices to establish the theorem for $\\epsilon = \\epsilon_0 := 1\/2$.\n\nBy Lemma \\ref{Lem:EBnd},\n\\begin{equation}\n\\int_{B_2(0)} u^{\\frac{2n}{n-2}} \\,dx \\leq C_6.\n\t\\label{Eq:EBnd}\n\\end{equation}\nThe constant $\\bar m$ in the result can be selected to be the least integer satisfying\n\\[\n\\bar m \\geq 2C_6 \\Big(\\int_{B_1} U^{\\frac{2n}{n-2}}\\,dx\\Big)^{-1}.\n\\]\n(Clearly, this is an obvious upper bound for $m$ if the $x^i$'s satisfies (iii).)\n\n\nLet $\\delta_3$ and $C_3$ be the constants in Lemma \\ref{lem:Ressmallenergy}. Fix some $N_0 > \\frac{C_1}{\\delta_3}$. \nThen there is some $r_0 \\in (3\/2, 2)$ such that\n\\[\n\\int_{r_0 < |x| < r_0 + \\frac{1}{2N_0} } u^{\\frac{2n}{n-2}}\\,dx \\leq  \\delta_3.\n\\]\nBy Lemma \\ref{lem:Ressmallenergy}, this implies that\n\\begin{equation}\nu(x) \\leq C_3\\,(8N_0)^{\\frac{n-2}{2}} =: C_{7} \\text{ for all } r_0 + \\frac{1}{8N_0} < |x| < r_0 + \\frac{3}{8N_0}.\n\t\\label{Eq:G0Barrier}\n\\end{equation}\n\nLet ${C_{0}}$ and ${\\delta_{0}}$ be as in Proposition \\ref{prop:BbUp1} (corresponding to $\\epsilon = \\epsilon_0$). We can assume without loss of generality that \n\\begin{equation}\n{C_{0}} > 2 \\text{ and }{\\delta_{0}} < 1.\n\\label{Eq:PreRC*d*}\n\\end{equation}\nWe now declare\n\\begin{equation}\n{C_*} = \\max\\Big(2C_7, {C_{0}} (2{\\delta_{0}}^{-1})^{\\frac{\\bar m(n-2)}{2}}(4N_0)^{-\\frac{n-2}{2}}\\Big).\n\t\\label{Eq:C*'Choice}\n\\end{equation}\nThis choice of ${C_*}$ will become clear momentarily.\n\n\nLet $U_1 = B_{r_0 + \\frac{3}{8N_0}}(0)$ and $V_1 = B_{r_0 + \\frac{1}{8N_0}}(0) \\subset U_0$. By \\eqref{Eq:C*'Choice}, ${C_*} \\geq 2C_7$, and so, by \\eqref{Eq:G0Barrier}, there is some $x^1 \\in V_1 \\subset B_2(0)$ such that\n\\[\nu(x^1) = \\sup_{U_1} u \\geq {C_*}.\n\\]\n\n\n\nLet $R_1 = \\frac{1}{4N_0}$, then \\eqref{Eq:C*'Choice} gives\n\\[\n{C_*} \\geq {C_{0}}\\,R_1^{-\\frac{n-2}{2}}.\n\\]\nHence, an application of Proposition \\ref{prop:BbUp1} to $u$ on the ball $B_{R_1}(x^1)$ leads to\n\\[\n|u(x) - U^{x^1, u(x^1)}(x)|\n\\le \\epsilon_0\\,U^{x^1, u(x^1)}(x),\n\\ \\ \\forall \\ x\\in B_{{\\delta_{0}} R_1}(x^1).\n\\]\nIn particular, for $\\frac{{\\delta_{0}} R_1}{2} \\leq |x - x^1| \\leq {\\delta_{0}}\\,R_1$, \n\\begin{equation}\nu(x) \\leq 2U^{x^1, u(x^1)}(x) \\leq \\frac{2}{u(x^1) |x - x^1|^{n-2}} \\leq \\frac{2^{n-1}}{{C_*} ({\\delta_{0}} R_1)^{n-2}} \\leq \\frac{{C_*}}{2},\n\t\\label{Eq:G1Barrier}\n\\end{equation}\nwhere we have used \\eqref{Eq:C*'Choice} in the last estimate.\n\nLet $U_2 = U_1 \\setminus B_{{\\delta_{0}}\\,R_1\/2}(x^1)$ and $V_2 = V_1 \\setminus B_{{\\delta_{0}} R_1}(x^1) \\subset U_1$. If \n\\[\n\\sup_{U_2} u \\leq {C_*},\n\\]\nwe stop. Otherwise, in view of \\eqref{Eq:G1Barrier}, there is some $x^2 \\in V_2$ such that\n\\[\nu(x^2) = \\sup_{U_2} u \\geq {C_*}.\n\\]\nWe then let $R_2 = \\frac{{\\delta_{0}} R_1}{2}$ so that \\eqref{Eq:C*'Choice} implies\n\\[\n{C_*} \\geq {C_{0}}\\,R_2^{-\\frac{n-2}{2}}.\n\\] \nHence, by Proposition \\ref{prop:BbUp1},\n\\[\n|u(x) - U^{x^2, u(x^2)}(x)|\n\\le \\epsilon_0 U^{x^2, u(x^2)}(x),\n\\ \\ \\forall \\ x\\in B_{{\\delta_{0}} R_2}(x^2).\n\\]\nWe then repeat the above process to define $U_3$, $V_3$, and to decide if a local max $x^3$ can be selected in $V_3$, etc. As explain above, the number $m$ of times this process can be repeated cannot exceed $\\bar m$.\n\nWe have obtained the set of local maximum points $\\{x^1, \\cdots, x^m\\}$ of $u$ and have verified (i) and (iv) for\n\\[\n{\\delta_*} =  \\Big(\\frac{{\\delta_{0}}}{2}\\Big)^{\\bar m} \\frac{1}{2N_0} \\leq {\\delta_{0}}\\,R_{m}.\n\\]\n(vi) is readily seen as\n\\[\ndist(x^i, \\partial U_i) \\geq R_i \\geq {\\delta_*}.\n\\]\n(ii) is also clear for\n\\[\n{K} \\geq  \\Big(\\frac{2}{{\\delta_{0}}}\\Big)^{\\bar m} 4N_0 \\geq \\frac{2}{{\\delta_{0}}\\,R_{m-1}}.\n\\]\nFrom construction, we have\n\\[\n\\sup_{U_{m+1}} u \\leq {C_*}.\n\\]\nBy Theorem \\ref{TheoremB}, this implies that\n\\begin{equation}\n|\\nabla \\ln u(x)| \\leq C_8 \\text{ for all } x \\in V_m = V_0 \\setminus \\cup_{i = 1}^m B_{{\\delta_{0}} R_i}(x^i).\n\t\\label{Eq:HarOutsideCore}\n\\end{equation}\nAlso, note that, for ${\\delta_*} < |x - x^i| < {\\delta_{0}} R_i$, we have\n\\[\n\\frac{1}{u(x^i)} \\Big(\\frac{1}{({C_*})^{-\\frac{4}{n-2}} + ({\\delta_{0}}\\,R_i)^2.}\\Big)^{\\frac{n-2}{2}}\n\t\\leq U^{x^i, u(x^i)}(x) \\leq \\frac{1}{u(x^i)} ({\\delta_*})^{-(n-2)} , \n\\]\nand so \n\\begin{equation}\n\\frac{1}{C_9u(x^i)} \\leq u(x) \\leq \\frac{C_9}{u(x^i)} ({C_*})^{-2}\\,({\\delta_*})^{-(n-2)}\n\t\\label{Eq:HarMargin}\n\\end{equation}\nIt is now clear that (iii) and (v) hold for ${K}$ sufficiently large. The proof is complete.\n\\end{proof}\n \n\n\n\n\n\n\n\n\\section{A quantitative Liouville theorem}\\label{sec:tQL}\n\n\\begin{proof}[Proof of Theorem \\ref{quantitativeliouville}]\nAssume by contradiction that, for some $\\epsilon \\in (0,1\/2]$, there exist $v_k \\in C^2(B_{3R_k}(0)$, $R_k \\rightarrow \\infty$, such that $f(\\lambda(A^{v_k})) = 1$ in $B_{3R_k}(0)$ and $v_k \\geq \\gamma$ in $B_{r_1}(0)$ but, for each $k$,\n\\begin{equation}\n\\text{\\eqref{AppB3} and \\eqref{AppB4} cannot simultaneously hold for any $\\bar x$.}\n\\label{Eq:QLCAs}\n\\end{equation}\n\nDefine\n\\[\nu_k(y) = R_k^{\\frac{n-2}{2}}v_k(R_k\\,y) \\text{ for } |y| \\leq 3.\n\\]\nThen $f(\\lambda(A^{u_k})) = 1$ in $B_3(0)$ and\n\\begin{equation}\nu_k \\geq R_k^{\\frac{n-2}{2}} \\gamma \\text{ in } B_{r_1\/R_k}(0). \n\t\\label{Eq:ukNondeg}\n\\end{equation}\nThus, by applying Theorem \\ref{theorem4} and after passing to a subsequence, we can select sets of local maximum points $\\{x_k^1, \\cdots, x_k^m\\}$ of $u_k$ such that assertions (i)-(vi) in Theorem \\ref{theorem4} hold. We can also assume that $x_k^i \\rightarrow x_*^i$, $1 \\leq i \\leq m$.\n\nBy assertions (i), (iv) and (v) of Theorem \\ref{theorem4}, $u_k$ converges locally uniformly to zero in $B_1 \\setminus \\{x_*^1, \\cdots, x_*^m\\}$. Thus, in view of \\eqref{Eq:ukNondeg}, we must have $x_*^{i_0} = 0$ for some (unique) $1 \\leq i_0 \\leq m$. Clearly, $B_{r_1\/R_k}(0) \\subset B_{{\\delta_*}}(x_k^{i_0})$ for large $k$.\n\nRecalling assertions (iv), (vi) and returning to the original sequence $v_k$ we get, for $\\bar x_k = R_k\\,x_k^{i_0}$ and $\\bar\\mu_k = v_k(\\bar x_k) = \\sup_{B_{{\\delta_*} R_k}(\\bar x_k)} v_k$, that $B_{r_1}(0) \\subset B_{{\\delta_*} R_k}(\\bar x_k)$, $\\bar\\mu_k \\geq \\gamma$ and\n\\[\n(1 - \\epsilon) U^{\\bar x_k, \\bar\\mu_k} \\leq v_k \\leq (1 + \\epsilon) U^{\\bar x_k, \\bar\\mu_k} \\text{ in } B_{{\\delta_*} R_k}(\\bar x_k).\n\\]\nWe then have\n\\[\n\\gamma \\leq v_k(0) = 2\\Big(\\frac{\\bar \\mu_k^{\\frac{2}{n-2}}}{1 + \\bar\\mu_k^{\\frac{4}{n-2}}|\\bar x_k|^2}\\Big)^{\\frac{n-2}{2}}\n\\leq \n\\frac{2}{\\bar\\mu_k\\,|\\bar x_k|^{n-2}}  \\leq \\frac{2}{\\gamma\\,|\\bar x_k|^{n-2}} \n\\]\nThis implies \n\\[\n|\\bar x_k| \\leq 2^{\\frac{1}{n-2}}\\,\\gamma^{-\\frac{2}{n-2}}.\n\\]\nOn the other hand, since $B_{r_1}(0) \\setminus B_{r_1\/2}(\\bar x_k) \\neq \\emptyset$, we can select some $y_k \\in B_{r_1}(0)$ such that\n\\[\n|\\bar x_k - y_k| \\geq \\frac{r_1}{2}.\n\\]\nThis implies that\n\\[\n\\gamma \\leq v(y_k) \\leq 2U^{\\bar x_k, \\bar\\mu_k}(y_k) \\leq \\frac{2}{\\bar\\mu_k\\,|y_k - \\bar x_k|^{n-2}} \\leq \\frac{2^{n-1}}{\\bar \\mu_k\\,r_1^{n-2}},\n\\]\nand so\n\\[\n\\bar\\mu_k \\leq \\frac{2^{n-1}}{\\gamma\\,r_1^{n-2}}.\n\\]\nWe have thus shown that $\\bar x = \\bar x_k$ satisfies both \\eqref{AppB3} and \\eqref{AppB4}, which contradicts \\eqref{Eq:QLCAs}.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nNichols algebras play a crucial role in the classification results for pointed\nHopf algebras over abelian groups, which vastly generalize the theory of\nquantized enveloping algebras. They appear in several recent investigations of\npointed Hopf algebras over nonabelian groups. Nichols algebras are seen to be\nfundamental objects, appearing in the study of the cohomology of flag\nmanifolds. See e.g., \\cite{AG2003a},\\cite{AS2002},\\cite{GHV2011\n,\\cite{HLV2012},\\cite{FK99} and references therein.\n\nA Nichols algebra $B(V)$ is constructed from a braided vector space $V,$ where\n$V$ in turn is, in the most studied cases, a Yetter-Drinfeld module over some\ngroup $G$. The Nichols algebra depends only on the braiding $c:V\\otimes\nV\\rightarrow V\\otimes V$ and there may be many groups yielding the braiding.\nIn this paper, we study groups that can arise given a fixed Yetter-Drinfeld\nmodule $V\\i\n\\genfrac{}{}{0pt}{}{G}{G\n\\mathcal{YD}$.\n\nWe assume that $V$ is a link-indecomposable Yetter-Drinfeld module over a\ngroup $G$, and that $X$ is the corresponding rack with cocycle $q:X\\times\nX\\rightarrow\\mathbb{\\Bbbk}^{\\times}$. $V\\ $also takes the form $\\oplus\n_{i}M(g_{i},\\rho_{i})\\cong(\\mathbb{\\Bbbk}X,c^{q})$ as braided vector spaces.\nThat is, we are assuming that $V$ is of \\textit{rack type} with braiding\n$c=c^{q}:V\\otimes V\\rightarrow V\\otimes V.$ Here $g_{i}\\in G$ and $\\rho_{i}$\nis a one-dimensional representation of the centralizer of $g_{i}$ in $G.$ The\nequivalence of the two constructions of $V$ is explained by \\cite[Theorem\n4.14]{AG2003a} (in greater generality)$.$\n\nThe theory of coalgebra coverings in \\cite{Chin2010} yields indecomposable\ncoalgebra coverings of $B(V)$. Such coverings take the form $B(V)\\rtimes\\Bbbk\nG\\rightarrow B(V)$ where $G$ is a homomorphic image of the universal coalgebra\ncovering group $\\tilde{G}$ for $B\\left(  V\\right)  $ and $\\rtimes$ is the\nsmash coproduct (or \"co-smash product\") coalgebra. Let $G_{X}$ denote the\nenveloping group of the rack $X$ (see \\cite{AG2003a},\\cite{GHV2011}). We have\nin general for a braiding of rack type that $V\\i\n\\genfrac{}{}{0pt}{}{\\Bbbk G_{X}}{\\Bbbk G_{X}\n\\mathcal{YD}$ and therefore there is a surjection $\\tilde{G}\\rightarrow G_{X}$\nand a corresponding coalgebra surjection $B(V)\\rtimes\\Bbbk\\tilde{G}\\rightarrow\nB(V)\\rtimes\\Bbbk G_{X}.$ We show in Corollary \\ref{uni} that a certain\nquotient $B(V)\\#\\hat{G}$ of $B(V)\\#\\tilde{G}$ serves as the universal Hopf\ncovering of $B(V)\\#G$. Thus we have a \\textit{Hopf algebra covering} of the\ngraded Hopf algebra the sense exhibited in \\cite{Lentner2013} arising from a\ncentral extension\n\\[\n1\\rightarrow Z\\rightarrow\\hat{G}\\rightarrow G\\rightarrow1.\n\\]\nIt is known by \\cite[Lemma 3.4]{AFGV2011} that $G_{X}$ also is universal for\n$V$ so we have in fact that $\\hat{G}=G_{X}.$\n\nThe corresponding rack braiding $c:X\\times X\\rightarrow X\\times X$ decomposes\ninto $c$-orbits $\\mathcal{O=O}_{x,y}$. Recall that with $c=c^{q}:V\\otimes\nV\\rightarrow V\\otimes V,$ $\\ker(1+c)$ defines the quadratic term $B(V)(2)$.\nWhen $\\ker(1+c^{{}})\\cap V_{\\mathcal{O}}\\mathcal{\\neq}0$ for each nontrivial\n$c$-orbit $\\mathcal{O}$, we say that $B(V)$ has \\textit{a full set of\nquadratic relations. (}We are ignoring the orbits of the form $\\{(x,x)\\}.)$ We\nshow in Theorem \\ref{full} that, for a Nichols algebra with a full set of\nquadratic relations, $G_{X}=\\tilde{G}.$ Thus $B(V)\\#G_{X}\\rightarrow B(V)$ is\nthe universal coalgebra covering, which factors through the Hopf algebra\nsurjection $B(V)\\#\\Bbbk G_{X}\\rightarrow B(V)\\#\\Bbbk G$. This means that every\ngroup $G$ such that $B(V)\\rtimes\\Bbbk G$ is an indecomposable coalgebra arises\nas a homomorphic image of $G_{X}.$ We have thus a universal Hopf covering for\nthe Nichols algebra $B(V).$ The use of $G_{X}$ as the \"principal\" grading\ngroup is mentioned in \\cite[Lemma 3.4]{AFGV2011} and suggested in\n\\cite{GaV2014}.\n\nIn case $G_{X}=\\tilde{G},$ we remark that the relations in the definition of\n$\\tilde{G}$ of degree greater than two are superfluous. In all known examples\nwith finite dimensional $B(V)$ we find that $G_{X}=\\tilde{G},$ and we\nconjecture that this is always the case.\n\nTo compare with \\cite{GHV2011}, recall that $B(V)$ is said to have\n\\textit{many quadratic relations} if the $\\dim\\ker(1+c)\\geq d(d-1)\/2$ where\n$d=\\dim V.$ The two notions pertaining to quadratic relations do not appear to\nbe comparable, see for example \\ref{FK Ex}.\n\nWe are interested in finite racks $X$ that are unions of conjugacy classes of\nthe group $G$ and we assume that $X$ generates $G$. In this latter case the\nYetter-Drinfeld module is said to be \\textit{link-indecomposable}, so the\ncovering Hopf algebras will also be link-indecomposable (i.e. indecomposable\nas coalgebra). We provide families of link-indecomposable Hopf algebras\ncorresponding to the covering Hopf algebras of finite-dimensional Nichols\nalgebras. These Hopf algebras are finite-dimensional when the covering group\nis finite. This addresses the question posed in \\cite{Montgomery95} by\nincluding all finite dimensional covering Hopf algebras of known examples of\nlink indecomposable bosonized Nichols algebras. Specifically, the families of\nfinite-dimensional Hopf algebras produced arise from the finite homomorphic\nimages $G_{X}$ by finite index subgroups of its center $Z(G_{X}).$ Some of\nthese examples are given in \\cite[Section 6]{AG2003a}.\n\nOur main references for racks, Nichols algebras, and pointed Hopf algebras are\n\\cite{AG2003a},\\cite{AS2002}; also see \\cite{GHV2011}, \\cite{HLV2012}.\n\nIt would be interesting to try to remove the condition that $\\dim\\rho=1$ from\nour results. Another direction would be to examine more general situations in\nwhich $\\tilde{G}=G_{X}.$ It might also be of interest to extend results here\nto liftings to nongraded pointed Hopf algebras.\n\n\\section{Path coalgebras and pointed coalgebras}\n\nWe refer to \\cite{Chin2004},\\cite{Chin2010} for basics of pointed coalgebras\nand path coalgebras. The \\textbf{path coalgebra} $\\Bbbk Q$ of a quiver $Q$ is\ndefined to be the span of all paths in $Q$ with coalgebra structure\n\\\n\\begin{array}\n[c]{c\n\\Delta(p)=\\sum_{p=p_{2}p_{1}}p_{2}\\otimes p_{1}+t(p)\\otimes p+p\\otimes s(p)\\\\\n\\varepsilon(p)=\\delta_{|p|,0\n\\end{array}\n\\]\nwhere $p_{2}p_{1}$ is the concatenation $a_{t}a_{t-1}...a_{s+1}a_{s\n...a_{1}\\,$of the subpaths $p_{2}=a_{t}a_{t-1}...a_{s+1}$ and $p_{1\n=a_{s}...a_{1}$ ($a_{i}\\in Q_{0}$)$.$ Here $|p|=t$ denotes the length of $p$\nand the starting vertex of $a_{i+1}\\,$ is required to be the end of $a_{i}.$\nThus the vertices $Q_{0}$ are group-like elements, and if $a$ is an arrow\n$g\\leftarrow h$, with $g,h\\in Q_{0},$ then $a$ is a $(g,h)$- skew primitive,\ni.e., $\\Delta a=g\\otimes a+a\\otimes h.$ It follows that $\\Bbbk Q$ is pointed\nwith coradical $(kQ)_{0}=kQ_{0}\\,$and the degree one term of the coradical\nfiltration is $(\\Bbbk Q)_{1}=\\Bbbk Q_{0}\\oplus\\Bbbk Q_{1}.$ Moreover the\ncoradical grading $\\Bbbk Q=\\bigoplus\\limits_{n\\geq0}\\Bbbk Q_{n}$ is given by\npath length. The path coalgebra may be identified with the cotensor coalgebra\n$C(\\Bbbk Q_{1})=\\oplus_{n\\geq0}(\\Bbbk Q_{1})^{\\Box n}$ of the $\\Bbbk Q_{0\n$-bicomodule $\\Bbbk Q_{1},$ cf. \\cite{Nichols78}.\n\nDefine the (\\textit{Gabriel- }or\\textit{\\ Ext-}) \\textit{quiver} of a pointed\ncoalgebra $B$ to be the directed graph $Q=Q_{B}$ with vertices $Q_{0} $\ncorresponding to group-likes and $\\mathrm{\\dim}_{\\Bbbk}P_{h,g}$ arrows from\n$h$ to $g$, for all $h,g\\in Q_{0}$. We will view $B$ as a subcoalgebra of the\npath coalgebra of its Gabriel quiver, with the same degree one coradical term.\nIf $B$ has a unique group-like element with a space of primitives of dimension\n$n$, then the quiver is the $n$-loop quiver. The path coalgebra is the cofree\npointed irreducible coalgebra (and the path algebra is the free algebra on $n$ generators).\n\nThe indecomposable components (\"blocks\") of $B$ are coalgebras that are the\ndirect sums of injective indecomposables having socles in a given graph\ncomponent. Therefore $B$ is indecomposable as a coalgebra if and only if it is\n\"link-indecomposable\", if and only if its quiver is connected as a graph. In\n\\cite{Montgomery95} it is shown that a pointed coalgebra is a crossed product\nover the principal block subcoalgebra, i.e. the one containing $1_{G(B)}.$\n\n\\section{Coverings}\n\nWe first summarize results from \\cite{Chin2010} concerning coverings of\npointed coalgebras. An analogous version for bound quivers finite-dimensional\nalgebras can be found in \\cite{Green83} and \\cite{Martinez83}.\n\nLet $B\\subseteq\\Bbbk Q$ be a subspace. Let $b=\\sum_{i\\in I}\\lambda_{i}p_{i}\\in\nB(x,y)$ with $x,y\\in Q_{0}$ and distinct paths $p_{i}.$ We say that $b$ is a\n\\textit{minimal element\\ }of $B$ if $\\sum_{i\\in I^{\\prime}}\\lambda_{i\np_{i}\\notin B(x,y)$ for every nonempty proper subset $I^{\\prime}\\subset I$ and\n$|I|\\geq2.$ Clearly every element of $B$ is a linear combination of paths and\nminimal elements. Let $\\min(B)$ denote the set of minimal elements of $B$.\n\nFor an admissible ideal $I$ of a path algebra, let $A=\\Bbbk^{Q}\/I$ denote an\nadmissible quotient of the path algebra $\\Bbbk^{Q}$ with ideal of relations\n$I$. Let $B\\subseteq\\Bbbk Q$ be an admissible subcoalgebra of the path\ncoalgebra $\\Bbbk Q$.\n\nFix a base vertex $x_{0}\\in Q_{0}$. We define a symmetric relation $\\sim$ on\npaths by declaring $p\\sim q$ if there is a minimal element $b=\\sum_{i\\in\nI}\\lambda_{i}p_{i}\\in B(x,y)$ where the $p_{i}$ are distinct paths,\n$\\lambda_{i}\\in\\Bbbk$, $x,y\\in Q_{0}$ and $p=p_{1}$, $q=p_{2}.$ We define\n$N(B,x_{0})$ to be the subgroup of $\\pi_{1}(B,x_{0})$ generated by equivalence\n(homotopy) classes of walks $w^{-1}p^{-1}qw$ where $p,q$ are paths in $Q(x,y)$\nwith $p\\sim q$ and $w$ is a walk from $x_{0}$ to $x.$\n\nIt is easy to see that $N(B,x_{0})$ is a normal subgroup of $\\pi_{1\n(B,x_{0}).$ Explicitly, if $w^{-1}p^{-1}qw$ is closed walk as above and\n$[v]\\in\\pi_{1}(B,x_{0})$ where $v$ is a closed walk at $x_{0}$, then $wv$ is a\npath from $x_{0}$ to $x$ and $[v^{-1}w^{-1}p^{-1}qwv]=[(wv)^{-1}p^{-1}q\\left(\nwv\\right)  ]\\in N(B,x_{0}).$\n\nConsider a Galois covering $F:\\tilde{Q}\\rightarrow Q$ of quivers with\nautomorphism group $G$ and lifting $L$. Let $\\tilde{B}$ denote the $\\Bbbk\n$-span of $\\{L(b)|L$ a lifting, $b\\in B$ a minimal element or a path\\}. We say\nthat the restriction $F:\\tilde{B}\\rightarrow B$ is a \\textit{Galois coalgebra\ncovering} if every minimal element of $B$ can be lifted to $\\tilde{B}$ in the\nfollowing sense: for every minimal element \\ $b\\in B(x,y)$ with $x,y\\in Q_{0}$\nand $\\tilde{x}\\in F^{-1}(x),$ there exists $\\tilde{y}\\in\\tilde{Q}_{0}$ and a\nminimal element $\\tilde{b}\\in\\tilde{B}(\\tilde{x},\\tilde{y})$ such that\n$F(\\tilde{b})=b.$ All coverings in this paper are assumed to be Galois.\n\nLet $B\\subseteq\\Bbbk Q$ be a pointed coalgebra. Then there exists a Galois\ncoalgebra covering $F:\\tilde{B}\\rightarrow B$ $,$ the \\textit{universal\ncoalgebra covering} of $B\\subseteq\\Bbbk Q,$ such that for every Galois\ncoalgebra covering $F^{\\prime}:B^{\\prime}\\rightarrow B$, there exists a Galois\ncoalgebra covering $E:\\tilde{B}\\rightarrow B^{\\prime}$ such that the following\ndiagram commutes\n\n\\\n\\begin{array}\n[c]{ccccc\n\\tilde{B} &  & \\overset{E}{\\xrightarrow{\\hspace*{.75cm}}} &  & B^{\\prime}\\\\\n& \\underset{F}{\\searrow} &  & \\underset{F^{\\prime}}{\\swarrow} & \\\\\n&  & B &  &\n\\end{array}\n\\]\n\n\nThe fundamental example of a covering is given as follows. Let $B\\subseteq\n\\Bbbk Q$ be a homogenous admissible subcoalgebra with respect to the grading\ngiven by an arrow weighting $\\delta:Q_{1}\\rightarrow G$. If $b=\\sum_{i\\in\nI}\\lambda_{i}p_{i}\\in B(x,y)$ is a minimal element, then it is necessarily\nhomogeneous in the $G$-grading. Consider the canonical map $F:\\Bbbk\nQ\\rtimes\\Bbbk G\\rightarrow\\Bbbk Q$ defined by $F(p\\rtimes g)=p$ and consider\nthe restriction to $B\\rtimes\\Bbbk G\\rightarrow B.$ Then under the\nidentification of $\\Bbbk Q\\rtimes\\Bbbk G$ with $\\Bbbk\\tilde{Q}$ we easily see\nthat $\\tilde{B}=B\\rtimes\\Bbbk G.$ The liftings of minimal element $b\\in B$ are\ngiven by $b\\rtimes g$ with $g\\in G.$\n\n\\begin{theorem}\n[\\cite{Chin2010}]\\label{Cov}The following are equivalent for a subcoalgebra\n$B\\subseteq\\Bbbk Q $ and Galois quiver covering $F:\\tilde{Q}\\rightarrow\nQ.$\\newline(a) $B$ is a homogeneous subcoalgebra of $\\Bbbk Q.$\\newline(b)\n$N(B,x_{0})\\subseteq F_{\\ast}(\\pi_{1}(\\tilde{Q},\\tilde{x}_{0}))$ for all\n$x_{0}\\in Q,$ $\\tilde{x}_{0}\\in\\tilde{Q}$ with $F(\\tilde{x}_{0})=x_{0\n.$\\newline(c) $F:\\tilde{B}\\rightarrow B$ is a Galois coalgebra\ncovering.\\newline(d) $B$ is a homogenous subcoalgebra of $\\Bbbk Q$ and the\ngrading is connected.\n\\end{theorem}\n\n\\begin{theorem}\n[\\cite{Chin2010}]The universal covering of the coalgebra $B\\subseteq\\Bbbk Q$\nis isomorphic to $B\\rtimes\\Bbbk\\tilde{G}\\rightarrow B$ where $\\tilde{G\n=\\frac{\\pi_{1}(\\tilde{Q},\\tilde{x}_{0})}{N(B,x_{0})},$ $(\\tilde{Q},\\tilde\n{x}_{0})$ is the universal covering quiver of $(Q,x_{0})$ (with base vertices\n$\\tilde{x}_{0}$ and $x_{0}$), and $B\\rtimes\\Bbbk\\tilde{G}$ is indecomposable\nas a coalgebra if $B$ is.\n\\end{theorem}\n\n\\subsection{Hopf coverings}\n\nThe idea of Hopf covering comes from the following simple observation.\n\n\\begin{lemma}\nLet $B(V)$ be a Nichols algebra with a link-indecomposable Yetter-Drinfeld\nmodule $V\\i\n\\genfrac{}{}{0pt}{}{\\Bbbk G}{\\Bbbk G\n\\mathcal{YD}.$ Let $Z$ be a normal subgroup of $G$ acting trivially on $V$.\nThen $Z$ is central in $G$. Let $\\bar{V}$ be a copy of $V$ graded by $\\bar\n{G}=G\/Z$ via $\\bar{V}_{gZ}=\\oplus_{t\\in gZ}V_{t}$ for $gZ\\in\\bar{G},$ and make\n$\\bar{V}$ a $\\bar{G}$-module by factoring by $Z$. Then $\\bar{V}\\i\n\\genfrac{}{}{0pt}{}{\\bar{G}}{\\bar{G}\n\\mathcal{YD}$ and $V$ is isomorphic to $\\bar{V}$ as a braided vector space.\nTherefore $B(V)$ and $B(\\bar{V})$ are isomorphic as braided Hopf algebras and\nthere is a surjection $B(V)\\#\\Bbbk G\\rightarrow B(\\bar{V})\\#\\Bbbk\\bar{G}$ on\nbosonized Nichols algebras given by $b\\#g\\mapsto b\\#gZ,$ for all $b\\in B(V),$\n$g\\in G.$\n\\end{lemma}\n\n\\begin{proof}\nWe just point out that $Z$ being central is equivalent to the condition\n$z.V_{g}=V_{zgz^{-1}}=V_{g}$ for all $g\\in G,$ $z\\in Z.$\n\nThe conclusions of the lemma hold even in the case that $V$ is not\nfinite-dimensional. Also if $V$ is not assumed to be link-indecomposable in\nthe lemma above, then we may replace \"$Z$\\text{ is central in }$G\"$ by \"$Z$ is\nin the centralizer of $\\{g\\in G|V_{g}\\neq0\\}\".$\n\\end{proof}\n\n\\begin{definition}\nWhen $\\tilde{A}=\\oplus_{n\\geq0}\\tilde{A}_{n}=R\\#\\Bbbk\\tilde{G}$ and\n$A=\\oplus_{n\\geq0}A_{n}=R\\#\\Bbbk G$ are coradically graded pointed Hopf\nalgebras, which are bosonizations of the braided graded Hopf algebra $R$ as in\n\\cite{AS2002}, and $f:\\tilde{G}\\rightarrow G$ is a group surjection, we say\nthat the Hopf algebra map $F:\\tilde{A}\\rightarrow A$ is a \\textit{Hopf\ncovering} if $F(r\\#\\tilde{g})=r\\#f(g)$ for all $r\\in R$ and $\\tilde{g}.$ We\nsay that $\\tilde{A}$ is a \\textit{covering Hopf algebra} of $A,$ with covering\ngroup $\\tilde{G}.$ A universal Hopf covering is one that is universal among\nHopf coverings of $A$.\n\\end{definition}\n\nThe following result specifies the Hopf coverings of a bosonization of a\nNichols algebra.\n\n\\begin{theorem}\nLet $B(V)\\#\\Bbbk G$ be the bosonization for a link-indecomposable $V\\i\n\\genfrac{}{}{0pt}{}{\\Bbbk G}{\\Bbbk G\n\\mathcal{YD}.$ Let $\\tilde{G}$ the universal coalgebra covering group of\n$B(V)$ and write $G=\\tilde{G}\/N$ where $N\\vartriangleleft\\tilde{G}$. The\ncovering Hopf algebras of $B(V)\\#\\Bbbk G$ are of the form $B(V)\\#\\Bbbk\n\\tilde{G}\/M$ where $M\\vartriangleleft\\tilde{G}$ and $[N,\\tilde{G}]\\subseteq\nM\\subseteq N.$\n\\end{theorem}\n\n\\begin{proof}\nWe have the universal coalgebra covering\n\\[\nB(V)\\rtimes\\Bbbk\\tilde{G}\\rightarrow B(V)\\rtimes\\Bbbk G\\rightarrow B(V)\n\\]\nby \\cite{Chin2010}, see Theorem \\ref{Cov} and \\cite{Chin2010}. So we may write\n$G=\\tilde{G}\/N$, $N\\vartriangleleft\\tilde{G}.$ Now consider the set of\nhomomorphic images $H=\\tilde{G}\/M$ of $\\tilde{G}$ where $M\\in\\mathcal{C}$, and\n$\\mathcal{C}$ is defined as\n\\begin{align*}\n\\mathcal{C}  &  \\mathcal{=}\\mathcal{\\{}M\\vartriangleleft\\tilde{G}|N\\supseteq\nM\\text{ and }N\/M\\text{ central in }\\tilde{G}\/M\\}\\\\\n&  =\\mathcal{\\{}M\\vartriangleleft\\tilde{G}|[N,\\tilde{G}]\\subseteq M\\}\n\\end{align*}\nusing the group theoretic commutator. It is evident that the unique minimal\nelement of $\\mathcal{C}$ is just $[N,\\tilde{G}].$ Now for such a group $H,$ it\nfollows from the Lemma that $B(V)\\#\\Bbbk H$ is a bosonization where the action\nis lifted from the action of $G,$ and the grading is inherited from the\n$\\tilde{G}$ grading. Thus we obtain the Hopf coverings $B(V)\\#\\Bbbk\\tilde\n{G}\/M$ of $B(V)\\#\\Bbbk G.$\n\\end{proof}\n\n\\begin{corollary}\n\\label{uni}Let $B(V)\\#\\Bbbk G$ be the bosonization for link-indecomposable\n$V\\i\n\\genfrac{}{}{0pt}{}{\\Bbbk G}{\\Bbbk G\n\\mathcal{YD}.$ Let $\\tilde{G}$ be the universal coalgebra covering group of\n$B(V)$ and write $G=\\tilde{G}\/N$ where $N\\vartriangleleft\\tilde{G}$. The\nuniversal covering Hopf algebra of $B(V)\\#\\Bbbk G$ is $B(V)\\#\\Bbbk\\tilde\n{G}\/[N,\\tilde{G}].$\n\\end{corollary}\n\n\\section{Racks and Nichols algebras}\n\nLet $B(V)$ be a Nichols algebra generated by the Yetter-Drinfeld module $V\\i\n\\genfrac{}{}{0pt}{}{\\Bbbk G}{\\Bbbk G\n\\mathcal{YD}$. We utilize the description of $B(V)$ as a subcoalgebra of the\ncotensor coalgebra $C(V)$ of the bicomodule $V$. Namely the graded component\n$B(V)(n)$ is the image of the quantum symmetrizer $\\mathfrak{S}_{n\n=\\sum_{\\sigma\\in\\mathbb{S}_{n}}\\hat{\\sigma}$ where $\\symbol{94}$ denotes the\nMatsumoto section $\\mathbb{S}_{n}\\rightarrow\\mathbb{B}_{n}$. In particular,\nthe quadratic summand $B(V)(2)\\subset V\\otimes V$ is the image of $1+c.$ For a\nfixed basis of $V$, we obtain an embedding $B\\hookrightarrow\\Bbbk Q$ where\n$Q=Q_{V}$ is the $\\dim V$-loop quiver with arrows labelled by basis elements\nof $V$ and single vertex $1_{B(V)}$. In fact, we choose a basis of\n$G$-homogeneous elements. Fixing this basis of homogeneous elements of $V$, we\nthus identify $C(V)$ with the path coalgebra $\\Bbbk Q.$ When $\\dim\\rho_{i}=1$\nfor all $i,$ the basis can be chosen to correspond to a union of conjugacy\nclasses of $G.$\n\nAs in \\cite{Rosso98},\\cite{Schau96}; cf. \\cite{AS2002} $B(V)$ can be\nconstructed as both the subcoalgebra $\\bigoplus\\limits_{n\\geq0\n\\operatorname{Im}\\mathfrak{S}_{n}$ of the cotensor coalgebra $\\Bbbk\nQ=C(V)=k1\\oplus V\\oplus V\\otimes V\\oplus\\cdots,$ and as a homomorphic image of\nthe tensor coalgebra $T(V)=\\Bbbk^{Q}$ modulo the ideal $\\bigoplus\n\\limits_{n\\geq0}\\ker\\mathfrak{S}_{n}.$\n\n\\subsection{Coverings of Nichols algebras}\n\nWe assume that $V=(V,c=c^{q})=\\Bbbk X$ is a Yetter-Drinfeld module with finite\nrack $X=(X,\\vartriangleright),$ with rack structure map $\\vartriangleright\n:X\\times X\\rightarrow X$ and $2$-cocycle $q:X\\otimes X\\rightarrow\\Bbbk\n^{\\times}$ is as in \\cite{AG2003a}, where\\textit{\\ we insist on a\none-dimensional image for }$q$. By \\cite[Theorem 4.14]{AG2003a} such a braided\nvector space arises as a Yetter-Drinfeld module a one-dimensional module\n$\\rho$ over the centralizer $G_{g}$ of an fixed chosen element $g\\in G$. Fix a\nbasis $\\{v_{x}|x\\in X\\}$ for $V$ where $v_{x}\\in V_{x}$ for all $x.$ Note that\nthe assumption entails that the subpaces $V_{x}=\\Bbbk v_{x},$ $x\\in X$ are\none-dimensional. The braiding $c=c^{q}:V\\otimes V\\rightarrow V\\otimes V$ is\ndefined by $c(v_{x}\\otimes v_{y})=q(x,y)v_{y}\\otimes v_{x}.$ We shall use the\nsame symbol for the map $c:X\\times X\\rightarrow X\\times X,$\n$c(x,y)=(x\\vartriangleright y,x),$ as in \\cite{GHV2011}. The Yetter-Drinfeld\nmodule is thus $\\oplus_{i}M(g_{i}$,$\\rho_{i})=\\oplus_{i}\\Bbbk G\\otimes_{\\Bbbk\nG_{g_{i}}}\\rho_{i}.$ The group $G$ can be chosen to be finite if the subgroup\nof $\\Bbbk^{\\times}$ generated by the $q(x,y)$ is finite and $X$ is finite, cf.\n\\cite[Theorem 4.14]{AG2003a}. We shall investigate groups that give rise to\nbraided vector spaces $(V,c^{q})$ and the Nichols algebra $B(V)$.\n\nWe follow the set-up as in \\cite{GHV2011}. We have the braided vector space\n$V=(\\Bbbk X,c),$ and we let $\\mathcal{O}$ $(=\\mathcal{O}_{x,y})$ denote\n$c$-orbit (of $(x,y)$) in $X\\times X$ . Set ~$V_{\\mathcal{O}}=\\sum\n_{(s,t)\\in\\mathcal{O}}V_{s}\\otimes V_{t}$ and note that $\\theta_{i\n:=c^{i}(v_{x}\\otimes v_{y}),$ $i=0,1,..m-1$ is a basis for $V_{\\mathcal{O}}.$\n\nThe \\textit{enveloping group} $G_{X}$ of a rack $X$ is defined to be the\nquotient of the free group on generators $\\{g_{x}|x\\in X\\}$ by the relations\n\\[\ng_{x}g_{y}=g_{x\\vartriangleright y}g_{x\n\\]\n$x,y\\in X.$\n\nLet $G$ be a group with link-indecomposable $V\\i\n\\genfrac{}{}{0pt}{}{\\Bbbk G}{\\Bbbk G\n\\mathcal{YD}.$ Then $V=(\\Bbbk X,c)$ for a rack $X\\subset G$ where $X$ is a\nunion of conjugacy classes of $G$ and Supp$_{G}V=$ $X$ generates $G$. Since\nthe defining relations of $G_{X}$ hold in $G,$ there is a surjective group\nhomomorphism $G_{X}\\rightarrow G.$\n\n\\begin{theorem}\n\\label{full}Let $B(V)$ be a Nichols algebra with a full set of quadratic\nrelations. Then \\newline(a) $G_{X}$ is the coalgebra covering group of\n$B(V)$\\newline(b) $B(V)\\#\\Bbbk G_{X}$ is the universal covering coalgebra of\n$B(V)\\subset\\Bbbk Q_{V}$ and \\newline(c) $B(V)\\#\\Bbbk G_{X}\\rightarrow\nB(V)\\#\\Bbbk G$ is the universal Hopf covering of $B(V)\\subset\\Bbbk Q_{V}.$\n\\end{theorem}\n\n\\begin{proof}\nAs mentioned above, $B(V)$ has quadratic component $B(V)(2)=\\operatorname{Im\n(1+c)\\subseteq V\\otimes V.$ For $v_{x}\\in V_{x},v_{y}\\in V_{y}$ with $x\\neq\ny\\in X$ we have $c(x\\otimes y)=q(x,y)(x\\triangleright y)\\otimes x.$ We need to\nsee that\n\\[\n(1+c)(v_{x}\\otimes v_{y})=v_{x}\\otimes v_{y}+q(x,y)v_{x\\triangleright\ny}\\otimes v_{x\n\\]\nis a minimal element of Im$(1+c)$. For then we obtain the relation $\\left[\nv_{x}\\right]  \\left[  v_{y}\\right]  =\\left[  v_{x\\triangleright y}\\right]\n\\left[  v_{x}\\right]  $ in $\\pi_{1}(Q,1_{B(V)})$ per the definition of\n$\\tilde{Q}_{B(V)}=\\frac{\\pi_{1}(B(V),1)}{N(B(V),1)}.$ By \\cite[Lemma\n3.4]{AFGV2011} $V\\i\n\\genfrac{}{}{0pt}{}{\\Bbbk G_{X}}{\\Bbbk G_{X}\n\\mathcal{YD}$ so there is a group surjection $\\tilde{G}\\rightarrow G_{X}$ with\n$\\left[  v_{x}\\right]  \\longmapsto g_{x}.$ It follows that $\\tilde{G}\\simeq\nG_{X}$\n\nWe claim that $1+c:V_{\\mathcal{O}}\\rightarrow V_{\\mathcal{O}}$ is onto (and\nthus bijective) if and only if $c^{m}(v_{i}\\otimes v_{j})\\neq(-1)^{m\n(v_{i}\\otimes v_{j}).$ Let $m$ be the order of $c$ in $\\mathcal{O\\subset\n}X\\mathcal{\\times}X.$ The restriction of $1+c$ to $V_{\\mathcal{O}}$ is given\nby the $m\\times m$ matrix\n\\\n\\begin{bmatrix}\n1 & 0 & \\cdots & 0 & \\lambda\\\\\n1 & 1 & 0 & \\cdots & 0\\\\\n0 & 1 & \\ddots & 0{}\\hspace{2pt}\\cdots & 0\\\\\n0 & \\ddots & \\ddots & 1 & 0\\\\\n0 & 0 & 0 & 1 & 1\n\\end{bmatrix}\n\\]\n$\\allowbreak$with respect to the basis $\\{\\theta_{i}\\}$, where $\\lambda\n\\in\\Bbbk^{\\times}$ is such that $c^{m}(v_{i}\\otimes v_{j})=\\lambda\nv_{i}\\otimes v_{j}.$ One can see that the determinant is $1+\\left(  -1\\right)\n^{m-1}$ $\\lambda$, so $1+c$ is bijective if and only if $\\lambda\\neq\\left(\n-1\\right)  ^{m-1}.$ When $1+c$ is not a bijection, then it is easy to see that\nthe $m-1$ elements\n\\[\n\\theta_{0}+\\theta_{1},\\theta_{1}+\\theta_{2},\\cdots,\\theta_{m-2}+\\theta_{m-1\n\\]\nform a basis for the image of $1+c$ restricted to $V_{\\mathcal{O}}.$ It\nfollows that the $\\theta_{i}$ are minimal elements.\n\\end{proof}\n\n\\begin{remark}\nWhen $\\lambda=\\left(  -1\\right)  ^{m-1}$, $\\ker(1+c)=\\Bbbk\\sum_{n=0\n^{m-1}\\left(  -1\\right)  ^{n}\\theta_{n}\\ $as noted in $\\cite{GHV2011}.$ When\n$\\lambda\\neq\\left(  -1\\right)  ^{m-1}$ then $\\ker(1+c)|_{\\mathcal{O}}=0.$\n\\end{remark}\n\n\\begin{remark}\nThe universal covering Hopf algebra was seen to be $B(V)\\#\\tilde{G}\/[\\tilde\n{G},N]$ in Theorem \\ref{uni}. Therefore $\\tilde{G}\/[\\tilde{G},N]\\simeq G_{X}$\nfor all choices of $N$ and it follows that $[\\tilde{G},N]=\\ker(\\tilde\n{G}\\rightarrow G_{X}).$ In case $N$ is central, we get the result\n$G_{X}=\\tilde{G}$ as in the conclusion of the Theorem. But if there is not a\nfull set of quadratic relations, we may still get $G_{X}=\\tilde{G},$ as can be\nshown for Nichols algebras of finite Cartan type, e.g. last example in\n$\\ref{nonabelian ex}.$\n\\end{remark}\n\n\\section{Examples}\n\n\\subsection{rank 1}\n\nLet $G=C_{m}=<K>$ be the cyclic group of order $m$ and let $q$ be an $m^{th}$\nroot of $1$. Let $H$ be the Hopf algebra with generators $E,$ $K$, where $K$\nis group-like and $E$ is $(K,1)$ skew-primitive, and with relations $KE=qEK,$\n$E^{m}=0.$ Then $H$ is the bosonization $B(\\Bbbk E)\\#\\Bbbk C_{m}$ where\n$B(\\Bbbk E)=\\Bbbk\\lbrack E]\/(E^{m})$ with $G$-comodule structure given by\n$\\delta(E)=K\\otimes E$ and $G$-module structure given by $KE=qE$. Then $\\Bbbk\nE\\i\n\\genfrac{}{}{0pt}{}{\\Bbbk G}{\\Bbbk G\n\\mathcal{YD}$ and $B(\\Bbbk E)\\#C_{m}$ is a Hopf algebra. Its quiver is a\ndirected cycle $Q_{m}$ of length $m.$ The finite Hopf coverings of $B(\\Bbbk\nE)\\#\\Bbbk C_{m}$ of have underlying coalgebra coverings $B(\\Bbbk\nE)\\#C_{n}\\rightarrow$ $B(\\Bbbk E)\\#C_{m}$ with $m|n.$ Setting $n=\\infty$ (so\n$G=<g>$ is infinite cyclic) results in the universal covering Hopf algebras\nwhose quiver is of type $\\mathbb{A}_{\\infty}^{\\infty}$ with unidirectional\narrows. These Hopf algebras appear in \\cite{DIN2013}.\n\nOne may also consider the path coalgebras of the cyclic quivers $Q_{n},$ and\nquivers of type $\\mathbb{A}_{\\infty}^{\\infty}$ with unidirectional arrows for\n$m\\in\\mathbb{Z}^{+}\\cup\\{\\infty\\}.$ As in \\cite{CiRo2002} the path coalgebras\ncan be furnished with a Hopf algebra structure depending on an $m$th root of\nunity $q$ where where $m|n$ or $n=\\infty$. The Hopf coverings for fixed $m$\n(and an $m$th root of 1) are $\\Bbbk Q_{n}\\rightarrow\\Bbbk Q_{m}$ where $m|n$\nor $n=\\infty.$ The Hopf algebras in the previous paragraph are the\nbosonizations of Nichols algebras for these infinite dimensional Hopf algebras.\n\n\\subsection{Fomin-Kirillov algebras \\label{FK Ex}}\n\nLet $X$ be the rack of transpositions of $\\mathbb{S}_{n},$ $n>2$ and consider\nthe versions of $V=(\\Bbbk X,c^{q})$ for the cocycles as in \\cite{GHV2011}.\nThen the quadratic version $\\hat{B}_{2}(V)$ is the algebra from \\cite{FK99\n,\\cite{MiS2000}; cf. \\cite{AS2002}. It is known that $B(V)=\\hat{B}_{2}(V)$ is\nfinite dimensional for $n\\leq5,$ but this problem has been open for $n>5$ for\nmore than a decade.\n\nOne can enumerate the orbits (including orbits of size 1) of $c$ on $X\\times\nX$ as follows\n\n\\\n\\begin{tabular}\n[c]{|c|c|}\\hline\nOrbits of size & \\#orbits\\\\\\hline\n1 & $\\binom{n}{2}=d$\\\\\\hline\n2 & $\\frac{1}{2}\\binom{n}{2}\\binom{n-2}{2}$\\\\\\hline\n3 & $2\\binom{n}{3}$\\\\\\hline\n\\end{tabular}\n\\]\n\n\nThe total number of $c$-orbits is\n\\begin{align*}\nf(n)  &  =\\frac{n\\left(  n-1\\right)  }{2}+\\frac{n\\left(  n-1\\right)  \\left(\nn-2\\right)  \\left(  n-3\\right)  }{8}+\\frac{n\\left(  n-1\\right)  \\left(\nn-2\\right)  }{3}\\\\\n&  =\\allowbreak\\frac{1}{24}n\\left(  3n^{3}-10n^{2}+21n-14\\right)  .\n\\end{align*}\n\n\nThe number of orbits in excess of $\\binom{d}{2}$ i\n\\begin{align*}\nf(n)-\\binom{d}{2}  &  =\\frac{1}{24}n\\left(  3n^{3}-10n^{2}+21n-14\\right) \\\\\n&  -\\allowbreak\\frac{1}{8}n\\left(  n-1\\right)  \\left(  n-2\\right)  \\left(\nn+1\\right) \\\\\n&  =\\allowbreak-\\frac{1}{6}n\\left(  n-1\\right)  \\left(  n-5\\right)\n\\end{align*}\n\n\nThere are fewer quadratic relations (=\\#orbits) than $\\binom{d}{2}$ (recall\n$d=\\binom{n}{2})$ for $n>5.$ There is a full set of quadratic relations, but\nnot \"many\" in the sense of \\cite{GHV2011}\n\n\\\n\\begin{tabular}\n[c]{|c|c|c|}\\hline\n$n$ & \\#orbits in $X\\times X=f(n)$ & \\#orbits$-\\binom{d}{2}$\\\\\\hline\n$3$ & $5$ & $2$\\\\\\hline\n$4$ & $17$ & $2$\\\\\\hline\n$5$ & $45$ & $0$\\\\\\hline\n$6$ & $100$ & $-5$\\\\\\hline\n\\end{tabular}\n\\]\n\n\n\\subsection{Nonabelian group type\\label{nonabelian ex}}\n\nWe adopt the list of some racks from e.g. \\cite{GHV2011},\\cite{GVZoo\n,\\cite{MiS2000},\\cite{HLV2012}. We will consider the racks $\\mathcal{A\n$,$\\mathcal{B}$,$\\mathcal{C}$,$\\mathcal{T}$ and the affine racks\n$\\mathrm{Aff}(5,2),\\mathrm{Aff}(5,3),\\mathrm{Aff}(7,5),\\mathrm{Aff}(7,3)$ with\n$2$-cocyles as in the references which result in finite-dimensional Nichols algebras.\n\nLet $\\mathcal{S}_{n}$ denote the rack of transpositions in $\\mathbb{S}_{n},$\n$n\\geq3$ (as above). Let $\\mathcal{B}$ be the rack of $4$-cycles in\n$\\mathbb{S}_{4}.$ Let $\\mathcal{D}_{4}$ denote the rack of $4$ reflections\n(transpositions) in the dihedral group $\\mathbb{D}_{4}$ (order $8$). Hence\n$\\mathcal{D}_{3}=\\mathcal{S}_{3}$, $\\mathcal{A=S}_{4}$ and $\\mathcal{C\n=\\mathcal{S}_{5}.$ Let $d=\\dim V$. All but the last two rows have\nindecomposable racks corresponding to irreducible Yetter-Drinfeld modules. All\nbut the last row have Nichols algebras with a full set of quadratic relations.\nThe last two rows have decomposable racks. The example over $\\mathbb{D}_{4}$\nis from \\cite[Example 6.5]{MiS2000}; since the center of $D_{4}$ acts\ntrivially on the Yetter -Drinfeld module $V,$ the braiding reduces to the\nKlein $4$-group $\\mathbb{V}$ and the $\\mathbb{D}_{4}$ bosonization is a double\ncover of the smaller Hopf algebra over $\\mathbb{V}$.\n\nThe two newer examples in \\cite[ Prop. 32, 36]{HLV2012} (over $\\mathcal{D\n_{3}$ and $\\mathcal{T}$ in rows 2 and 5 in table below, respectively) do have\na full set of quadratic relations, but do not have relations of form $x^{2}$.\n\nIn the last row, the Nichols algebra of type finite Cartan type of rank $2$ is\nseen to have no quadratic relations (where the root of unity has order\n$>\n$2$) because the Serre relations are cubic. It can be shown that $G_{X\n=\\tilde{G}$ and is free abelian for Nichols algebras of finite Cartan type.\n\nThe computation of the enveloping groups and there centers was done with the\naid of GAP, or done by hand. The fact concerning $G_{\\mathbb{S}_{n}}$and its\ncenter are from \\cite[Prop. 3.2]{AFGV2011}\n\n\\\n\\begin{tabular}\n[c]{|r|r|r|r|r|r|}\\hline\nRack $X$ & rank $d$ & $Z(G_{X})$ & $G_{X}\/Z(G_{X})$ & $\\#$orbits &\n\\#QR\\\\\\hline\n$\\mathcal{S}_{n}$ & $\\binom{n}{2}$ & $C_{\\infty}$ & $\\mathbb{S}_{n}$ & $f(n) $\n& $f(n)$\\\\\\hline\n$\\mathcal{D}_{3}$ & $3$ & $C_{\\infty}$ & $\\mathbb{S}_{n}$ & $5$ & $2$\\\\\\hline\n$\\mathcal{B}$ & $6$ & $C_{\\infty}$ & $\\mathbb{S}_{4}$ & $17$ & $17$\\\\\\hline\n$\\mathcal{T}$ & $4$ & $C_{\\infty}\\times C_{2}$ & $\\mathbb{A}_{4}$ & $8$ & $8\n$\\\\\\hline\n$\\mathcal{T}$ & $4$ & $C_{\\infty}\\times C_{2}$ & $\\mathbb{A}_{4}$ & $8$ & $4\n$\\\\\\hline\n$\\mathrm{Aff}(5,2)$ & $5$ & $C_{\\infty}$ & $C_{5}\\rtimes C_{4}$ & $10$ & $10\n$\\\\\\hline\n$\\mathrm{Aff}(5,3)$ & $5$ & $C_{\\infty}$ & $C_{5}\\rtimes C_{4}$ & $10$ & $10\n$\\\\\\hline\n$\\mathrm{Aff}(7,3)$ & $7$ & $C_{\\infty}$ & $C_{7}\\rtimes C_{6}$ & $21$ & $21\n$\\\\\\hline\n$\\mathrm{Aff}(7,5)$ & $7$ & $C_{\\infty}$ & $C_{7}\\rtimes C_{6}$ & $21$ & $21\n$\\\\\\hline\n$\\mathcal{D}_{4}$ & $4$ & $C_{\\infty}\\times C_{\\infty}\\times C_{2}$ &\n$C_{2}\\times C_{2}$ & $4$ & $4$\\\\\\hline\nrank 2 & $2$ & $C_{\\infty}\\times C_{\\infty}$ &  & $2$ & $0$\\\\\\hline\n\\end{tabular}\n\\]\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Contents}\n\t\\leavevmode\n\t\\par\n\t\\listoftoc*{toc}\n}\n\n\\renewcommand{\\topfraction}{1}\n\\renewcommand{\\textfraction}{0}\n\\renewcommand{\\bottomfraction}{1}\n\\renewcommand{\\floatpagefraction}{1}\n\\begin{document}\n\\maketitle\n\\nonumberfootnote{\n\tPublished in Journal of Algebra, \n\tdoi: \\href{http:\/\/dx.doi.org\/10.1016\/j.jalgebra.2021.02.024}{10.1016\/j.jalgebra.2021.02.024}.\n}\n\n\n\\begin{abstract}\n\t\\noindent\n\t{\\bfseries Abstract}\\enspace\n\tFor a semisimple complex Lie algebra $\\mathfrak g$,\n\tthe BGG category $\\mathcal{O}$ is of particular interest in representation theory.\n\tIt is known that Irving's shuffling functors $\\Sh{w}$,\n\tindexed by elements $w\\in W$ of the Weyl group,\n\tinduce an action of the braid group $B_W$ associated to $W$\n\ton the derived categories $D^\\mathrm{b}(\\mathcal{O}_\\lambda)$ of blocks of $\\mathcal{O}$.\n\t\n\tWe show that for maximal parabolic subalgebras $\\mathfrak{p}$ of $\\mathfrak{sl}_n$\n\tcorresponding to the parabolic subgroup $W_\\mathfrak{p}=S_{n-1}\\times S_1$ of $S_n$,\n\tthe derived shuffling functors $\\LSh{s_i}$\n\tare instances of Seidel and Thomas' spherical twist functors.\n\tNamely, we show that certain parabolic indecomposable projectives $P^\\mathfrak{p}(w)$\n\tare spherical objects,\n\tand the associated twist functors are naturally isomorphic to $\\LSh{w}\\hShift{1}$\n\tas auto-equivalences of $D^\\mathrm{b}(\\mathcal{O}^\\mathfrak{p})$.\n\t\n\tWe give an overview of the main properties of the BGG category $\\mathcal{O}$,\n\tthe construction of shuffling and spherical twist functors,\n\tand give some examples how to determine images of both.\n\tTo this end, we employ the equivalence of blocks of $\\mathcal{O}$ \n\tand the module categories of certain path algebras.\n\\end{abstract}\n\n\\thispagestyle{empty}\n\\enlargethispage*{4cm}\n\\tableofcontents\n\n\\pagebreak\n\n\\section{Introduction}\nConsider a finite dimensional semisimple complex Lie algebra $\\mathfrak g$\nwith Cartan subalgebra $\\mathfrak h$ that gives rise to a root system $\\Phi$\nand a root space decomposition $\\mathfrak g = \\bigoplus_{\\alpha \\in \\Phi} \\mathfrak g_\\alpha$.\nA choice of simple roots $\\Delta$ fixes the positive roots $\\Phi^+$,\nthe corresponding subalgebra $\\mathfrak n \\coloneqq \\bigoplus_{\\alpha \\in \\Phi^+} \\mathfrak g_\\alpha$\nand the corresponding Borel subalgebra $\\mathfrak b = \\mathfrak h \\oplus \\mathfrak n$.\n\nRepresentations of $\\mathfrak g$ are equivalent to modules\nover the universal enveloping algebra $U(\\mathfrak g)$ \\autocite[\u00a7V]{Humphreys:Lie}.\nThe \\emph{BGG category} $\\mathcal{O}$ of $\\mathfrak g$\nis the full subcategory of $\\Mod{U(\\mathfrak g)}$\nconsisting of modules that\n\\begin{thmlist*}[label=($\ud835\udcde$\\arabic*)]\n\t\\item are finitely generated,\n\t\\item have a weight space decomposition $M=\u2a01_{\ud835\udf06\u2208\ud835\udd25^*} M_\ud835\udf06$ and\n\t\\item are locally $\ud835\udd2b$-finite; \\ie, for every $v\u2208M$, the orbit $U(\ud835\udd2b^+)v$ is finite dimensional.\n\\end{thmlist*}\n\n\n\\subsection{Category \\texorpdfstring{$\\mathcal{O}$}{\ud835\udcde}, blocks and shuffling functors}\nDenote the Weyl group of $\\mathfrak g$ by $W$.\nThe half-sum of positive roots $\\rho \\coloneqq \\frac{1}{2}\\sum_{\\alpha \\in \\Phi^+} \\alpha$\ngives rise to the dot-action $w\\cdot\\colon \\lambda \\mapsto w(\\lambda+\\rho)-\\rho$.\nThe $\\Complex$-span of $\\Phi$ is a Euclidean space\nand its inner product $(-,-)$ fixes the coroots $\\check{\\alpha} \\coloneqq \\frac{2\\alpha}{(\\alpha,\\alpha)}$\nand the \\term{fundamental weights} $\\varpi_{\\alpha}$ for $\\alpha \\in \\Delta$\ndefined by $\\frac{(\\varpi_\\alpha, \\beta)}{(\\beta, \\beta)}$ for $\\alpha, \\beta \\in \\Delta$.\nWe call a weight $\\lambda \\in \\Complex\\Phi$ \\emph{integral}\nif $\\frac{2(\\lambda, \\alpha)}{(\\alpha,\\alpha)} \\in \\mathbf{Z}$\nand denote by $\\Lambda$ the set of all integral weights.\nWe call a $\\mathbf{Z}_{\\geq 0}$-linear combination of the $\\varpi_{\\alpha}$'s \\emph{dominant}\nand denote the set of all dominant weights by $\\Lambda^+$.\nWe call a weight $\\lambda$ \\term{$\\rho$-dominant}\nif $\\frac{2(\\lambda + \\rho, \\alpha)}{(\\alpha, \\alpha)} \\notin \\mathbf{Z}_{<0}$ for all $\\alpha \\in \\Delta$.\nTo a weight $\\lambda$ we associate the subgroup\n$W_\\lambda \\coloneqq \\langle s_\\alpha \\mid \\frac{2(\\lambda + \\rho, \\alpha)}{(\\alpha,\\alpha)} \\in \\mathbf{Z} \\rangle \\leq W$;\nSee \\autocites{Humphreys:CatO}{Jantzen:Moduln-mit-hoechstem-Gewicht} for details.\n\nThe category $\\mathcal{O}$ has a decomposition\n$\\mathcal{O}=\\bigoplus_{\\lambda}\\mathcal{O}_\\lambda$ into \\term{blocks} $\\mathcal{O}_\\lambda$,\nindexed by the $\\rho$-dominant weights $\\lambda$,\neach of which consists of the modules of highest weight in $W_\\lambda\\cdot\\lambda$\n\\autocite[thm\\ 4.9]{Humphreys:CatO}%\n\\footnote{Humphreys indexes blocks by $\\rho$-antidominant weights, which results in slightly different formulation for some statements.}.\nIn particular, Each block $\\mathcal{O}_\\lambda$ contains the simple modules $L(w\\cdot \\lambda)$,\nthe indecomposable projectives $P(w\\cdot\\lambda)$\nand the Verma modules $M(w\\cdot\\lambda)$\nof highest weight $w\\in W\/W_\\lambda$.\nIf a block $\\mathcal{O}_\\lambda$ is fixed,\nwe just write $L(w)$, $P(w)$ and $M(w)$ for the respective objects therein.\nEach block $\\mathcal{O}_\\lambda$ is Morita equivalent \nto modules over a quasi-hereditary algebra \\autocite{BGG}.\n\nA weight $\\lambda$ is called \\term{regular} \nif its \\term{stabilizer subgroup} $W_\\lambda$ \\wrt\\ the dot-action is trivial;\n\\ie, if $\\lambda$ does not lie on any reflection plane.\nAll blocks $\\mathcal{O}_\\lambda$ associated to regular weights are equivalent as categories;\nin the following we shall thus work in the block $\\mathcal{O}_0$\ncontaining the trivial $\\mathfrak g$-representation $L(e\\cdot 0)=\\Complex$,\nwhich is called the \\term{principal block}.\n\n\\begin{definition}\n\tA \\emph{Coxeter system} consists of a group $W$, a fixed set $S$ of generators and a presentation \n\t$W=\u27e8s\u2208S \\mid s^2 = e, sts\\dotsm = tst\\dotsm \u27e9$ with $m_{st}$ factors $s,t$ on both sides.\n\tThe $s\\in S$ are called \\term{simple reflections}.\n\tThe matrix $(m_{st})_{s,t\\in W}$ is called the \\term{Coxeter matrix} of $W$.\n\tTo $W$, there is the associated \\term{braid group}\n\t$B_W=\u27e8s\u2208S \\mid sts\\dotsm = tst\\dotsm\u27e9$,\n\tsuch that there is a natural quotient map $B_W\\onto W$.\n\tA finite Coxeter system has a \\term{length function} $\\ell\\colon W \\to \\mathbf{N}_0$,\n\twhich assigns to an element $w \\in W$ the length of a shortest expression for $w$\n\tin terms of $S$.\n\tWith respect to $\\ell$, there is a unique longest element $w_0 \\in W$.\n\\end{definition}\n\n\\begin{example}\n\tThe Weyl group of $\\mathfrak g$ is a Coxeter group.\n\tIn particular, the symmetric group $S_n$,\n\twhich is the Weyl group of $\\mathfrak{sl}_n$,\n\tis a Coxeter group, \n\tgenerated by the simple reflections $s_1,\\dotsc,s_{n-1}$.\n\tIts Coxter matrix has entries\n\t$m_{s_i,s_j} = \\begin{smallcases}\n\t1 & \\text{if $i=j$}\\\\ 3 & \\text{if $\\lvert i-j\\lvert = 1$}\\\\2 & \\text{otherwise}\n\t\\end{smallcases}$.\n\tFor the symmetric group, $B_n \u2254 B_{S_n}$ is the well-known Artin braid group.\n\\end{example}\n\nFor weights $\\lambda, \\mu \\in \\mathfrak{h}^*$ with $\\lambda - \\mu \\in \\Lambda$,\nthere is a unique $\\nu \\in W(\\mu-\\lambda)$ (\\wrt\\ the ordinary $W$-action)\nsuch that the simple module $L(\\nu)$ is finite-dimensional,\nwhich is the case if and only if $\\nu$ is dominant\n\\autocite[thm.\\ 1.6]{Humphreys:CatO}.\n\n\\begin{definition}[{\\autocite[54]{Jantzen:Moduln-mit-hoechstem-Gewicht}}]\n\tThe \\term{translation functor} $T_\\lambda^\\mu\\colon \\mathcal{O}_\\lambda \\to \\mathcal{O}_\\mu$\n\tassigns to $M$ the direct summand of $M \\otimes L(\\nu)$ lying in $\\mathcal{O}_\\mu$.\n\\end{definition}\n\nThe functor $T_\\lambda^\\mu$ is exact, preserves projectives, commutes with duality,\nand is biadjoint to $T_\\mu^\\lambda$.\nFrom now on, we assume that $\\lambda, \\mu$ are integral.\nFor most of what we need, this is stricter than necessary,\nbut sufficient for our needs;\nsee \\autocite[\u00a77]{Humphreys:CatO} for a more general treatment\nand for an overview of properties of $T_\\lambda^\\mu$.\n\n\\begin{definition}\n\tLet $s\\in W$ be a simple reflection\n\tand $\\mu$ be an integral weight with stabiliser $W_\\mu=\\{e,s\\}$.\n\tThe \\term{translation through the $s$-wall} is the composition \n\t$\\Theta_s\\coloneqq T_\\mu^0 T_0^\\mu\\colon \\mathcal{O}_0 \\to \\mathcal{O}_0$.\n\\end{definition}\n\nAs notation suggests, $\\Theta_s$ is independent of the choice of $\\mu$.\nIt is an exact self-adjoint auto-equivalence of the block $\\mathcal{O}_0$\n\\autocite[\u00a72.10]{Jantzen:Moduln-mit-hoechstem-Gewicht}.\nIt is uniquely determined by the existence of short exact sequences\n\\begin{equation}\n\t\\label{eqn:translation-defining-ses}\n\t0\\to M(w)\\to \\Theta_s M(w) \\to M(ws) \\to 0\n\t\\quad\\text{and}\\quad\n\t\\Theta_s M(w)\\cong\\Theta_s M(ws)\n\\end{equation}\nfor $w<ws$ \\autocite[Satz 2.10]{Jantzen:Moduln-mit-hoechstem-Gewicht}.\nFurthermore, $\\Theta_s^2 = \\Theta_s \u2295 \\Theta_s$ \\autocite[\u00a77.14]{Humphreys:CatO}.\n\n\\begin{definition}[{\\autocites[\u00a72]{Carlin:Extensions-of-Vermas}[\u00a73]{Irving:Shuffled-Verma-Modules}}]\n\t\\label{def:shuffling-functor}\n\tFrom the adjunctions $T_\\mu^0 \\dashv T_0^\\mu$ and $T_0^\\mu \\dashv T_\\mu^0$ \n\twe get adjunction maps $\\eta_s\\colon \\id \\Rightarrow \\Theta_s$ \n\tand $\\varepsilon_s\\colon \\Theta_s \\Rightarrow \\id$,\n\tand we define the mutually adjoint \\term{shuffling} and the \\term{coshuffling functor} $\\Sh{s}$ and $\\Csh{s}$ as\n\t\\begin{equation*}\n\t\t\\Sh{s}  \u2254 \\coker(\\eta_s)\\colon \\mathcal{O}_0 \\to \\mathcal{O}_0,\n\t\t\\qquad\n\t\t\\Csh{s} \u2254 \\ker(\\varepsilon_s)\\colon \\mathcal{O}_0 \\to \\mathcal{O}_0.\n\t\\end{equation*} \n\\end{definition}\n\nIn particular, \\cref{eqn:translation-defining-ses} implies that for $w < ws$,\n\\begin{equation}\n\t\\label{eq:shuffling-on-Vermas}\n\t\\Sh{s} M(w) = M(ws) \\quad \\text{and} \\quad \\Csh{s} M(ws) = M(w).\n\\end{equation}\nBy the snake lemma,\n$\\Sh{s}$ is right exact and $\\Csh{s}$ is left exact,\nso we can consider the mutually inverse \\autocite[thm.\\ 5.7]{stroppel:translation-and-shuffling} derived functors\n$\\LSh{s}$ and $\\RCsh{s}$.\n\nIn general, we can see an adjunction morphism $\\eta_s$ on an abelian category $\\mathcal C$\nas a functor from $\\mathcal C$ to the arrow category $\\mathcal C^{[1]}$.\nBy the snake lemma, the cokernel is a right exact functor functor $\\mathcal C^{[1]} \\to \\mathcal C$,\nwhose derived functor $\\mathbf L \\coker\\colon D^\\mathrm{b}(\\mathcal C^{[1]}) \\to D^\\mathrm{b}(\\mathcal{C})$\nis the mapping cone \\autocite[thm\\ 3.5.6]{groth:derivators}.%\n\\footnote{%\n\tAs the inclusion $D^\\mathrm{b}(\\mathcal C^{[1]}) \\into D^\\mathrm{b}(\\mathcal C)^{[1]}$ is no equivalence in general,\n\tone has to pay attention to the fact that\n\tthe mapping cone is a functor on coherent diagrams $D^\\mathrm{b}(\\mathcal C^{[1]})$,\n\tbut not necessarily on general diagrams $D^\\mathrm{b}(\\mathcal C)^{[1]}$.\n}\nWe thus obtain that $\\LSh{s} \\cong\\cone(\\eta_s)$ \nand dually that $\\RCsh{s}\\cong\\cocone(\\varepsilon_s) = \\cone(\\varepsilon_s)\\hShift{1}$.\nThe snake lemma also implies that $\\mathbf L^i\\Sh{s} = 0\\ \\text{for $i\\neq 0,1$}$ \nand $\\mathbf R_j \\Csh{s} = 0\\ \\text{for $j\\neq -1,0$}$.\n\n\\begin{definition}\n\tWe say that an object $M \\in \\mathcal C$ is $F$-acyclic\n\tfor a left or right exact functor $F\\colon \\mathcal C \\to \\mathcal D$ of abelian categories\n\tif $\\mathbf R_i F M = 0$ or $\\mathbf L^i M = 0$ for $i \\neq 0$, respectively.\n\\end{definition}\n\nProjective (respectively, injective) objects are acyclic for every right (resp., left) exact functor.\nSince the adjunction unit $\\eta_s\\colon M(w) \\to \\Theta_s$ \nis injective for all $w \\in W$ \\autocite[thm\\ 12.2]{Humphreys:CatO},\nevery $M(w)$ is both $\\Sh{s}$-acyclic.\n\n\n\\subsection{\\texorpdfstring{Actions of $W$ on $K_0(\\mathcal{O}_0)$ and $D^\\mathrm{b}(\\mathcal{O}_0)$}{Actions of \ud835\udc4a on \ud835\udc3e\u2080(\ud835\udcde\u2080) and D\u1d47(\ud835\udcde\u2080)}}\n\\begin{definition}\n\tThe \\term{Grothendieck group} $K_0(\\mathcal C)$ of an abelian category $\\mathcal C$\n\tis the abelian group generated by symbols $[M]$ for the objects $M\\in\\mathcal C$,\n\tsubject to the relation $[E]=[A]+[B]$ \n\twhenever $E$ is an extension of $A$ by $B$.\n\tSimilarly, the Grothendieck group $K_0(\\mathcal T)$ of a triangulated category $\\mathcal T$\n\tis the abelian group generated by symbols $[M]$ for the objects $M\\in\\mathcal{T}$,\n\tsubject to the relation $[E]=[A]+[B]$ whenever $A \\to E \\to B \\to A\\hShift{-1}$ is a triangle.\n\\end{definition}\n\nAn exact (respectively, triangulated) functor induces a group homomorphism\non $K_0(\\mathcal C)$ (respectively, $K_0(\\mathcal T)$).\nThe inclusion $\\mathcal C \\into D^\\mathrm{b}(\\mathcal C)$\ninduces an isomorphism $K_0(\\mathcal C)\\cong K_0(D^\\mathrm{b}(\\mathcal C))$ \nof abelian groups \\autocite{Grothendieck:Tohoku}.\n\nFor the category $\\mathcal{O}_0$, the group $K_0(\\mathcal{O}_0)$ is free,\nand each of the collections $\\{L(w)\\}_{w\\in W}$, $\\{M(w)\\}_{w\\in W}$ and $\\{P(w)\\}_{w\\in W}$\nis a $\\mathbf Z$-basis of it.\nFor the simple modules, this follows from the fact\nthat every object an $\\mathcal{O}_0$ has a composition series\n(see \\cref{sec:composition-series}),\nand for the others from the fact that each $[L(w)]$\ncan be written uniquely in terms of the $M(u)$'s or $P(v)$'s,\ndue to the composition series of the latter.\n\nThe shuffling functors induce a right action of $W$ on $K_0(\\mathcal{O}_0)$,\ngiven for a $w \\in W$ and a simple reflection $s \\in W$ by the assignment\n\\begin{equation}\n\tW \\longto \\operatorname{Aut}(K_0(\\mathcal{O}_0)),\\quad\n\ts \\longmapsto [\\Sh{s}]\\colon [M(w)] \\mapsto [M(ws)];\n\\end{equation}\nindeed, we even have $\\Sh{s} M(w) = M(ws)$ for $w \\leq ws$\nand from \\cref{eqn:translation-defining-ses} and from the $\\Sh{s}$-acyclicity of Verma modules\nwe learn that $M(w)$ and $\\Sh{s}(ws)$ have the same composition factors\nsuch that $[M(w)] = \\sum_{v} [M(w) : L(v)] [L(v)] = [M(ws)]$;\nhence, the above action is well-defined on $W$.\nThis action actually comes from an action on the derived category $\\mathcal{O}_0$:\n\n\\begin{theorem}[{\\autocite[thm.\\ 10.4]{rouquier:categorification}}]\n\t\\label{thm:roquier-braid-group-action}\n\tThe assignment \n\t\\begin{equation}\n\t\tB_W\\to\\operatorname{Aut}\\bigl(D^\\mathrm{b}(\\mathcal{O}_0)\\bigr),\\quad\n\t\ts\\mapsto\\LSh{s},\\quad \n\t\ts^{-1}\\mapsto\\RCsh{s}\n\t\\end{equation}\n\tdefines a weak action of $B_W$ on the derived category $D^\\mathrm{b}(\\mathcal{O}_0)$,\n\tmeaning that for simple reflections $s,t \\in W$,\n\tthere are natural isomorphisms\n\t$\\LSh{s} \\LSh{t} \\cong \\LSh{t} \\LSh{s}$ whenever $st = ts$\n\tand $\\LSh{s} \\LSh{t} \\LSh{s} \\cong \\LSh{t} \\LSh{s} \\LSh{t}$ whenever $sts = tst$,\n\tand the following square commutes:\n\t\\begin{equation*}\n\t\t\\begin{tikzcd}\n\t\t\tB_W \\ar[d, \"\\LSh{(-)}\"'] \\ar[r, \"\\can\", two heads]  & W \\dar[\"{[\\Sh{(-)}]}\"] \\\\\n\t\t\t\\operatorname{Auteq}(D^\\mathrm{b}(\\mathcal{O}_0)) \\ar[r, \"{[-]}\"'] & \\operatorname{Aut}(K_0(\\mathcal{O}_0)).\n\t\t\\end{tikzcd}\n\t\\end{equation*}\n\\end{theorem}\n\n\n\\subsection{Seidel and Thomas' spherical twist functors}\nLet $k$ be a field and $\\mathcal C$ be a $k$-linear abelian category $\\mathcal C$ of finite global dimension.\nSeidel and Thomas have constructed\nan action of the braid group $B_n\\coloneqq B_{S_n}$ of the symmetric group $S_n$ on $\\mathcal C$\nin terms of \\term{spherical objects} \\autocite{seidel-thomas:braid-group-actions}.\nWe give a short summary of their construction.\n\n\\begin{definition}\n\tLet $X, Y\\in\\Ch(\\mathcal C)$ be chain complexes.\n\tWe employ the convention that the homological shift upwards; \\ie, $X\\hShift{1}_j \\coloneqq X_{j-i}$.\n\tWe define the graded $k$-vector space $\\Hom^*_{D^\\mathrm{b}(\\mathcal C)}(X,Y)$\n\twith graded components $\\Hom^j_{D^\\mathrm{b}(\\mathcal C)}(X,Y) \\coloneqq \\Hom_{D^\\mathrm{b}(\\mathcal C)}(X\\hShift{j}, Y)$.\n\t\n\tWe denote by $\\hom^\\bullet_\\mathcal C(X, Y) \\in \\Ch(\\VS{k})$ the chain complex with homological components\n\t$\\hom^j_\\mathcal C(X, Y) \\coloneqq \\Hom_k(X\\hShift{j}, Y)$ the graded $k$-linear morphisms $X\\to Y$ \n\t(not necessarily chain maps),\n\tand differential $\\mathrm d_{\\hom}$\n\tdefined on homogeneous elements of degree $j$\n\tby $\\mathrm d_{\\hom}(f)=\\mathrm d_Y f-(-1)^j f \\mathrm d_X$.\n\tIts cohomology $H^j \\hom^\\bullet_\\mathcal C(X,Y) = \\Hom_{K^\\mathrm{b}(\\mathcal C)}(Y\\hShift{j}, Y)$\n\tis the (honest) the Hom-space in the bounded homotopy category $K^\\mathrm{b}(\\mathcal C)$ \\autocite[\u00a72.7.5]{Weibel:Hom-Alg}.\n\\end{definition}\n\n\\begin{definition}\n\tLet $X \\in \\Ch(\\mathcal C)$ and $V \\in \\Ch(\\VS{k})$.\n\tSince $\\mathcal C$ is a $k$-linear category,\n\twe can regard $X$ as a complex of vector spaces,\n\ton which a $\\lambda \\in k$ acts  by $\\lambda x \\coloneqq (\\lambda\\id_X)(x)$ for $x \\in X$.\n\tWe define the complex $\\lin^\\bullet(V, X) \\in \\Ch(\\mathcal C)$ \n\tof \\term{$k$-linear maps} from $V$ to $X$\n\twith differential $(\\mathrm d_{\\lin}f) v \\coloneqq (-1)^{\\deg v}[\\mathrm dfv - f\\mathrm dv]$\n\tand the \\term{tensor product} complex $V \\otimes X$\n\twith differential $\\mathrm d_{V \\otimes X}(v \\otimes x) \\coloneqq \\mathrm dv \\otimes x + (-1)^{\\deg v} v \\otimes \\mathrm dx$ for homogeneous $v$.\n\\end{definition}\n\n\\begin{remark}\n\t\\label{rmk:lin-and-tensor}\n\tFor a homogeneous basis $B$ of $V$ and $v \\in V$,\n\tlet $\\beta_b(v)$ be the coefficients such that\n\t$v = \\sum_{b \\in B} \\beta_b(v) b$.\n\tThere are homogeneous isomorphisms of graded vector spaces\n\t\\begin{align*}\n\t\tV \\otimes X           &\\stackrel\\cong\\longto \\textstyle\\bigoplus_{b \\in B} X\\hShift{-\\deg b}_b,&\n\t\t\\lin^\\bullet(V, X)    &\\stackrel\\cong\\longto \\textstyle\\prod_{b \\in B} X \\hShift{\\deg b}_b, \\\\\n\t\t%\n\t\tv\\otimes x            &\\longmapsto \\textstyle\\sum_b \\beta_b(v) x_b &\n\t\tf                     &\\longmapsto (f(b))_b;\n\t\\end{align*}\n\tThe differential on $V \\otimes X$ and $\\lin^\\bullet(V, X)$ is specified by\n\t\\begin{align*}\n\t\t\\mathrm d_{V \\otimes X}(x_b) &= \n\t\t\t\\textstyle\\sum_{b' \\in B} \\beta_{b'}(\\mathrm db) x_{b'} + (-1)^{\\deg b} (\\mathrm d x)_b,\\\\\n\t\t\\mathrm d_{\\lin^\\bullet(V, X)}\\bigl((x_b)_{b \\in B}\\bigr) &= \\textstyle\n\t\t\t\\bigl((-1)^{\\deg b} \\bigl(\\mathrm dx_b - (\\sum_{b' \\in B}\\beta_{b}(\\mathrm db') x_b) \\bigr) \\bigr)_{b \\in B};\n\t\\intertext{\n\t\tif $V$ is degree-wise finite dimensional,\n\t\tthe complex $\\lin(V,X) \\cong V^*\\otimes X$\n\t\tis spanned by maps $x_b \\colon b' \\mapsto \\delta_{bb'} x$\n\t\t(where $\\delta$ denotes the Kronecker-$\\delta$),\n\t\tand we get\n\t}\n\t\t\\mathrm d_{\\lin^\\bullet(V, X)}(x_b) &= \\textstyle\n\t\t\t(-1)^{\\deg b} \\bigl( (\\mathrm dx_b) - \\sum_{b' \\in B}\\beta_{b}(\\mathrm db') x_{b'}\\bigr).\n\t\\end{align*}\n\\end{remark}\n\n\n\\begin{definition}\n\t\\label{def:spherelike-and-spherical}\n\t\\Needspace{3\\baselineskip}\n\tAn object $E\\inD^\\mathrm{b}(\\mathcal C)$ is called \\term{$d$-spherelike} for $d\\geq 0$\n\tif\n\t\\begin{thmlist}[label={(S\\arabic*)}, ref={\\thetheorem(S\\arabic*)}]\n\t\t\\item\\label{def:spherical-object:finite-total-dimension}\n\t\t\tfor any $F\u2208D^\\mathrm{b}(\ud835\udcd2)$, \n\t\t\tthe spaces $\\Hom^*_{D^\\mathrm{b}(\ud835\udcd2)}(E, F)$ and $\\Hom^*_{D^\\mathrm{b}(\ud835\udcd2)}(F, E)$ are of finite total dimension, and\n\t\t\\item\\label{def:spherical-object:spherical-cohomology}\n\t\t\t$\\Hom^*_{D^\\mathrm{b}(\ud835\udcd2)}(E, E) \u2245 k[x]\/(x^2)$ as graded algebras,\n\t\t\twhere $x$ is a morphism of degree $d$.\n\t\\end{thmlist}\n\tIt is called \\term{$d$-spherical} if it also has the \\term{Calabi-Yau-property}: \n\t\\begin{thmlist}[resume*]\n\t\t\\item\\label{def:spherical-object:calabi-yau}\n\t\t\tFor all $F$ and $i$,\n\t\t\tcomposition of morphisms gives a non\\-\/degenerate pairing\\\\\n\t\t\t$\\Hom^i_{D^\\mathrm{b}}(F, E) \u2297 \\Hom^{d-i}_{D^\\mathrm{b}}(E, F) \\to \\Hom^d_{D^\\mathrm{b}}(E, E) \\to kx$.%\n\t\t\t\\footnote{\n\t\t\t\tIf $d\\neq 0$, we already have $\\Hom^d_{D^\\mathrm{b}}(E, E) = kx$,\n\t\t\t\tbut if $d=0$, we have $\\Hom^d_{D^\\mathrm{b}}(E, E) = kx \\oplus k\\id_E$.\n\t\t\t\tIn this case, we mean that \n\t\t\t\t$\\circ\\colon\\Hom^i_{D^\\mathrm{b}}(F, E) \u2297 \\Hom^{d-i}_{D^\\mathrm{b}}(F, E) \\to \\Hom^d_{D^\\mathrm{b}}(E, E)\/k\\id_E \u2245 k$\n\t\t\t\tbe non\\-\/degenerate.\n\t\t\t}\n\t\\end{thmlist}\n\\end{definition}\n\n\\begin{definition}\n\t\\label{def:twist-cotwist}\n\tFor $\\Xi_E F \u2254 \\hom^\\bullet_\\mathcal C(E, F)\u2297E $ \n\tand $\\Xi'_E F \\coloneqq \\lin^\\bullet(\\hom^\\bullet_\\mathcal C(F, E), E)$,\n\twe consider the evaluation and coevaluation maps\n\t\\begin{align*}\n\t\t\\mathit{ev}\\colon \\Xi_E F &\\to F,\\ \\phi\u2297f  \\mapsto\\phi(f); &\n\t\t\\mathit{ev}'\\colon F &\\to \\Xi'_E F,\\ f \\mapsto\\phi\\mapsto\\phi(f).\n\t\\end{align*}\n\tIf $E$ is $d$-spherelike,\n\twe define its \\term{spherical twist} and \\emph{cotwist functor}\n\t\\begin{align*}\n\t\t\tT_E &\u2254 \\cone(\\mathit{ev}\\colon\\Xi_E \\Rightarrow\\id_\\mathcal C), &\n\t\t\tT'_E &\u2254 \\cocone(\\mathit{ev}'\\colon\\id_\\mathcal C\\Rightarrow\\Xi'_E);\n\t\\end{align*}\n\there, the grading of the (co)cone is such that $\\id_\\mathcal C$\n\tis in degree zero.\n\\end{definition}\nIsomorphic objects in $D^\\mathrm{b}(\\mathcal C)$ give rise\nto naturally isomorphic twist and cotwist functors \\autocite[prop.\\ 2.6]{seidel-thomas:braid-group-actions},\nhence $T_E$ and $T'_E$ are well-defined.\nThere is an adjunction $T'_E\\dashv T_E$ \\autocite[lem.\\ 2.8]{seidel-thomas:braid-group-actions}.\n\n\\begin{remark}\n\t\\label{rmk:evaluation-in-terms-of-shifted-copies}%\n\t\\label{rmk:induced-map-in-terms-of-shifted-copies}%\n\tIn terms of the descriptions of $\\lin^\\bullet$ and $\\otimes$\n\t\\wrt\\ fixed bases $A$ and $B$ of $\\hom^\\bullet_\\mathcal{C}(E,F)$\n\tand $\\hom^\\bullet_\\mathcal{C}(F,E)$ from \\cref{rmk:lin-and-tensor},\n\tthe evaluation and coevaluation map are the homogeneous maps\n\t\\begin{align*}\n\t\t\\mathit{ev}\\colon \\Xi_E F = \\bigoplus_{\\mathclap{a \\in A}} E \\hShift{\\deg a}_a & \\longto F\n\t\t& \\mathit{ev}' \\colon F & \\longto  \\prod_{\\mathclap{b \\in B}} F \\hShift{-\\deg b}_b = \\Xi'_E F\n\t\t%\n\t\t\\\\\n\t\t(e_a)_a &\\longmapsto \\textstyle \\sum_a a(e_a)\n\t\t& v &\\longmapsto (b(v))_b.\n\t\\end{align*}\n\tSimilarly, a morphism $f\\colon F \\to F'$ of complexes induces maps\n\t\\begin{align*}\n\t\t  f_*\\colon \\Xi_E F &\\longto \\Xi_E F'\n\t\t& f_*\\colon \\Xi'_E F &\\longto \\Xi'_E F'\n\t\t\\\\\n\t\tg\\otimes e &\\longmapsto fg\\otimes e &\n\t\t\\psi &\\longmapsto h \\mapsto \\psi(hf), \\\\\n\t\\intertext{which correspond to}\n\t\t \\bigoplus_{a \\in A}E\\hShift{\\deg a}\n\t\t&\\longto \\bigoplus_{\\mathclap{a' \\in A'}} E\\hShift{\\deg a'}\n\t\t&\\prod_{\\mathclap{b \\in B}} E\\hShift{-\\deg a}\n\t\t&\\longto \\prod_{\\mathclap{b' \\in B'}} E\\hShift{-\\deg b'}\n\t\t\\\\\n\t\te_a &\\longmapsto \\textstyle\\sum_{a' \\in A'} (\\beta_{a'}(af) e_{a})_{a'}\n\t\t& (e_{b})_{b}\n\t\t&\\longmapsto \\bigl(\\textstyle\\sum_{b \\in B} \\beta_b(b'f) e_b\\bigr)_{b'}.\n\t\\end{align*}\n\\end{remark}\n\n\\begin{proposition}[{\\autocite[prop.\\ 2.10]{seidel-thomas:braid-group-actions}}]\n\tIf $E$ is $d$-spherical, then $T_E$ and $T'_E$ are mutually inverse auto-equivalences of $D^\\mathrm{b}(\\mathcal C)$.\n\\end{proposition}\n\n\\begin{notation}\n\t\\label{notation:total-complexes}\n\tGiven a double (or triple) complex $X^{\\bullet\\bullet}$, we denote its $\u2295$-total complex by $\\{X^{\\bullet\\bullet}\\}$.\n\tIn particular, since we can regard a morphism $f\\colon X\\to Y$ of chain complexes\n\ttrivially as a double complex with $Y$ placed in degree zero,\n\tthe mapping cone can be written as $\\cone(f)=\\{f\\colon X\\to Y\\}$.\n\tWe occasionally write a zero below the complex placed in degree zero, so $\\cone(f)=\\{f\\colon X\\to \\degZero{Y}\\}$.\n\\end{notation}\n\n\n\\begin{remark}\n\t\\label{rmk:twisting-E-by-itself}\n\tSince for every $d$-spherical object $E$,\n\t%\n\t$\\hom^\\bullet_\\mathcal C(E, E) \\cong \u27e8\\id\u27e9_k\u2295\u27e8x\u27e9_k \\hShift{d}$ as complexes of vector spaces,\n\twe have\n\t\\begin{equation*}\n\t\tT'_E E = \\{\\begin{psmallmatrix}1\\\\x\\end{psmallmatrix}\\colon \\degZero{E}\\to E\u2295E \\hShift{-d}\\} \\simeq E\\hShift{1-d}.\n\t\\end{equation*}\n\\end{remark}\n\n\\begin{definition}\n\tA collection $\\{E_1,\\dotsc, E_n\\}$ of $d$-spherical objects is an \\term{$\\mathrm{A}_n$-configuration}\n\tif $\\dim\\Hom^*_{D^\\mathrm{b}(\\mathcal C)}(E_i, E_j) = \\begin{smallcases}1 & \\text{if $|i-j|=1$},\\\\0 & \\text{otherwise}\\end{smallcases}$.\n\\end{definition}\n\n\\begin{theorem}[{\\autocite[thm.\\ 2.17]{seidel-thomas:braid-group-actions}}]\n\tGiven an $\\mathrm{A}_n$-configuration $\\{E_1,\\dotsc, E_n\\}$ of $d$-spherical objects,\n\tthe assignment\n\t\\begin{equation*}\n\t\tB_n\\to\\operatorname{Auteq}(D^\\mathrm{b}(\\mathcal C)),\\quad s_i \\mapsto T_{E_i}\n\t\\end{equation*}\n\tdefines a weak action of the braid group.\n\\end{theorem}\n\n\\subsection{Objective}\nThe category $\\mathcal{O}_0$ is a $\\Complex$-linear category of finite global dimension\nand hence satisfies the requirements of \\autocite{seidel-thomas:braid-group-actions}.\nFor $\\mathfrak g=\\mathfrak{sl}_n$, we want to understand\nwhether there is an $\\mathrm{A}_n$-configuration in $D^\\mathrm{b}(\\mathcal{O}_0)$\nsuch that the associated twist functors relate to the shuffling functors.\n\nWe shall show in \\cref{thm:shuffling-is-twisting:sl2}\nthat for $\\mathfrak g = \\mathfrak{sl}_2$, derived shuffling functors are in instance\nof spherical twists for suitable objects,\nand we shall develop the following generalisation.\n\nFor a parabolic subalgebra $\\mathfrak{p} \\subseteq \\mathfrak{sl}_n$\ncorresponding to a parabolic subgroup $W_\\mathfrak{p} \\leq S_n$,\nwe write $\\mathcal{O}^\\mathfrak{p}_0$ for the principal block\nof the parabolic category $\\mathcal{O}$,\nsee \\cref{sec:parabolic-subalgebras}.\nWe shall prove the following:\n\n\\begin{theorem*}[{\\ref{thm:main-result}}]\n\tFor a maximal parabolic subalgebra $\\mathfrak{p}$ of $\\mathfrak{sl}_n$\n\tcorresponding to the parabolic subgroup $S_{n-1}\u00d7S_1\\leq S_n$,\n\tthe derived category $D^\\mathrm{b}(\\mathcal{O}^\\mathfrak{p}_0)$ contains\n\tan $\\mathrm{A}_{n-1}$-configuration of $0$-spherical objects\n\tsuch that the associated spherical twist functors\n\tand the restriction of $\\LSh{s_i}\\hShift{1}$ to $D^\\mathrm{b}(\ud835\udcde^\ud835\udd2d_0)$\n\tare naturally isomorphic auto\\-\/equivalences of $D^\\mathrm{b}(\ud835\udcde^\ud835\udd2d_0)$.\n\\end{theorem*}\n\n\\subsection*{Outline}\n\\Cref{sec:tools-for-O} gives an overview of some of the most important properties\nof blocks of $\\mathcal{O}$\nand the tools we employ.\nWe do not require any prior knowledge about $\\mathcal{O}$.\nWe include a short refresher on Kazhdan-Lusztig theory, quivers and graded algebras.\n\nIn \\Cref{sec:sl2-case}, we explain how to compute images of the shuffling functors\nfor the special case $\\mathfrak g=\\mathfrak{sl}_2$ explicitly.\nThe proof of \\cref{thm:shuffling-is-twisting:sl2} \nwill serve as a model for the proof of the more general \\cref{thm:main-result},\nwhich is worked out in \\cref{sec:sln-case}.\n\n\\section{Some theory of the category \\texorpdfstring{$\\mathcal{O}_\\lambda$}{\ud835\udcde}}\n\\label{sec:tools-for-O}\nIn this section, we collect the most important properties of $\\mathcal{O}_\\lambda$.\nFor simplicity, we always assume that $\\lambda$ is an integral $\\rho$-dominant weight.\n\n\\subsection{Composition series}\n\\label{sec:composition-series}\nEvery module $M\\in\\mathcal{O}_\\lambda$ admits a composition series\n\\begin{equation*}\n\t0\\subsetneq M_1\\subsetneq\\cdots\\subsetneq M_\\ell=M\n\\end{equation*}\nwith subquotients $M_i\/M_{i-1}$ isomorphic to the simple modules $L(w\\cdot \\lambda)$ \n\\autocite[Satz 1.13]{Jantzen:Moduln-mit-hoechstem-Gewicht}.\nAccording to the Jordan-H\u00f6lder theorem, any composition series for $M$\ninvolves the same isomorphism classes of the simple factors,\nup to their order of appearance.\nThe multiplicity of $L(w\\cdot \\lambda)$ in any composition series for $M$\nis denoted by $[M:L(w\\cdot\\lambda)]$.\nThere is another important filtration:\n\n\\begin{definition}\n\t\\label{def:loewy-filtration}\n\tA \\term{Loewy filtration} of a module $M\\in\\mathcal{O}_0$ is a filtration\n\tof minimal length with semisimple subquotients $M_i$.\n\tA module $M$ is called \\emph{rigid} if it has a unique Loewy filtration;\n\tfor instance, Verma modules are rigid \\autocite[thm.\\ 1]{Irving:socle-filtration}.\n\tThe \\term{socle} $\\soc M$ and the \\term{head} $\\operatorname{hd} M$ of a module $M$\n\tare its maximal semisimple submodule and quotient, respectively.\n\\end{definition}\n\n\\begin{notation}\n\tWe write Loewy filtrations diagrammatically \n\tby putting all simple summands of each semisimple subquotient into a common horizontal layer,\n\tfrom the socle in the bottom line to the head in the top line.\n\\end{notation}\n\n\\begin{example}\n\t\tFor $\\mathfrak g=\\mathfrak{sl}_2$, the non-simple Verma module has a composition series (and also Loewy filtration)\n\t\t$M(e)=\\begin{psmallmatrix}L(e)\\\\L(s)\\end{psmallmatrix}$,\n\t\twhich is just another way to say that there is a short exact sequence\n\t\t$0\\to\\nolinebreak[2] L(s)\\to M(e)\\to L(e)\\to 0$.\n\t\t\n\t\tFor $\\mathfrak g=\\mathfrak{sl}_3$, \n\t\tthe Verma module $M(s)=\\begin{psmallmatrix}L(s)\\\\L(st)\\quad L(ts)\\\\L(w_0)\\end{psmallmatrix}$\n\t\thas the simple quotient $L(s)$ and the simple submodule $L(w_0)$.\n\t\tIts non-trivial submodules have Loewy filtrations\n\t\t$L(w_0)$,\n\t\t$\\begin{psmallmatrix}L(st)\\\\L(w_0)\\end{psmallmatrix}$, $\\begin{psmallmatrix}L(ts)\\\\L(w_0)\\end{psmallmatrix}$\n\t\tand $\\begin{psmallmatrix}L(st)\\quad L(ts)\\\\L(w_0)\\end{psmallmatrix}$.\n\\end{example}\n\\begin{remark}\n\tFor a morphism $f\\colon M \\to N$ in $\\mathcal{O}$\n\tthere exist composition series of $M$ and $N$\n\tcontiguously containing a composition series of $\\im f$\n\tat the top and the bottom, respectively.\n\tThe cokernel and kernel of $f$ comprise the composition factors\n\tof $M$ and $N$ not belonging to $\\im f$, respectively.\n\tFor example, for $\\mathfrak g = \\mathfrak{sl}_2$,\n\twe can write the short exact sequence $0 \\to M(e) \\to P(s) \\to M(s) \\to 0$ diagrammatically as\n\t\\begin{equation*}%\n\t\t\\begin{matrix}%\n\t\t\t&& L(s) & \\onto & \\smash{\\underbrace{L(s)}_{M(s).}}\\\\\n\t\t\t\\underbrace{ \\begin{array}{@{}c@{}} L(e)\\\\L(s) \\end{array} }_{M(e)}%\n\t\t\t& \\into & \n\t\t\t\\underbrace{ \\begin{array}{@{}c@{}} L(e)\\\\ L(s) \\end{array} }_{P(s)}%\n\t\t\\end{matrix}%\n\t\\end{equation*}\n\tSince $\\dim\\Hom_\\mathcal{O}(L(v), L(w))=\\delta_{vw}$,\n\ta morphism is determined by a scalar\n\tfor each pair of composition factors from its domain and its codomain.\n\\end{remark}\n\nCertain modules, such as the indecomposable projectives $P(w\\cdot\\lambda)$,\nalso admit a \\term{standard filtration} whose subquotients are isomorphic to Verma modules $M(v\\cdot\\lambda)$.\nWe write $(P(w\\cdot\\lambda) : M(v\\cdot\\lambda))$ for the (unique) multiplicity\nof $M(v\\cdot\\lambda)$ in a standard filtration of $P(w\\cdot\\lambda)$.\n\n\\begin{theorem}[\\thmtitle{BGG reciprocity} \\autocite{BGG}]\n\t$(P(v\\cdot\\lambda):M(w\\cdot\\lambda))=[M(w\\cdot\\lambda):L(v\\cdot\\lambda)]$.\n\\end{theorem}\n\n\n\\subsection{Kazhdan-Lusztig theory}\n\\label{sec:kl-theory}\nThere is a $W^2$-parametrised collection $\\{p_{vw}\\}$\nof polynomials in $\\mathbf Z[q^{\\pm 1}]$,\ncalled \\term{Kazhdan-Lusztig polynomials},\nwhich occur as base change coefficients between the standard basis and the Kazhdan-Lusztig basis\n(usually denoted, respectively, by $\\{T_w\\}$ and $\\{C_w\\}$ for $w \\in W$)\nof the \\term{Iwahori-Hecke algebra} $\\mathsf{H}_q(W)$\n\\autocites{KL:Representations-of-Coxeter-groups}{Soergel:KL-polynomials},\nwhich is a $\\mathbf{Z}[q^{\\pm 1\/2}]$-algebra and a deformation of the group algebra $\\mathbf{Z}W$.\nThe following relation to composition factor multiplicities\nhas been conjectured in \\autocite[conjecture 1.5]{KL:Representations-of-Coxeter-groups}:\n\n\\begin{theorem}[\\thmtitle{Kazhdan-Lusztig conjecture} {\\autocites[\u00a74]{BB:localisation-de-g-modules}[\u00a78]{BK:KL-conjecture}}]\n\t\\label{thm:BGG-reciprocity}\n\tThe composition factor multiplicities in a block $\\mathcal{O}_\\lambda$\n\tfor a regular weight $\\lambda$ are given by\n\t\\begin{equation}\n\t\t\\label{eqn:BGG-reciprocity}\n\t\t(P(v\\cdot\\lambda):M(w\\cdot\\lambda)) = [M(w\\cdot\\lambda):L(v\\cdot\\lambda)] = p_{vw}(1).\n\t\\end{equation}\n\\end{theorem}\n\n\\begin{remark}\n\t\\label{rmk:conventions-kl-polynomials}\n\tIf one indexes blocks by $\\rho$-antidominant rather than $\\rho$-dominant weights,\n\t\\cref{eqn:BGG-reciprocity} takes the form $[M(w\\cdot\\lambda):L(v\\cdot\\lambda)] = p_{w_0v,w_0w}(1)$.\n\tDepending on whether one uses the basis elements $C_w$ or $C'_w$ for the Kazhdan-Lusztig basis%\n\t\\footnote{See \\autocite[after thm.\\ 1.1]{KL:Representations-of-Coxeter-groups} for their relation.},\n\t$T_w$ or $H_w = q^{\\ell(w)\/2} T_w$ for the natural basis,\n\t$q$ or $v = q^{-1\/2}$ for the formal indeterminate,\n\tand $p_{vw}$ or $h_{vw} = q^{-2(\\ell(w) - \\ell(v))} p_{vw}$ for the Kazhdan-Lusztig polynomials,\n\tthe precise statement exhibiting the $p_{vw}$'s as base change coefficients may look different\n\t(see also \\autocite[2.6]{Soergel:KL-polynomials}).\n\\end{remark}\n\n\n\\subsection{Gradings}\n\\label{sec:gradings}\nWe summarise how to pass from $\\mathcal{O}_\\lambda$\nto a graded category $\\mathcal{O}^{\\mathbf Z}_\\lambda$.\n\nThe category $\\mathcal{O}_\\lambda$ has a projective generator $P_\\lambda \u2254 \\bigoplus_{w\\in W\/W_\\lambda} P(w\\cdot\\lambda)$,\nfor which we define the algebra $A_\\lambda \\coloneqq \\End_\\mathcal{O}(P_\\lambda)$.\nBy Morita's theorem \\autocite[Thm.\\ II.1.3]{Bass:K-theory},\nthere is an equivalence of categories\n\\begin{equation}\n\t\\label{eqn:morita-eq}\n\t\\mathcal{O}_\\lambda\\xrightarrow{\\simeq}\\Mod{}[A_\\lambda],\\quad M \\mapsto\\Hom_\\mathcal{O}(P_\\lambda, M).\n\\end{equation}\n\nThere exists a natural grading on $A_\\lambda$, given as follows\n\\autocite[\\S\\S 1.3, 2]{stroppel:gradings-and-translation}.\n\n\\begin{theorem}[\\thmtitle{Struktursatz}{ \\autocite[thm.\\ 2]{Soergel:perverse-Garben}}]\n\t\\label{thm:struktursatz}\n\tThe functor $\\mathbf V_\\lambda\\colon\\mathcal{O}_\\lambda \\to\\Mod{}[\\End_\\mathcal{O}(P(w_0 \\cdot \\lambda))]$,\n\tcalled the \\emph{combinatoric functor}, is fully faithful on projectives.\n\\end{theorem}\n\nThe algebra $\\End_\\mathcal{O}(P(w_0 \\cdot \\lambda))$ can be described explicitly:\nThe ordinary (non-dotted) action of the Weyl group $W$ on $\\mathfrak{h}$\ngives rise to an action of $W$ on the symmetric algebra $\\Complex[\\mathfrak h^*]$.\nLet $(\\Complex[\\mathfrak h^*]_+^W)$ denote its ideal generated \nby $W$-invariant polynomials without constant term\nand $C\u2254\\Complex[\\mathfrak h^*]\/(\\Complex[\\mathfrak h^*]_+^W)$ \nbe the \\emph{coinvariant algebra} of $\\mathfrak g$.\n\n\\begin{theorem}[\\thmtitle{Endomorphismensatz} {\\autocite[thm.\\ 3]{Soergel:perverse-Garben}}]\n\t$\\End_\\mathcal{O}(P(w_0\\cdot\\lambda))$ is isomorphic to the invariant subalgebra $C^{W_\\lambda}$.\n\\end{theorem}\n\nThe images $\\mathbf{V}P(w\\cdot\\lambda)$ of the indecomposable projectives\nare particular direct summands of the $C$-module $C \\otimes_{C^{s_{i_n}}} \\cdots \\otimes_{C^{s_{i_1}}} \\Complex$\nfor a reduced expression $w = s_{i_1} \\dotsm s_{i_n}$\n\\autocite[thm.\\ 10]{Soergel:perverse-Garben}.\n\nBy putting $\\mathfrak{h}^*$ in degree two, $C[\\mathfrak h^*]$ becomes an evenly graded algebra.\nSince $(\\Complex[\\mathfrak h^*]_+^W)$ is a homogeneous ideal,\nalso $\\End_\\mathcal{O}(P(w_0\\cdot\\lambda)) \\cong C$ is a graded algebra.\nTensor products and direct summands of graded modules with indecomposable complement inherit a grading, \nhence the images $\\mathbf{V} P(w\\cdot\\lambda)$\nand the spaces $\\Hom_{\\mathcal{O}}(P(v\\cdot\\lambda), P(w\\cdot\\lambda)) \\cong \\Hom_C(\\mathbf{V}P(v\\cdot\\lambda), \\mathbf{V}P(w\\cdot\\lambda))$ are graded as well,\nwhich establishes a grading on $A_\\lambda \\cong \\End_C(\\mathbf{V} P_\\lambda)$\n\\autocite[thm\\ 2.1]{stroppel:gradings-and-translation}.\nThe equivalence in \\eqref{eqn:morita-eq} motivates the following:\n\n\\begin{definition}\n\tLet $\\mathcal{O}^{\\mathbf Z}_\\lambda \u2254 \\Mod[g]{}[A_\\lambda]$ \n\tbe the category of graded $A_\\lambda$-modules.\n\\end{definition}\n\n\\begin{notation}\n\tWe denote the $k$-th graded component of a graded $A$-module $M$ by $M_k$.\n\tWe employ the convention that the grading shift $\\langle-\\rangle$ shifts upwards;\n\t\\ie, $M\u27e8i\u27e9_k \u2254 M_{k-i}$.\n\\end{notation}\n\n\\begin{remark}\n\t\\label{rmk:graded-lifts}\n\tA module $M\\in \\mathcal{O}_\\lambda$ is \\term{gradable} if there is a graded module $\\tilde M\\in\\mathcal{O}^{\\mathbf Z}_\\lambda$\n\tsuch that forgetting the grading yields $\\mathrm f\\tilde M=M$.\n\tIn particular, simple, Verma and indecomposable projective modules\n\tare gradable \\autocite[\u00a7\u00a72f.]{stroppel:gradings-and-translation}.\n\tUp to grading shifts and isomorphisms, an indecomposable module has a unique graded lift.\n\tIn the following, we shall not distinguish notationally between these modules and their graded lifts.\n\\end{remark}\n\nWe may fix the shift for the graded lifts of $M(w)$, $L(w)$ and $P(w)$\nsuch that each of these lifts has its non-zero graded component\nof least degree in degree zero.\nFor a graded module $M$\nthe multiplicity $[M_i : L(w)]$ of $L(w)$ in $M_i$, seen as ungraded modules,\ntherefore is precisely the multiplicity $[M : L(w)\u27e8i\u27e9]$ of graded modules.\n\nSince $A_\\lambda$ is non-negatively graded,\na graded module $M$ is filtered by the submodules $M_{\\geq k} \\coloneqq \\bigoplus_{l \\geq k} M_l$,\nand its graded components $M_k = M_{\\geq k}\/M_{\\geq k+1}$ are subquotients.\nFor Verma and projective modules, these are precisely the Loewy layers\n\\autocites[302]{stroppel:gradings-and-translation}[\u00a71.1]{Irving:socle-filtration}.\nThe graded component a simple composition factor belongs to\nis encoded in the exponents of Kazhdan-Lusztig polynomials:\n\n\\begin{theorem}[\\thmtitle{generalised KL-theorem} {\n\t\\autocites[thm.\\ 2]{Irving:socle-filtration}%\n\t\t[cor.\\ 7.1.3]{Irving:filtered-category-OS}%\n\t\t[thm.\\ 3.1.4%\n\t\t\\protect\\footnote{\n\t\t\tBlocks by are indexed by dominant weights,\n\t\t\t$n_{xy}$ is used for Soergel's $h_{xy}$.\n\t\t}]{BGS:Koszul-Duality}%\n}]\n\t\\label{thm:generalised-kl}\n\tThe composition factor multiplicities\n\tin the graded components of Verma modules satisfy\n\t\\begin{equation}\n\t\t\\label{eq:generalised-kl-theorem}\n\t\tp_{vw}=\\textstyle \\sum_\\ell [M(v)_{\\ell} : L(w) ] q^{(\\ell(w) - \\ell(v) - \\ell)\/2};\n\t\\end{equation}\n\tin other words, each summand $q^k$ of $p_{vw}$ corresponds to a factor $L(w)$\n\tin the $\\ell = (\\ell(w) - \\ell(v) - 2k)$-th layer of the Loewy filtration for $M(v)$, \n\twith the zeroth layer at the top. \n\\end{theorem}\n\n\\begin{remark}\n\tIf one prefers to work with Soergel's $h_{vw}$\n\t(see \\cref{rmk:conventions-kl-polynomials}), one gets\n\t$h_{vw} = \\sum_\\ell [M(v)_\\ell : L(w)] v^\\ell$.\n\tIf one indexes blocks of $\\mathcal{O}$ by antidominant weights instead,\n\t$v$ and $w$ have to be replaced by $w_0v$ and $w_0w$, respectively,\n\tand \\cref{eq:generalised-kl-theorem} becomes\n\t$p_{w_0v,w_0w} = \\sum_\\ell [M(w_0v)_{\\ell} : L(w_0w) ] q^{(\\ell(w_0v) - \\ell(w_0w) - \\ell)\/2}$.\n\tThis is equivalent to \\cref{eq:generalised-kl-theorem}\n\tsince $w_0^2 = e$ and $\\ell(w_0w) = \\ell(w_0) - \\ell(w)$ for all $w$.\n\\end{remark}\nA graded analogue of the BGG reciprocity theorem \\ref{thm:BGG-reciprocity} holds\n\\autocite[thm.\\ 7.6]{stroppel:gradings-and-translation}.\n\\begin{definition}\n\t\\label{def:graded-translation}\n\tOn $\\mathcal{O}^{\\mathbf Z}_\\lambda$,\n\tthe graded \\term{translation through the $s$-wall} $\\Theta_s$\n\tis uniquely defined by short exact sequences\n\t\\begin{equation}\n\t\t\\label{eq:definition-of-graded-translation}\n\t\t0\\to M(w)\u27e81\u27e9 \\xto{\\eta}\\Theta_s M(w) \\to M(ws)\\to 0\n\t\t\\quad\\text{and}\\quad\n\t\t\\Theta_s M(ws) = \\Theta_s M(w)\u27e8-1\u27e9\n\t\\end{equation}\n\tfor $w<ws$.\n\tBoth adjunction maps are of degree one,\n\tand $\\Theta_s^2 \\cong \\Theta_s\u27e8-1\u27e9 \u2295 \\Theta_s\u27e81\u27e9$.\n\tAnalogously to the non-graded case (see \\cref{def:shuffling-functor}),\n\tthe graded (co)shuffling functors $\\Sh{s}$ and $\\Csh{s}$ are defined by\n\t$\\Sh{s} M  = \\coker(\\eta\\colon \\id\u27e81\u27e9 \\Rightarrow \\Theta_s)$ and \n\t$\\Csh{s} M = \\coker(\\Theta_s \\Rightarrow \\id\u27e8-1\u27e9)$.\n\\end{definition}\n\n\\begin{remark}\n\tThe grading shift $\\langle 1\\rangle$ renders\n\tthe Grothendieck group $K_0(\\mathcal{O}^\\mathbf Z_0)$ becomes a $\\mathbf Z[v^{\\pm 1}]$-module\n\tby $v[M]\u2254[M\u27e81\u27e9]$.\n\tWith $v = q^{-1\/2}$,\n\tthere is an isomorphism of $\ud835\udc19[v^{\u00b11}]$-modules\n\t\\begin{equation*}\n\t\t\tK_0\\bigl(\ud835\udcde^\\mathbf{Z}_0(\\mathfrak{sl}_n)\\bigr) \\to  \\mathsf{H}_v(S_n),\\quad\n\t\t\t[M(w)\u27e8v\u27e9]\t\\mapsto vH_w,\\quad\n\t\t\t[P(w)\u27e8v\u27e9]\t\\mapsto vC_w,\n\t\\end{equation*}\n\tfor the Iwahori-Hecke algebra $\\mathsf{H}_v(S_n)$ (see \\cref{sec:kl-theory}).\n\tExplicitly, the basis $\\{H_w\\}_{w \\in W}$ is subject to the relations\n\t$H_w H_s = H_{ws}$ and $H_{ws} H_s = H_w - (v - v^{-1})H_{ws}$ for $w < ws$.\n\t\n\tFrom the defining short exact sequence of $\\Theta_s$\n\tin \\cref{eq:definition-of-graded-translation} we see for $w < ws$ that\n\t\\begin{align*}\n\t\t[\\Theta_s M(w)] &= v[M(w)] + [M(ws)]\\\\\n\t\t[\\Sh{s} M(w)] &= [\\Theta_s M(w)] - [M(w)\u27e8v\u27e9] = M(ws)\\\\\n\t\t[\\Sh{s} M(ws)] &= [\\Theta_s M(w)\u27e8-1\u27e9] - [M(ws)\u27e81\u27e9] = [M(ws)] - (v-v^{-1})[M(w)],\n\t\\end{align*}\n\thence $[\\Theta_s]$ and $[\\Sh{s}]$ respectively correspond\n\tto the right multiplication $\\cdot(v + H_s)$ and $\\cdot H_s$\n\tunder the above isomorphism\n\t\\autocite[thm.\\ 7.1]{stroppel:gradings-and-translation}\n\\end{remark}\n\n\n\\subsection{Quivers}\n\\label{sec:quivers}\nRecall that we defined the algebra $A_\\lambda \\coloneqq \\End_{\\mathcal{O}}(P_\\lambda)\n= \\End_{\\mathcal{O}}(\\bigoplus_w P(w\\cdot\\lambda))$\nin \\cref{sec:gradings}.\nThis algebra $A_\\lambda$ arises as the path algebra of a quiver as follows.\n\n\\begin{definition}\n\tThe \\term{Ext-quiver} $Q_A$ associated to a basic algebra $A$\n\tover an alg.\\ closed field\n\thas vertices corresponding to the isoclasses of simple $A_\\lambda$-modules $L$\n\tand $\\dim \\Ext^1(L, L') $ many arrows $L' \\from L$.\n\\end{definition}\n\n\\begin{remark}\n\tEquivalently, for a complete set $\\{e_i \\mid i \\in I\\}$ of primitive orthogonal idempotents of $A$,\n\t$Q_A$ can be defined as the quiver\n\twith vertex set $I$ corresponding to the idempotents\n\tand $\\dim (e_i (\\rad A)e_j \/ e_i(\\rad^2 A) e_j)$ many arrows $i \\from j$\n\t\\autocites[prop.\\ 1.14]{ARS:1995}[prop.\\ 2.4.3]{Benson:1995}[lem.\\ 2.12]{ASS:2006}.\n\\end{remark}\n\nBy Gabriel's theorem \n\\autocites[thm.\\ 4.1.7]{Benson:1995}[thm.\\ 3.7]{ASS:2006}[\u00a74.3]{Gabriel:Auslander-Reiten-Sequences},\nthere exists an admissible ideal $\\mathfrak{a}$\nsuch that $k Q_A \/ \\mathfrak{a} \\cong A$.\nUnder this isomorphism,\nthe idempotent $e_i$ of $A$ corresponds to the trivial path $\\trivpath{i}$ in $kQ_A$.\nMore generally, every finite dimensional algebra over an alg.\\ closed field\nis Morita-equivalent to a basic algebra \\autocite[cor.\\ 6.10]{ASS:2006}\nand thus to a path algebra quotient.\n\nFor the basic $\\Complex$-algebra $A_\\lambda$\nwith complete set $\\{\\id_{P(w)}\\}$ of primitive orthogonal idempotents,\nwe denote the Ext-quiver quiver by $Q_\\lambda \\coloneqq Q_{A_\\lambda}$\nand the admissible ideal by $\\mathfrak{a}_\\lambda$,\nsuch that $A_\\lambda \\cong Q_\\lambda \/ \\mathfrak{a}_\\lambda$.\n\n\\begin{notation}\n\tWe denote the composition of arrows $v_2\\xleftarrow{a}v_1$ and $v_3\\xleftarrow{b}v_2$ of a quiver by\n\t$v_3\\xleftarrow{ba}v_1$.\n\tWe denote trivial the path associated to a vertex $v$ by $\\trivpath{v}$.\n\\end{notation}\n\nThe path algebra of any quiver $Q_\\lambda$ is non-negatively graded by the length of a path in terms of arrows.\nOne can show that $\\mathfrak a_\\lambda$ is a homogeneous ideal;\nhence $A_\\lambda$ is graded as well.\nThis grading coincides with the grading induced by $\\mathbf V_\\lambda$ \n\\autocite{stroppel:gradings-and-translation} that we explained in \\cref{sec:gradings}.\n\nTo summarise, we have that $\\mathcal{O}_\\lambda\\simeq\\Mod{}[A_\\lambda]$ and $\\mathcal{O}_\\lambda^\\mathbf Z\\simeq\\Mod[g]{}[A_\\lambda]$.\nThe equivalence $\\mathcal{O}_\\lambda\\simeq\\Mod{}[A_\\lambda]$ maps indecomposable projectives $P(w)$ in $\\mathcal{O}_\\lambda$\nto the indecomposable projectives in $\\Mod{}[A_\\lambda]$;\nthese are precisely the right ideals $\\trivpath{w} A_\\lambda$ of all paths ending in $w$\n\\autocite[cor.\\ 4.18, rmk.\\ 4.20]{Barot:Rep-Theory}.\n\nOne can choose a $\\Complex$-basis of the projective module $\\trivpath{w} A_\\lambda$\nconsisting of paths ending in $w$,\neach of which corresponds to a simple composition factor $L(v)\u27e8i\u27e9$\nfor its source vertex $v$ and length $i$.\nHence as a vector space, $\\trivpath{w}A$ is a direct sum\n$\\trivpath{w}A = \\bigoplus_{v \\in W} \\trivpath{w}A\\trivpath{v}$ of graded vector spaces,\nwhose $i$-th graded component has dimension\n$\\dim (\\trivpath{w}A\\trivpath{v})_i = [P(w)_i : L(v)\u27e8i\u27e9]$.\n\t\nFrom now on, we shall stick to the principal block $\\mathcal{O}_0$\nand omit all the subscript-$\\lambda$'s from $A$ and $Q$.\n\n\n\\subsection{Parabolic subalgebras}\n\\label{sec:parabolic-subalgebras}\nLet $(W,S)$ be a Coxeter system\nand $S_\\mathfrak{p}\\subseteq S$ be any subset of the simple reflections of $W$.\n\n\\begin{definition}\n\tThe associated \\term{parabolic subgroup} $W_\\mathfrak{p}\\leq W$\n\tis the subgroup $W_\\mathfrak{p}=\u27e8S_\\mathfrak{p}\u27e9$ of $W$.\n\tEvery left coset in $W_\\mathfrak{p}\\backslash W$ has a unique representative of minimal length \\autocite[\u00a71.10]{Humphreys:Coxeter-Groups}.\n\tWe denote the set of such representatives by $W^\\mathfrak{p}$.\n\\end{definition}\n\nTo the quiver $Q$ we associate the full subquiver $Q^\\mathfrak{p}$ of $Q$\nwith vertex set $Q^\\mathfrak{p}=W^\\mathfrak{p}$\nand define the respective algebra\n$A^\\mathfrak{p}\u2254A\/(\\trivpath{v})_{v\\in W_\\mathfrak{p}}$.\nThe category $\\mathcal{O}^\\mathfrak{p}_0 \u2254 \\Mod{}[A^\\mathfrak{p}]$\nis equivalent to the smallest Serre subcategory of $\\mathcal{O}^\\mathfrak{p}_0$\ncontaining all simple modules $L(w)$ for $w\\in W^\\mathfrak{p}$\n\\autocite[\u00a72]{kildetoft-mazorchuk:parabolic-projective-functors}.\nThe quotient map $A\\onto A^\\mathfrak{p}$ induces an induction-restriction-adjunction\n\\begin{equation}\n\t\\Ind^{\\mathfrak{p}}\\colon \\mathcal{O}_0 \\xtofrom{\\;\\dashv\\;}  \\mathcal{O}^\\mathfrak{p}_0 \\noloc \\Res_{\\mathfrak{p}},\\quad\n\tM \\mapsto M \\otimes_A A^\\mathfrak{p}, \\quad\n\tN_A \\mapsfrom N,\n\\end{equation}\nwhere $N_A$ denotes the module $N$ on which $A$ acts via $A\\onto A^\\mathfrak{p}$.\nThe functor $\\Res_\\mathfrak{p}$ is fully faithful\nand thus renders $\\mathcal{O}^\\mathfrak{p}_0$ a subcategory of $\\mathcal{O}_0$.\nIt has a left adjoint $Z^\\mathfrak{p}\u2254\\Ind^\\mathfrak{p}$, called \\term{Zuckerman functor},\nand a right adjoint $Z_\\mathfrak{p}$, called \\term{dual Zuckerman functor},\nwhich respectively assign to a module $M$ its largest quotient and its largest submodule\nthat have simple composition factors $L(w)$ for $w \\in W^\\mathfrak{p}$ \n\\autocite[Thm.\\ 6.1]{mazorchuk:categorification}.\nLet $P^\\mathfrak{p}(w)\u2254Z^\\mathfrak{p}(P(w))=\\trivpath{w}A^\\mathfrak{p}$ and $M^\\mathfrak{p}(v) \u2254 Z^\\mathfrak{p}(M(v))$.\nAs notation suggests, the $P^\\mathfrak{p}(w)$'s are the indecomposable projectives in $\\mathcal{O}^\\mathfrak{p}_0$ \\autocite[\u00a74.6]{mazorchuk:categorification}.\n\n\\begin{remark}\n\tThe constructions and statements from \\crefrange{sec:composition-series}{sec:quivers}\n\thave parabolic analogues; namely:\n\t\\begin{thmlist}\n\t\t\\item The category $\\mathcal{O}^\\mathfrak{p}_0$ has a projective generator $P^\\mathfrak{p}=\\bigoplus_{w\\in W^\\mathfrak{p}}P^\\mathfrak{p}(w)$.\n\t\t\\item The analogously constructed graded version $\\mathcal{O}_0^{\\mathbf Z,\\mathfrak{p}}$ of $\\mathcal{O}_0^\\mathfrak{p}$\n\t\t\tcontains graded lifts of simples, parabolic Vermas and indecomposable projectives.\n\t\t\\item The $P^\\mathfrak{p}(w)$'s have parabolic standard filtrations with subquotients isomorphic to $M^\\mathfrak{p}(v)$'s.\n\t\t\\item The respective multiplicities satisfy a parabolic BGG reciprocity theorem\n\t\t\t\\autocite[Prop.\\ 4.5, Thm.\\ 6.1]{RC:splitting.criteria}.\n\t\t\\item Parabolic Kazhdan-Lusztig polynomials $p^\\mathfrak{p}_{vw}\\in\\mathbf Z[q^{\\pm 1}]$ \n\t\t\toccur as base change coefficients for parabolic Hecke modules\n\t\t\t\\autocites[\u00a73]{Deodhar:parabolic-KL}[rmk.\\ 3.2]{Soergel:KL-polynomials}.\n\t\t\\item The generalised Kazhdan-Lusztig theorem \\ref{thm:generalised-kl}\n\t\t\thas the following parabolic analogue\n\t\t\tfor graded lifts of simple, parabolic Verma and indecomposable projective modules\n\t\t\t\\autocites[Cor.\\ 7.1.3]{Irving:filtered-category-OS}[Thm.\\ 1.3]{CC:parabolic-KL}:\n\t\t\t\\begin{equation}\n\t\t\t\tp^\\mathfrak{p}_{vw}\n\t\t\t\t= \\sum_{k\\geq 0}(P^\\mathfrak{p}(w)_\\ell : M^\\mathfrak{p}(v)) q^{(\\ell(w) - \\ell(v) - \\ell)\/2} = \\sum_{k\\geq 0} [M^\\mathfrak{p}(v)_\\ell : L(w)] q^{(\\ell(w) - \\ell(v) - \\ell)\/2}.\n\t\t\t\\end{equation}\n\t\t\tIn other words, each summand $q^k$ corresponds,\n\t\t\tfor $\\ell - (\\ell(w) - \\ell(v) - 2k)$,\n\t\t\tto an $M^\\mathfrak{p}(v)\\langle\\ell\\rangle$ in $P^\\mathfrak{p}(w)$\n\t\t\tand an $L(w)\\langle\\ell\\rangle$ in $M^\\mathfrak{p}(v)$,\n\t\t\twith the top of the modules in degree zero.\n\t\\end{thmlist}\n\\end{remark}\n\n\\begin{remark}\n\t\\label{rmk:parabolic-translation-and-shuffling}\n\tThe functor $Z^\\mathfrak{p}$ commutes with projective functors,\n\tin particular with $\\Theta_s$\n\t\\autocite[Thm.\\ 6.1]{mazorchuk:categorification}.\n\tThis implies that both $\\Theta_s$ and $\\Sh{s}$ restrict to $\\mathcal{O}^\\mathfrak{p}_0$\n\tand $\\mathcal{O}^{\\mathbf{Z},\\mathfrak{p}}_0$,\n\tand the restriction of $\\Theta_s$ is uniquely characterised \n\tby the short exact restrictions\n\t\\begin{equation}\n\t\t\\label{eq:ses-parabolic-wall-crossing}\n\t\t0 \\to M^\\mathfrak{p}(w)\u27e81\u27e9 \\xto{\\eta} \\Theta_s \\to M^\\mathfrak{p}(ws)\\to 0\n\t\t\\quad\\text{and}\\quad\n\t\t\\Theta_s M^\\mathfrak{p}(ws) = \\Theta_s M^\\mathfrak{p}(w)\u27e8-1\u27e9,\n\t\\end{equation}\n\tfor $w<ws$,\n\tof the short exact sequences \\cref{eqn:translation-defining-ses} defining $\\Theta_s$.\n\\end{remark}\n\n\\begin{caveat}\n\t\\label{caveat:parabolic-projectives-not-projective}\n\tThe inclusion $\\mathcal{O}^\\mathfrak{p}_0\\subseteq \\mathcal{O}_0$ does not preserve projectives,\n\tas we shall see in \\cref{rmk:inclusion-of-parabolic-subalgebradoesnt-preserve-projectives}.\n\\end{caveat}\n\t\n\n\n\\section{\\texorpdfstring{$B_n$-actions for $\\mathfrak{sl}_2$}{\ud835\udc35n-actions for \ud835\udd30\ud835\udd29\u2082}}\n\\label{sec:sl2-case}\nConsider the Lie algebra $\\mathfrak{g}=\\mathfrak{sl}_2$ and its Weyl group $S_2=\\{e, s\\}$.\nSince the Kazhdan-Lusztig polynomials for $S_2$ are\n$p_{vw} = \\begin{smallcases}\n\t1 & \\text{if $v \\leq w$,}\\\\\n\t0 & \\text{otherwise,}\n\\end{smallcases}$\nthe Verma modules and the indecomposable projectives\nhave composition series and standard filtrations\n\\begin{equation}\n\tP(e) \n\t\t= M(e) \n\t\t= \\begin{psmallmatrix}L(e)\\\\L(s)\\end{psmallmatrix},\n\t\\qquad\n\tP(s) \n\t\t= \\begin{psmallmatrix}M(s)\\\\M(e)\\end{psmallmatrix} \n\t\t= \\begin{psmallmatrix} L(s)\\\\L(e)\\\\L(s)\\end{psmallmatrix},\n\t\\qquad M(s) = L(s).\n\\end{equation}\nThe category $\\mathcal{O}_0$ thus is equivalent to $\\Mod{}[A]$\nfor the path algebra quotient $A=\\Complex Q\/(ba)$ \nof the quiver $Q = (a\\colon e \\rightleftarrows s \\noloc b)$.%\nThe arrows $a,b$ of $Q$ and their relations\ncorrespond to the unique (up to scalars) non-trivial morphisms $a\\colon P(e)\\into P(s)$ and $b\\colon P(s)\\to P(e)$\n\\autocite{stroppel:gradings-and-translation}.\n\n\\subsection{The action of \\texorpdfstring{$\\Sh{s}$}{Sh\u209b}}\n\\label{sec:shuffling-sl2}\nWe obtain the images of the indecomposable projectives and Verma modules under $\\Sh{s}$\nas follows:\n\\begin{enumerate}\n\t\\item $M(e)=P(e)$: from the short exact sequence \\cref{eqn:translation-defining-ses} we obtain that\n\t\t$\\Theta_s M(e) = \\begin{psmallmatrix}M(s)\\\\M(e)\\end{psmallmatrix}=P(s)$\n\t\tand $\\Sh{s} M(e)= M(s)$.\n\t\\item $M(s)$: Up to scalars, there is a unique morphism $M(s) \\into M(e)$;\n\t\thence we obtain that\n\t\t$\\Sh{s} M(s)\n\t\t\\cong \\begin{psmallmatrix}M(s)\\\\M(e)\/M(s)\\end{psmallmatrix} \n\t\t=     \\begin{psmallmatrix}L(s)\\\\L(e)\\end{psmallmatrix}=M^\\vee(s)$,\n\t\t\\ie, the \\term{dual Verma module};\n\t\tsee \\autocite[\\S 3.2]{Humphreys:CatO} for the definition.\n\t\\item $P(s)$: using $\\Theta_s^2\\cong\\Theta_s\\oplus\\Theta_s$, \n\t\tit follows from $P(s)=\\Theta_s M(e)$\n\t\tthat $\\Sh{s} P(s) \\cong P(s)$.\n\\end{enumerate}\nFrom the naturality diagram\n\\begin{equation}\n\t\t\\label{eqn:adj-of-Ps}\n\t\\begin{tikzcd}[ampersand replacement=\\&]\n\t\tP(e) \\rar[\"a\"]\\dar[\"\\eta_{P(e)}\"']\\& P(s) \\rar[\"b\"]\\dar[\"\\Mtrx{1\\\\x}\", \"\\eta_{P(s)}\"'] \\& P(e) \\dar[\"\\eta_{P(s)}\"]\\\\\n\t\tP(s) \\rar[\"\\Mtrx{1\\\\0}\"] \\& P(s)\u2295P(s) \\rar[\"\\Mtrx{0 & 1}\"]\\rar \\& P(s)\n\t\\end{tikzcd}\n\\end{equation}\nof $\\eta\\colon\\id\\Rightarrow\\Theta_s$ we get that\nthe morphisms $a\\colon P(e)\\into P(s)$ and $b\\colon P(s)\\to P(e)$ have images \n\\begin{equation}\n\t\\label{eq:shuffling-sl2:images-of-generating-morphisms}\n\t\\Sh{s} a\\colon M(s)\\into P(s)\\quad\\text{and}\\quad\\Sh{s} b\\colon P(s)\\onto M(s).\n\\end{equation}\nSimilar arguments show that $\\Csh{s} P(s) \\cong P(s)$\nand $\\RCsh{s} P(e)$ is the mapping cone $\\RCsh{s} P(e)\\simeq\\{\\degZero{P(s)} \\to P(e)\\}$\n(recall the notation of mapping cones from \\cref{notation:total-complexes});\nso we have\n\\begin{equation}\n\t\\label{eq:shuffling-sl2:imaegs-of-projectives}\n\t\\begin{aligned}\n\t\t\\LSh{s} P(e) &= \\{P(e)\\to \\degZero{P(s)}\\} \\simeq M(s)  &\n\t\t\\RCsh{s} P(e) &= \\{\\degZero{P(s)}\\to P(e)\\} \\\\\n\t\t\\LSh{s} P(s) &= P(s) &\n\t\t\\RCsh{s} P(s) &= P(s),\n\t\\end{aligned}\n\\end{equation}\nand in terms of the projective resolution $\\{P(e) \\to P(s)\\}$ of $M(s)$,\nthe morphisms from \\cref{eq:shuffling-sl2:images-of-generating-morphisms} become\n\\begin{align}\n\t\\label{eq:shuffling-sl2:images-of-generating-morphisms:lifts}\n\t\\LSh{s} a&\\colon \\left\\{\n\t\t\\begin{tikzcd}[cramped, sep=small, baseline=(ab.base), ampersand replacement=\\&]\n\t\t\t\\{P(e) \\rar \\& P(s) \\dar[\"ab\" name=ab]\\} \\\\\n\t\t\t\\& P(s),\n\t\t\\end{tikzcd}\n\t\\right. &\n\t\\LSh{s} b&\\colon \\left\\{\n\t\t\\begin{tikzcd}[sep=small, cramped, baseline=(id.base), ampersand replacement=\\&]\n\t\t\t\\& P(s) \\dar[\"\\id\" name=id]\\\\\n\t\t\t\\{P(e) \\rar \\& P(s)\\},\n\t\t\\end{tikzcd}\n\t\\right.\\\\[1ex]\n\t\\RCsh{s} a&\\colon \\left\\{\n\t\t\\begin{tikzcd}[cramped, sep=small, baseline=(ab.base), ampersand replacement=\\&]\n\t\t\t\\{P(s) \\dar[\"\\id\" name=id] \\rar \\& P(s) \\} \\\\\n\t\t\tP(s),\n\t\t\\end{tikzcd}\n\t\\right.\n\t&\n\t\\RCsh{s} b&\\colon \\left\\{\n\t\t\\begin{tikzcd}[sep=small, cramped, baseline=(id.base), ampersand replacement=\\&]\n\t\t\tP(s) \\dar[\"ab\" name=ab]\\\\\n\t\t\t\\{P(s) \\rar \\& P(s) \\}.\n\t\t\\end{tikzcd}\n\t\\right.\n\\end{align}\nSince $\\mathcal{O}_0$ has finite global dimension \\cite[prop.\\ 6.9]{Humphreys:CatO},\n$D^\\mathrm{b}(\\mathcal{O}_0)$ is equivalent to the bounded homotopy category of projectives\n(cf.\\ \\autocite[thm.\\ 10.4.8]{Weibel:Hom-Alg})\nand therefore generated as a triangulated category\nby the indecomposable projectives;\n\\ie, the smallest triangulated subcategory of $D^\\mathrm{b}(\\mathcal{O}_0)$\ncontaining the indecomposable projectives\nis $D^\\mathrm{b}(\\mathcal{O}_0)$ itself \\autocite[\\S 3.3]{Krause:Derived-Categories}.\nThese above data thus suffices to determine $\\LSh{s}$.\nPhrasing the observations differently, we have shown the following:\n\\begin{proposition}\n\tThe action on $D^\\mathrm{b}(\\mathcal{O}_{0,\\mathfrak{sl}_2})$ of the Artin braid group $B_1$\n\t(which is an infinite cyclic group)\n\tby $s \\mapsto \\LSh{s}$ and $s^{-1} \\mapsto \\RCsh{s}$\n\tis given explicitly by\n\t\\begin{align*}\n\t\tB_1 &\\longto \\operatorname{Aut}(D^\\mathrm{b}(\\mathcal{O}_0))\\\\\n\t\ts^n &\\longmapsto\n\t\t\t\\underset{\\strut}{\n\t\t\t\\begin{aligned}[t] \n\t\t\t\t& P(s)\\mapsto P(s),\\\\\n\t\t\t\t& P(e)\\mapsto \n\t\t\t\t\t\\begin{cases}\n\t\t\t\t\t\t\\bigl\\{P(e)\\into \\smash[t]{\\overbrace{P(s)\\xto[ab]{} P(s)\\to \\dotsb\\to \\degZero{P(s)}}^{\\text{$n$ times}}}\\bigr\\} & \\text{if $n\\geq 0$},\\\\\n\t\t\t\t\t\t\\bigl\\{\\smash{\\underbrace{\\degZero{P(s)}\\xto{ab} P(s)\\to \\dotsb\\to P(s)}_{\\text{$-n$ times}}} \\to P(e) \\bigr\\} & \\text{if $n\\leq 0$,}\n\t\t\t\t\t\\end{cases}\n\t\t\t\\end{aligned}\n\t\t\t}\n\t\\end{align*}\n\twith homological degree $0$ as indicated.\n\\end{proposition}\n\n\\begin{example}\n\tTo illustrate why this proposition is clear from what we derived before,\n\tlet us compute the action of, say, $s^2$, which acts by $\\LSh{s}^2$.\n\tIn $D^\\mathrm{b}(\\mathcal{O}_0)$, the module $M(s)$ is isomorphic\n\tto its projective resolution $\\{P(e) \\xto{a} P(s)\\}$,\n\tto which we apply $\\LSh{s}^2$.\n\tFrom the images in \\cref{eq:shuffling-sl2:imaegs-of-projectives,eq:shuffling-sl2:images-of-generating-morphisms}\n\twe get that $\\LSh{s} M(s)$ is isomorphic to the double complex\n\t\\begin{equation*}\n\t\t\\LSh{s} M(s) \\simeq\n\t\t\\left\\{\n\t\t\t\\begin{tikzcd}[sep=small, slightly cramped, baseline=(b.base)]\n\t\t\t\tP(e) \\dar[\"b\" name=b] \\\\\n\t\t\t\tP(s) \\rar{\\id} & P(s)\n\t\t\t\\end{tikzcd}\n\t\t\\right\\}\n\t\t\\simeq \n\t\t\\{P(e) \\to P(s) \\to \\degZero{P(s)}\\}.\n\t\\end{equation*}\n\tApplying $\\LSh{s}$ once again shows that\n\t$\\LSh{s} M(s) \\simeq \\{P(e) \\to P(s) \\to P(s) \\to \\degZero{P(s)}\\}$.\n\\end{example}\n\n\\subsection{Spherical objects}\n\\label{sec:spherical-objects-sl2}\nTo check that a spherelike object is spherical indeed,\n\\ie, that the composition pairing from the Calabi-Yau-property\n(\\cref{def:spherical-object:calabi-yau}) is non-degenerate,\nit suffices to check non-degeneracy \\wrt\\ indecomposable projectives:\n\n\\begin{lemma}\n\tLet $\\mathcal{C}$ be an abelian category of finite global dimension\n\tand $E \\in D^\\mathrm{b}(\\mathcal{C})$ a $d$-spherelike object.\n\tIf the composition pairing\n\t$\\Hom^i_{D^\\mathrm{b}}(P, E) \u2297 \\Hom^{d-i}_{D^\\mathrm{b}}(P, E) \\to kx$\n\tis non-degenerate for all $i$\n\tand all indecomposable projective objects $P \\in \\mathcal{C}$,\n\tseen as complexes in $D^\\mathrm{b}(\\mathcal{C})$ concentrated in a single degree,\n\tthen\n\t$\\Hom^i_{D^\\mathrm{b}}(F, E) \u2297 \\Hom^{d-i}_{D^\\mathrm{b}}(F, E) \\to kx$\n\tis non-degeneate for all $i$ and $F \\in D^\\mathrm{b}(\\mathcal{C})$.\n\\end{lemma}\n\\begin{proof}\n\tSince $\\mathcal{C}$ is of finite global dimension,\n\tevery object of $D^\\mathrm{b}(\\mathcal C)$ is quasi-isomorphic to a bounded complex of projectives;\n\tin other words, for the full subcateogires\n\t$S_0 \\coloneqq \\Proj{\\mathcal C}$\n\tand $S_s \\coloneqq \\{\\cone f \\mid f \\in S_{s-1}\\}$ of $D^\\mathrm{b}(\\mathcal C)$,\n\twe have $D^\\mathrm{b}(\\mathcal C) = \\bigcup_{s \\geq 0} S_s$.\n\tAssume for some $s \\geq 0$\n\tthat the pairing $\\Hom^i_{D^\\mathrm{b}}(F, E) \u2297 \\Hom^{d-i}_{D^\\mathrm{b}}(F, E) \\to kx$ is non-degenerate\n\twhenever $F \\in S_s$.\n\tFor a distinguished triangle $F' \\xto{f} F \\to F'' \\to F'\\hShift{-1}$\n\twith $F, F' \\in S_s$,\n\twe thus get that in the diagram\n\t\\begin{equation*}\n\t\t\\begin{tikzcd}[sep=small, cramped]\n\t\t\t\\cdots \\rar & \\Hom_{D^\\mathrm{b}}(E, F') \\dar{\\cong} \\rar & \\Hom_{D^\\mathrm{b}}(E, F) \\dar \\rar & \\Hom_{D^\\mathrm{b}}(E, F'') \\dar{\\cong} \\rar & \\cdots\\\\\n\t\t\t\\cdots \\rar & \\Hom_{D^\\mathrm{b}}(F', E\\hShift{-d})^* \\rar & \\Hom_{D^\\mathrm{b}}(F, E\\hShift{-d})^* \\rar & \\Hom_{D^\\mathrm{b}}(F'', E\\hShift{-d})^* \\rar & \\cdots\n\t\t\\end{tikzcd}\n\t\\end{equation*}\n\twhose rows are long exact sequences,\n\tthe indicated vertical maps are isomorphisms.\n\tBy the five lemma, it follows that also\n\t$\\Hom^i_{D^\\mathrm{b}}(F'', E) \u2297 \\Hom^{d-i}_{D^\\mathrm{b}}(F'', E) \\to kx$ is non-degenerate.\n\tThe claim follows by induction.\n\\end{proof}\n  \n\\begin{lemma}\n\t$P(s)$ is a $0$-spherical and $L(e)$ is a $2$-spherical object\n\tof $D^\\mathrm{b}(\\mathcal{O}_0)$ for $\\mathfrak{sl}_2$.\n\\end{lemma}\n\\begin{proof}\n\tTo see that $P(s)$ is $0$-spherical,\n\tnote that the non-trivial endomorphism $x\\coloneqq ab$ of $P(s)$\n\tsatisfies $x^2 = 0$ and $\\End_{\\mathcal{O}}(P(s))\\cong\\Complex[x]\/(x)^2$,\n\tso $P(s)$ is 0-spherelike.\n\tSince $P(e)$ and $P(s)$ generate $D^\\mathrm b(\\mathcal O_0)$\n\tand since $x$ factors through $P(e)$,\n\tthe composition pairing is non-degenerate, hence $P(s)$ is spherical.\n\t\n\tTo see that $L(e)$ is $2$-spherical, consider its projective resolution\n\t$\\{P(e)\\to P(s)\\to P(e)\\}$.\n\tApart from the identity, the only non-trivial graded endomorphisms \n\tof this resolution are (scalar multiples of)\n\t\\bgroup\n\t\\tikzcdset{\n\t\tevery diagram\/.append style={\n\t\tsep=0.6em, cramped, font=\\footnotesize, baseline=(B.base),\n\t\tevery cell\/.append style={inner sep=0.1ex}\n\t}}\n\t\\begin{gather}\n\t\t\t\\biggl(\\,\n\t\t\t\t\\begin{tikzcd}[ampersand replacement=\\&]\n\t\t\t\t\tP(e) \\rar \\& P(s) \\dar[\"\\strut\" name=B] \\rar \\& P(e) \\dar \\ar[dl, equal]\\\\\n\t\t\t\t\t\\& P(e) \\rar \\& P(s) \\rar \\& P(e)\n\t\t\t\t\\end{tikzcd}\n\t\t\t\\,\\biggr), \n\t\t\t\\biggl(\\,\n\t\t\t\t\\begin{tikzcd}[ampersand replacement=\\&]\n\t\t\t\t\t\\& P(e) \\rar \\dar[\"\\strut\" name=B] \\& P(s) \\dar \\ar[dl, equal] \\rar \\& P(e) \\\\\n\t\t\t\t\tP(e) \\rar \\& P(s) \\rar \\& P(e)\n\t\t\t\t\\end{tikzcd}\n\t\t\t\\,\\biggr),\n\t\t\t\\biggl(\n\t\t\t\\underbrace{\\,\n\t\t\t\t\\begin{tikzcd}[ampersand replacement=\\&]\n\t\t\t\t\t\\&\\& P(e) \\rar \\dar[equal, \"\\strut\" name=B] \\& P(s) \\rar \\& P(e), \\\\\n\t\t\t\t\tP(e) \\rar \\& P(s) \\rar \\& P(e)\n\t\t\t\t\\end{tikzcd}\n\t\t\t\\,}_{{}\\eqqcolon x}\n\t\t\t\\biggr)\n\t\\end{gather}\n\t\\egroup\n\tof which the first two are null-homotopic\n\twith the indicated chain homotopies.\n\tTherefore, $\\End_{D^\\mathrm{b}(\\mathcal{O})}(L(e)) = \\Complex[x]\/(x^2)$ with $x$ of degree two,\n\twhich renders $L(e)$ 2-spherelike.\n\tThe endomorphism $x$ factors through $P(e)$,\n\tand $\\Hom^*_{D^\\mathrm{b}(\\mathcal{O})}(P(s), L(e)) = 0 = \\Hom^*_{D^\\mathrm{b}(\\mathcal{O})}(L(e), P(s))$,\n\thence the composition pairing is non-degenerate.\n\\end{proof}\n\n\\begin{remark}\n\tSimple modules in $\\mathcal{O}_0$ only have trivial self-extensions \\autocite[Prop.\\ 3.1]{Humphreys:CatO}\n\tand thus cannot be $0$ or $1$-spherical.\n\\end{remark}\n\n\\subsection{Spherically twisting by \\texorpdfstring{$P(s)$}{P(s)}}\n\nThe images of projectives under the cotwisting functor $T'_{P(s)}$ are\n\\begin{equation*}\n\t\\begin{alignedat}{4}\n\t\tT'_{P(s)}P(s) &= \\{\\Mtrx{1\\\\x}\\colon \\degZero{P(s)} \u2192 P(s)\u2295P(s)\\}  \n\t\t\t&&\\simeq P(s)\\hShift{1} \n\t\t\t&&\\stackrel{\\text{\\cref{eq:shuffling-sl2:imaegs-of-projectives}}}{=} \n\t\t\t  \\Sh{s} P(s)\\hShift{1},\\\\\n\t\tT'_{P(s)}P(e) &= \\{a\\colon \\degZero{P(e)} \u2192 P(s)\\} \n\t\t\t&&\\simeq M(s)\\hShift{1} \n\t\t\t&&\\stackrel{\\phantom{\\text{\\cref{eq:shuffling-sl2:imaegs-of-projectives}}}}{=} \n\t\t\t  \\Sh{s} P(e)\\hShift{1}.\n\t\\end{alignedat}\n\\end{equation*}\nThis is a first step towards a proof of the following:\n\\begin{theorem}%\n\t\\label{thm:shuffling-is-twisting:sl2}\n\tThere is a natural isomorphism $\\LSh{s}\\hShift{1}\\cong T'_{P(s)}$ of autoequivalences of $D^\\mathrm{b}(\ud835\udcde_0(\\mathfrak{sl}_2))$.\n\\end{theorem}\n\\begin{lemma}[\\thmtitle{Morita}]\n\tLet $A$ and $B$ be rings.\n\tAny right exact functor $F\\colon  \\Mod{}[A] \u2192 \\Mod{}[B]$ that preserves arbitrary direct sums\n\tis isomorphic to tensoring with the $A$-$B$-bimodule $FA$\n\t\\autocite[Thm.\\ II.2.3]{Bass:K-theory}.\n\\end{lemma}\n\\newcommand{\\FBim}[1]{M_{#1}}\n\\begin{corollary}\n\t\\label{cor:morita-equivalence-and-induced-module}\n\tFor abelian categories $\ud835\udcd0$ and $\ud835\udcd1$ with projective generators $P_\ud835\udcd0$ and $P_\ud835\udcd1$,\n\tMorita's theorem \\eqref{eqn:morita-eq} \n\tallows us to identify $\ud835\udcd0$ and $\ud835\udcd1$ with $\\Mod{}[\\End_\ud835\udcd0(P_\ud835\udcd0)]$ and $\\Mod{}[\\End_\ud835\udcd1(P_\ud835\udcd1)]$ respectively.\n\tThen any right exact functor $F\\colon  \ud835\udcd0 \\to \ud835\udcd1$ that commutes with arbitrary direct sums \n\tcan be identified with the functor\n\t\\begin{equation*}\n\t\t-\u2297_{\\End_\ud835\udcd0(P_\ud835\udcd0)} \\FBim{F} \\colon \\Mod{}[\\End_\ud835\udcd0(P_\ud835\udcd0)] \\to \\Mod{}[\\End_\ud835\udcd1(P_\ud835\udcd1)],\n\t\\end{equation*}\n\twhere $\\FBim{F} \\coloneqq \\Hom_\ud835\udcd1(P_\ud835\udcd1, FP_\ud835\udcd0)$ \n\tis an $\\End_\ud835\udcd0(P_\ud835\udcd0)$-$\\End_\ud835\udcd1(P_\ud835\udcd1)$-bimodule\n\ton which elements $a\\in \\End_\ud835\udcd0(P_\ud835\udcd0)$ and $b\\in \\End_\ud835\udcd1(P_\ud835\udcd1)$\n\tact by $a\\mathbin{.} f\\mathbin{.} b = Fa\\circ f\\circ b$\n\tfor $f\\in \\FBim{F}$.\n\\end{corollary}\n\n\\begin{proof}[{Proof of \\cref{thm:shuffling-is-twisting:sl2}}]\n\tBy the corollary there are natural isomorpisms\n\t\\begin{alignat*}{2}\n\t\t\\LSh{s}\\hShift{1} &\\stackrel{\\text{def}}{=} \\{ \\degZero{\\id_\ud835\udcde} \\Rightarrow \ud835\udee9_s \\} &&\u2245 -\u2297_A \\{ A \\to \\FBim{\ud835\udee9_s} \\},\\\\*\n\t\tT'_{P(s)} &\\stackrel{\\text{def}}{=} \\{ \\degZero{\\id_\ud835\udcde} \\Rightarrow \\Xi'_{P(s)} \\} &&\u2245 -\u2297_A \\{ A \\to \\FBim{\\Xi'_{P(s)}} \\},\n\t\\end{alignat*}\n\tsuch that it suffices to show $\\FBim{\ud835\udee9_s} \\cong \\FBim{\\Xi'_{P(s)}}$.\n\tRecall that $\\Xi'_{P(s)} = \\lin(\\hom^\\bullet_\\mathcal{O}(-, P(s)), P(s))$.\n\tBy finite dimensionality,\n\tthere is an isomorphism $\\FBim{\\Xi'_{P(s)}} \u2245 P(s)^* \u2297_\ud835\udc02 P(s)$ of $A$-$A$-bimodules.\n\tConsider the canonical vector space basis $\\{\\trivpath{s}, s\\from e, s\\from e\\from s\\}$ of $P(s)=\\trivpath{s}A$\n\tand the dual basis of $P(s)^*$.\n\tA basis of $\\FBim{\\Xi'_{P(s)}}$ is given by\n\tthe nine pairwise tensor products in the schematic\n\t\\begin{equation}\n\t\t\\label{eq:shuffling-twisting-isomorphic:sl2-tensor}\n\t\tM_{\\Xi'} \\cong P(s)^* \u2297_\ud835\udc02 P(s)\\colon \\quad\n\t\t\\begin{tikzpicture}[mth, baseline=(B.base)]\n\t\t\t\\matrix (A) [\n\t\t\t\tmatrix of math nodes, \n\t\t\t\trow sep=0mm, \n\t\t\t\tcolumn sep=-2mm, \n\t\t\t\ttext height=2ex, text depth=.5ex, \n\t\t\t\tcolumn 2\/.style={anchor=base east},\n\t\t\t\tcolumn 4\/.style={anchor=base west}\n\t\t\t] {\n\t\t\t\t\t~& (s\u2190e\u2190s)^* \t\t\t&\t\t\t\t& \\trivpath{s}\t&~\\\\\n\t\t\t\t\t~& |[alias=B]| (s\u2190e)^* \t& \\quad\u2297\\quad\t& (s\u2190e)\t\t\t&~\\\\\n\t\t\t\t\t~& \\trivpath{s}^*\t\t\t&\t\t\t\t& (s\u2190e\u2190s),\t\t&~\\\\\n\t\t\t\t};\n\t\t\t\t\\node[braced box={T}, fit=(A-1-2) (A-3-2)] {} ;\n\t\t\t\t\\node[braced box'={T'},fit=(A-1-4) (A-3-4)] {} ;\n\t\t\t\t%\n\t\t\t\t\\draw\n\t\t\t\t\t(A-1-1) edge[|->, out=195, in=165] node[swap]{$(e\u2190[b]s)_*$} (A-2-1.160)\n\t\t\t\t\t(A-2-1.200) edge[|->, out=195, in=165] node[swap]{$(s\u2190[a]e)_*$} (A-3-1)\n\t\t\t\t\t(A-1-5)\tedge[|->, out=-15, in=15] node{$\u2218(s\u2190[a]e)$} (A-2-5.20)\n\t\t\t\t\t(A-2-5.340) edge[|->, out=-15, in=15] node{$\u2218(s\u2190[b]e)$} (A-3-5);\n\t\t\\end{tikzpicture}\n\t\\end{equation}\n\twith the indicated action on basis vectors.\n\tTo describe the $A$-$A$-bimodule action on $\\FBim{\\Theta_s} = \\Hom_A(P, \\Theta_s P)$\n\tin terms of a vector space basis,\n\twe introduce the following notation.\n\tRecall that $P=P(e)\u2295P(s)$ and $\\Theta_s P\\cong P(s)^{\u22953}$.\n\tWe enumerate the summands of $\\Theta_s P = P(s)_1\u2295P(s)_2\u2295P(s)_3$.\n\tWe then abbreviate, \\eg, the morphism $\\Mtrx{0&0\\\\0&0\\\\0&x} \\in \\Hom_A(P, \\Theta_s P)$,\n\tby  $P(s)_3 \\xleftarrow{x} P(s)$.\n\tFor $\\Theta_s$, the naturality diagram \\eqref{eqn:adj-of-Ps} of $\\eta\\colon\\id\\Rightarrow\\Theta_s$ \n\tshows that the images under $\\Theta_s$ of the morphisms $a$ and $b$ generating $\\End_\\mathcal{C}(P)$ are\n\t\\settowidth{\\algnRef}{${}\u2295{}$}\n\t\\begin{equation}\n\t\t\\begin{tikzcd}[column sep={\\the\\algnRef}, nodes={inner xsep=0pt}, row sep=small]\n\t\t\tP(e)\\rar[phantom, \"\u2295\"] & \n\t\t\tP(s)\\ar[dl, \"b\"'] &[4em] \n\t\t\tP(s)_1 \\rar[phantom, \"\u2295\"] & \n\t\t\tP(s)_2 \\rar[phantom, \"\u2295\"] & \n\t\t\tP(s)_3 \\ar[dll, \"\\id_{P(s)}\"' {near end, inner sep=0pt, outer sep=0pt}]\n\t\t\t\\\\\n\t\t\tP(e)\\rar[phantom, \"\u2295\"] \\ar[dr, \"a\"'] & \n\t\t\tP(s) \\ar[phantom, \"\\xmapsto{\\Theta_s}\", r]& \n\t\t\tP(s)_1 \\rar[phantom, \"\u2295\"] \\ar[dr, \"\\id_{P(s)}\"' {near start, inner sep=0pt, outer sep=0pt}] & \n\t\t\tP(s)_2 \\rar[phantom, \"\u2295\"] & P(s)_3\n\t\t\t\\\\\n\t\t\tP(e)\\rar[phantom, \"\u2295\"] & \n\t\t\tP(s) & \n\t\t\tP(s)_1 \\rar[phantom, \"\u2295\"] & \n\t\t\tP(s)_2 \\rar[phantom, \"\u2295\"] & \n\t\t\tP(s)_3\\mathrlap{.}\n\t\t\\end{tikzcd}\n\t\\end{equation}\n\tA vector space basis of $\\FBim{\ud835\udee9_s}$ is given \n\tby the nine ways to map a summand $P(w)$ of $P$ from the right of the following schematic\n\tto a summand $P(s)_i$ of $\\Theta_s P(s)$ on the left of\n\t\\begin{equation}\n\t\t\\label{eq:shuffling-twisting-isomorphic:sl2-hom}\n\t\t\\FBim{\ud835\udee9_s}\\colon\\quad\n\t\t\\begin{tikzpicture}[mth, ampersand replacement=\\&]\n\t\t\t\\matrix (A) [\n\t\t\tmatrix of math nodes, \n\t\t\trow sep={3.5ex,between origins},\n\t\t\tcolumn sep=10mm,\n\t\t\tcolumn 1\/.style={anchor=base east, text width=width(\"$P(s)_2$\")},\n\t\t\tcolumn 3\/.style={anchor=base west, text width=width(\"$P(s)$\")}\n\t\t\t] {\n\t\t\t\tP(s)_3 \\&  \\& P(s)\\\\\n\t\t\t\tP(s)_1 \\& P(s) \\& P(e)\\\\\n\t\t\t\tP(s)_2 \\& \\& P(s) \\\\\n\t\t\t};\n\t\t\t\\draw[->] (A-2-2)\n\t\t\tedge[->, out=180, in=0] (A-1-1)\n\t\t\tedge[->, out=180, in=0] (A-2-1)\n\t\t\tedge[->, out=180, in=0] (A-3-1)\n\t\t\tedge[<-, in=180, out=0, \"$\\id$\" very near end] (A-1-3)\n\t\t\tedge[-, in=180, out=0] (A-2-3)\n\t\t\tedge[-, in=180, out=0, \"$x$\"' very near end] (A-3-3);\n\t\t\t\\draw\n\t\t\t(A-1-1.west)  edge[|->, out=195, in=165] node[swap]{$\\Theta_s(P(e)\\xleftarrow{b} P(s))\\circ $} (A-2-1.170)\n\t\t\t(A-2-1.190)   edge[|->, out=195, in=165] node[swap]{$\\Theta_s(P(s)\\xleftarrow{a} P(e))\\circ$} (A-3-1.west)\n\t\t\t(A-1-3.east)  edge[|->, out=-15, in=15] node{$\u2218(P(s)\u2190[a]P(e))$} (A-2-3.10)\n\t\t\t(A-2-3.-10)   edge[|->, out=-15, in=15] node{$\u2218(P(e)\u2190[b]P(s))$} (A-3-3.east);\n\t\t\\end{tikzpicture}\n\t\\end{equation}\n\twith the bimodule action as indicated.\n\tComparison of \\eqref{eq:shuffling-twisting-isomorphic:sl2-tensor} and \\eqref{eq:shuffling-twisting-isomorphic:sl2-hom} shows\n\tthat the obvious isomorphism $\\FBim{\\Theta_s} \\cong \\FBim{\\Xi'_{P(s)}}$ of vector spaces\n\tis an isomorphism of $A$-$A$-bimodules.\n\\end{proof}\n\n\\subsection{Spherically twisting by \\texorpdfstring{$L(e)$}{L(e)}}\n\\label{sec:twisting-by-Le}\nWe first note that $\\LSh{s} L(e) = L(e)\\hShift{-1}$;\nnamely, from the images in \\cref{eq:shuffling-sl2:imaegs-of-projectives}\nwe get that the projective resolution $\\{P(e) \\to P(s) \\to \\degZero{P(e)}\\}$ of $L(e)$\nis mapped under $\\LSh{s} L(e)$ to $\\{M(s) \\to P(s) \\to \\degZero{M(s)}\\}$,\nwhich is quasi-isomorphic to $L(e)\\hShift{-1}$.\n\nSince $L(e)$ is 2-spherical as we have seen in \\cref{sec:spherical-objects-sl2},\n\\cref{rmk:twisting-E-by-itself} implies\nthat if there is any isomorphism $T'_{L(e)} \\cong \\LSh{s}\\hShift{k}$,\nthen the shift $k$ must be zero.\nWe shall now show that $T'_{L(e)} \\cong \\LSh{s}\\hShift{0}$ indeed.\n\nComputing $T'_{L(e)}$ involves $\\hom^\\bullet_{\\mathcal{O}}(-, L(e))$,\nfor which we employ the following notation.\nBetween angle brackets, we write down\n$\\Complex$-bases for the homological components of $\\hom^\\bullet_{\\mathcal{O}}(-, L(e))$.\nEvery basis element, which is a morphism of complexes, is written\nwith the codomain and the (shifted) domain written horizontally \nand the map of complexes vertically.\nThe horizontal arrows between the $\\langle\\cdots\\rangle$'s\ncarry matrix representations of the boundary map of $\\hom^\\bullet_{\\mathcal{O}}(-, L(e))$\n\\wrt\\ these bases.\nFrom the projective resolution $L(e)\\simeq\\{P(e)\\to P(s)\\to P(e)\\}$,\nwe obtain\n{%\n\\tikzset{\n\tampersand replacement=\\&,\n\tcommutative diagrams\/diagrams={\n\t\trow sep=small,\n\t\tcolumn sep=tiny,\n\t\tnodes={inner sep=1pt, font=\\scriptsize}\n\t}\n}%\n\\renewcommand{\\xto}[1]{\\xrightarrow{\\mathmakebox[\\widthof{\\scriptsize$\\!\\!\\!\\Mtrx{0\\\\1}\\!\\!\\!$}]{#1}}}%\n\\begin{align}\n\t\\mathmakebox[1em][l]{\\hom^\\bullet_\\mathcal{O}(P(s), L(e))} \\notag\\\\*%\n\t\t&= \\left\\{\n\t\t\t\\left\\langle\n\t\t\t\t\\begin{tikzcd}[baseline=(B.base)]\n\t\t\t\t\tP(s) \\ar[d, \"b\"' name=B] \\\\\n\t\t\t\t\tP(e) \\ar[r] \\& P(s) \\ar[r] \\& P(e)\n\t\t\t\t\\end{tikzcd}\n\t \t\t\\right\\rangle\n\t\t\t\\xto{\\Mtrx{0\\\\1}}\n\t\t\t\\left\\langle\n\t\t\t\t\\begin{tikzcd}[baseline=(B.base)]\n\t\t\t\t\t\\& P(s) \\ar[d, \"{\\id, x}\"' name=B]\\\\\n\t\t\t\t\tP(e) \\ar[r] \\& P(s) \\ar[r] \\& P(e)\n\t\t\t\t\\end{tikzcd}\n\t\t\t\\right\\rangle\n\t\t\t\\xto{(1, 0)}\n\t\t\t\\left\\langle\n\t\t\t\t\\begin{tikzcd}[baseline=(B.base)]\n\t\t\t\t\t\\&\\& P(s) \\ar[d, \"b\" name=B]\\\\\n\t\t\t\t\tP(e) \\ar[r] \\& P(s) \\ar[r] \\& P(e)\n\t\t\t\t\\end{tikzcd}\n\t\t\t\\right\\rangle\n\t\t\\right\\}\n\t\\notag\\\\*\n\t\t&\\simeq 0, \\label{eq:twisting-sl2-by-Le:Hom-Ps-Le}\n\t\\\\[2.5pt]\n\t%\n\t\\mathmakebox[1em][l]{\\hom^\\bullet_\\mathcal{O}(L(e), P(s))} \\notag\\\\*\n\t\t&= \\left\\{\n\t\t\\left\\langle\n\t\t\t\\begin{tikzcd}[baseline=(B.base)]\n\t\t\t\tP(e) \\ar[r] \\& P(s) \\ar[r] \\& P(e) \\ar[d, \"b\"' name=B]\\\\\n\t\t\t\t\\&\\& P(s)\n\t\t\t\\end{tikzcd}\n\t\t\\right\\rangle\n\t\t\\xto{\\Mtrx{0\\\\1}}\n\t\t\\left\\langle\n\t\t\t\\begin{tikzcd}[baseline=(B.base)]\n\t\t\t\tP(e) \\ar[r] \\& P(s) \\ar[r] \\ar[d, \"{\\id, x}\"' name=B] \\& P(e) \\\\\n\t\t\t\t\\& P(s)\n\t\t\t\\end{tikzcd}\n\t\t\\right\\rangle\n\t\t\\xto{(1, 0)}\n\t\t\\left\\langle\n\t\t\t\\begin{tikzcd}[baseline=(B.base)]\n\t\t\t\tP(e) \\ar[d, \"b\"' name=B] \\ar[r] \\& P(s) \\ar[r] \\& P(e)\\\\\n\t\t\t\tP(s)\n\t\t\t\\end{tikzcd}\n\t\t\\right\\rangle\n\t\t\\right\\}\n\t\\notag\\\\*\n\t&\\simeq 0 \\label{eq:twisting-sl2-by-Le:Hom-Le-Ps},\n\\intertext{\n\tWe thus get immediately from the  definition \\ref{def:twist-cotwist} of $T'$ and $T$\n\tthat $\\Xi'_{L(e)} P(s) = 0 = \\Xi_{L(e)} P(s)$\n\tand thus that $T'_{L(e)} P(s) = P(s) = T_{L(e)} P(s)$.\n\tFor $P(e)$, we obtain\n}\n\t\t\\mathmakebox[1em][l]{\\hom^\\bullet_\\mathcal{O}(P(e), L(e))} \\notag\\\\*%\n\t&= \\Biggl\\{\n\t\\left\\langle\n\t\\underbrace{\n\t\t\\begin{tikzcd}[baseline=(B.base)]\n\t\t\tP(e) \\ar[d, \"\\id\"' name=B]\\\\\n\t\t\tP(e) \\ar[r] \\& P(s) \\ar[r] \\& P(e)\n\t\t\\end{tikzcd}\n\t}_{\\eqqcolon i^{(\\bar{2})}}\n\t\\right\\rangle\n\t\\xto{1}\n\t\\left\\langle\n\t\\underbrace{\n\t\t\\begin{tikzcd}[baseline=(B.base)] \n\t\t\t\\& P(e) \\ar[d, \"a\"' name=B]\\\\\n\t\t\tP(e) \\ar[r] \\& P(s) \\ar[r] \\& P(e)\n\t\t\\end{tikzcd}\n\t}_{\\eqqcolon a^{(\\bar{1})}}\n\t\\right\\rangle\n\t\\xto{0}\n\t\\left\\langle\n\t\\underbrace{\n\t\t\\begin{tikzcd}[baseline=(B.base)]\n\t\t\t\\&\\& P(e) \\ar[d, name=B, \"\\id\"]\\\\\n\t\t\tP(e) \\ar[r] \\& P(s) \\ar[r] \\& P(e)\n\t\t\\end{tikzcd}\n\t}_{\\eqqcolon i^{(0)}}\n\t\\right\\rangle\n\t\\Biggr\\}\n\t\\label{eq:twisting-sl2:Hom-Pe-Le}\n\t\\\\*[-2ex]\n\t&\\simeq \\langle i^{(0)}\\rangle,\n\t\\notag\n\t\\\\[2.5pt]\n\t%\n\t\\mathmakebox[1em][l]{\\hom^\\bullet_\\mathcal{O}(L(e), P(e))} \\notag\\\\*%\n\t&= \\Biggl\\{\n\t\\left\\langle\n\t\\underbrace{\n\t\t\\begin{tikzcd}[baseline=(B.base)]\n\t\t\tP(e) \\ar[r] \\& P(s) \\ar[r] \\& P(e) \\ar[d, \"\\id\"' name=B]\\\\\n\t\t\t\\&\\& P(e) \n\t\t\\end{tikzcd}\n\t}_{\\eqqcolon i^{(0)}}\n\t\\right\\rangle\n\t\\xto{1}\n\t\\left\\langle\n\t\\underbrace{\n\t\t\\begin{tikzcd}[baseline=(B.base)]\n\t\t\tP(e) \\ar[r] \\& P(s) \\ar[r] \\ar[d, \"b\"' name=B] \\& P(e) \\\\\n\t\t\t\\& P(e)\n\t\t\\end{tikzcd}\n\t}_{\\eqqcolon b^{(1)}}\n\t\\right\\rangle\n\t\\xto{0}\n\t\\left\\langle\n\t\\underbrace{\n\t\t\\begin{tikzcd}[baseline=(B.base)]\n\t\t\tP(e) \\ar[r] \\ar[d, \"\\id\"' name=B]\\& P(s) \\ar[r] \\& P(e) \\\\\n\t\t\tP(e)\n\t\t\\end{tikzcd}\n\t}_{\\eqqcolon i^{(2)}}\n\t\\right\\rangle\n\t\\Biggr\\}\n\t\\label{eq:twisting-sl2:Hom-Le-Pe}\n\t\\\\*[-2ex]\n\t&\\simeq \\langle i^{(2)}\\rangle \\hShift{2},\n\t\\notag\n\\end{align}\n}%\nWe denote the morphisms of complexes as indicated in the above formulae.\nBefore we compute the images of $P(e)$ under $T'_{L(e)}$ and $T_{L(e)}$,\nwe state the following:\n\n\\begin{lemma}[(Elimination of trivial summands)]\n\t\\label{lem:gauss-elimination}\n\tIn any abelian category, \n\tif the map $d$ in the chain complex in the first line of the diagram\n\t\\[\n\t\\begin{tikzcd}[ampersand replacement=\\&, row sep=scriptsize]\n\t\t\\cdots \\rar \n\t\t\\& U \\rar{\\Mtrx{a\\\\b}}\\dar[equal] \n\t\t\\& V \\oplus W \\rar{\\Mtrx{c&d\\\\e&f}}\\dar{(1, 0)} \n\t\t\\& X \\oplus Y \\rar{(g, h)} \\dar{(-fd', 1)}\n\t\t\\& Z \\rar \\dar[equal] \\& \\cdots\\\\\n\t\t\\cdots \\rar\n\t\t\\& U \\rar[swap]{b}\n\t\t\\& V \\rar[swap]{e - fd'c}\n\t\t\\& Y \\rar[swap]{h}\n\t\t\\& Z \\rar \\& \\cdots\n\t\\end{tikzcd}\n\t\\]\n\tis an isomorphism with inverse $d'$,\n\tthe second line is a chain complex and the vertical map is a quasi-isomorphism.\n\\end{lemma}\n\n{%\n\\predisplaypenalty=1000\nWe now compute the images $T'_{L(e)}P(e)$ and $T_{L(e)}P(e)$ parallelly in two columns.\nThese are the respective total complexes of the triple complexes\n\\tikzcdset{every diagram\/.append style = {\n\trow sep={.7cm,between origins}, \n\tcolumn sep={1.0cm,between origins}, \n\tnodes={inner sep=1pt},\n\tampersand replacement=\\&,\n\tcramped,\n\tbaseline=(B.base)\n}}\n\\begin{align}\n\t\\notag\n\t&\\mathrel{\\phantom{=}} T'_{L(e)} P(e) &&\\mathrel{\\phantom{=}} T_{L(e)} P(e)\\\\*\n\t\\notag\n\t&=\\bigl\\{\n\t\t\\bgroup\\color{gray}\n\t\t\t\\degZero{P(e)}\n\t\t\t\\xto{\\mkern -5mu \\mathit{ev}' \\mkern-5mu}\n\t\t\\egroup\n\t\t\\underbrace{\\lin\\bigl(\\hom^\\bullet_\\mathcal{O}(P(e),L(e)), L(e)\\bigr)}_{\\Xi'_{L(e)}}\\bigr\\}\n\t&&= \\underbrace{\\bigl\\{\\hom^\\bullet_\\mathcal{O}\\bigl(L(e),P(e)\\bigr) \\otimes L(e)}_{\\Xi_{L(e)} P(e)}\n\t\t\\bgroup\\color{gray}\n\t\t\t\\xto{\\mkern -5mu \\mathit{ev} \\mkern-5mu}\n\t\t\t\\degZero{P(e)}\n\t\t\\egroup\n\t\t\\bigr\\}\\\\*\n\\intertext{\n\twith the gray $P(e)$ in degree $0$.\n\tRecall from \\cref{rmk:lin-and-tensor}\n\tthat the double complexes\n\t$\\lin(\\hom^\\bullet_{\\mathcal{O}}(P(e),L(e)), L(e))$ and $\\hom^\\bullet_{\\mathcal{O}}(P(e),L(e),P(e)) \\otimes L(e)$,\n\tare sums of shifted copies of $L(e)$, indexed by basis elements\n\tfrom \\cref{eq:twisting-sl2:Hom-Pe-Le,eq:twisting-sl2:Hom-Le-Pe}.\n\tWorking this out gives\n\tthat $T'_{L(e)} P(e)$ and $T_{L(e)} P(e)$ respectively\n\tare the triple complexes\n}\n\t\t&=\n\t\\label{eq:cotwist-of-P(e)-wrt-L(e)}\n\t\\left\\{\n\t\t\\begin{tikzcd}[row sep={.7cm,between origins}, column sep={1.0cm,between origins}, nodes={inner sep=1pt},ampersand replacement=\\&]\n\t\t\t     P(e)_{i^{(0)}}       \\ar[rr, \"a\"]\\ar[dd,\"0\"']\n\t\t\t\\&\\& P(s)_{i^{(0)}}       \\ar[rr,\"b\"]\n\t\t\t\\&\\& P(e)_{i^{(0)}}       \\ar[dd,\"0\"]\\\\\n\t\t\t\\& |[color=gray]| P(e)    \\ar[gray, dddl, \"\\id\"]\\ar[gray, dr, \"a\"]\\ar[gray, urrr, \"\\id\" very near start, dashed]\\\\\n\t\t\t     P(e)_{a^{(\\bar{1})}}   \\ar[rr, \"a\", cross line]\\ar[dd,\"\\id\"', dashed]\t\n\t\t\t\\&\\& P(s)_{a^{(\\bar{1})}}   \\ar[from=uu, cross line, \"0\"]\\ar[rr,\"b\"]\\ar[dd,\"\\id\", dashed] \t\n\t\t\t\\&\\& P(e)_{a^{(\\bar{1})}}   \\ar[dd,\"\\id\", dashed]\\\\\\\\\n\t\t\t     P(e)_{i^{(\\bar{2})}} \\ar[rr, \"a\"']\n\t\t\t\\&\\& P(s)_{i^{(\\bar{2})}} \\ar[rr,\"b\"']\n\t\t\t\\&\\& P(e)_{i^{(\\bar{2})}}\n\t\t\\end{tikzcd}\n\t\\right\\}\n\t&&=\n\t\\left\\{\n\t\\begin{tikzcd}\n\t\t\t     P(e)_{i^{(0)}}      \\ar[rr, \"a\"] \\ar[dd,\"\\id\"', dashed]\n\t\t\t\\&\\& P(s)_{i^{(0)}}      \\ar[rr,\"b\"] \n\t\t\t\\&\\& P(e)_{i^{(0)}}      \\ar[dd,\"\\id\", dashed]\\\\\\\\ |[alias=B]|\n\t\t\t     P(e)_{b^{(1)}}      \\ar[rr, \"a\"] \\ar[dd,\"0\"']\n\t\t\t\\&\\& P(s)_{b^{(1)}}      \\ar[from=uu, \"\\id\", dashed] \\ar[rr,\"b\"] \\ar[dd,\"0\"']  \t\n\t\t\t\\&\\& P(e)_{b^{(1)}}      \\ar[dd,\"0\"]\\\\\n\t\t\\&\\&\\& |[color=gray]| P(e) \\ar[gray, from=dlll, cross line, \"\\id\", dashed]\\ar[gray, from=ul, \"b\"]\\ar[gray, from=uuur, cross line, \"\\id\"' near start]\\\\\n\t\t\t     P(e)_{i^{(2)}}      \\ar[rr, \"a\"']\n\t\t\t\\&\\& P(s)_{i^{(2)}}      \\ar[rr,\"b\"'] \n\t\t\t\\&\\& P(e)_{i^{(2)}}\n\t\\end{tikzcd}\n\t\\right\\} \\\\\n\\intertext{\n\tstill with the gray $P(e)$ in degree zero.\n\tHere, the black $P(-)$'s are indexed by \n\tbasis elements from \\cref{eq:twisting-sl2:Hom-Pe-Le,eq:twisting-sl2:Hom-Le-Pe},\n\taccording to the copy of $L(e)$ they belong to.\n\tSince $\\hom^\\bullet_{\\mathcal{O}}(P(e),L(e))$, $\\hom^\\bullet_{\\mathcal{O}}(L(e),P(e))$\n\tand $L(e)$ are respectively concentrated \n\tin degrees $-2$ to $0$, $0$ to $2$ and $-2$ to $0$,\n\tthe two double complexes $\\Xi'_{L(e)} P(e)$ and $\\Xi_{L(e)} P(e)$ printed in black\n\thave their degree-zero summands on the secondary diagonal.\n\tAccording to \\cref{rmk:evaluation-in-terms-of-shifted-copies},\n\tthe non-zero components of the (co)evaluation maps therefore consist of the gray morphisms.\n\tUsing the elimination lemma \\ref{lem:gauss-elimination},\n\twe may eliminate all summands adjacent to one of the dashed identity morphisms;\n\tthis shows that there are quasi-isomorphisms\n}\n\t\\notag\n\t&\\simeq \\{P(e) \\xto{a} \\degZero{P(s)}\\},\n\t&&\\simeq \\{\\degZero{P(s)} \\xto{b} P(e)\\} \\\\*\n\t\\notag\n\t&\\simeq \\LSh{s} P(e),\n\t&& \\simeq \\RCsh{s} P(e),\n\\end{align}\n}%\nby comparison with \\eqref{eq:shuffling-sl2:imaegs-of-projectives}.\nHence, we already have shown half of the following:\n\n\\needspace{3\\baselineskip}\n\\begin{theorem}\n\tThere is a natural isomorphism $T'_{L(e)} \\cong \\LSh{s}$.\n\\end{theorem}\n\\begin{proof}\n\tTo show that both functors are isomorphic\n\twe have yet to show that the morphisms $\\id_{P(e)}, \\id_{P(s)}, a$ and $b$ generating $\\End_{\\mathcal{O}}(P)$\n\thave isomorphic images under both functors.\n\tRecall the images $\\Sh{s} a\\colon M(s)\\into P(s)$ and $\\Sh{s} b\\colon P(s)\\onto M(s)$ from \\eqref{eq:shuffling-sl2:images-of-generating-morphisms}.\n\tRecall from \\cref{rmk:induced-map-in-terms-of-shifted-copies}\n\thow the maps $a\\colon P(e) \\to P(s)$ and $b\\colon P(s) \\to P(e)$ induce morphisms\n\t$a_*: \\Xi'_{L(e)} P(e) \\to \\Xi'_{L(e)} P(s)$ and $b_*: \\Xi'_{L(e)} P(s) \\to \\Xi'_{L(e)} P(s)$.\n\tSince\n\t\\begin{align*}\n\t\ta^*\\colon \\hom^\\bullet_{\\mathcal{O}}(P(s), L(e)) &\\xtofrom[]{\\quad} \\hom^\\bullet_{\\mathcal{O}}(P(e), L(e)) \\noloc b^*\\\\\n\t\t\\begin{aligned}[t]\n\t\t\tb^{(0)}, x^{(\\bar{1})}, b^{(\\bar{2})}  &\\longmapsto 0,\\\\\n\t\t\ti^{(\\bar{1})}                          &\\longmapsto a^{(\\bar{1})}\\\\\n\t\t\\end{aligned}\n\t\t&\\phantom{{}\\xtofrom[]{\\quad}{}}\n\t\t\\begin{aligned}[t]\n\t\t\tb^{(0)}          & \\longmapsfrom i^{(0)}\\\\\n\t\t\tx^{(\\bar{1})}    & \\longmapsfrom a^{(\\bar{1})}\\\\\n\t\t\tb^{(\\bar{2})}    & \\longmapsfrom i^{(\\bar{2})},\n\t\t\\end{aligned}\n\t\\end{align*}\n\tthe morphisms $a_*$ and $b_*$ act\n\ton an element $m_{f}$ of a summand $L(e)\\hShift{\\deg f}_f$\n\tof $\\Xi'_{L(e)}P(e)$ or $\\Xi'_{L(e)}P(e)$ indexed by a basis element $f$ by\n\t\\begin{align*}\n\t\ta_*: \\Xi'_{L(e)} P(e) &\\xtofrom[]{\\quad} \\Xi'_{L(e)} P(s)\\noloc b_*\\\\\n\t\t\\begin{aligned}[t]\n\t\t\tm_{i^{(0)}}, m_{i^{(\\bar{2})}}   &\\longmapsto 0\\\\\n\t\t\tm_{a^{(\\bar{1})}}                &\\longmapsto m_{i^{(\\bar{1})}}\n\t\t\\end{aligned}\n\t\t&\\phantom{{}\\xtofrom[]{\\quad}{}}\n\t\t\\begin{aligned}[t]\n\t\t\tm_{i^{(0)}}                      &\\mapsfrom m_{b^{(0)}}\\\\\n\t\t\tm_{a^{(\\bar{1})}}                &\\mapsfrom m_{x^{(\\bar{1})}}\\\\\n\t\t\\end{aligned}\\quad\n\t\t\\begin{aligned}[t]\n\t\t\t0                                &\\mapsfrom m_{i^{(\\bar{1})}}\\\\\n\t\t\tm_{i^{(\\bar{2})}}                &\\mapsfrom m_{b^{(\\bar{2})}}.\n\t\t\\end{aligned}\n\t\\end{align*}\n\tThe maps $T'_{L(e)}a$ and $T'_{L(e)}b$ are therefore quasi-isomorphic to\n\t\\[\n\t\t\\begin{tikzcd}[ampersand replacement=\\&, row sep=normal]\n\t\t\tT'_{L(e)} P(s)\n\t\t\t\t\\ar[r, phantom, \"\\simeq\"]\n\t\t\t\t\\ar[d, shift right, \"T'_{L(e)} b\"'] \\&[-3.5ex]\n\t\t\t\\Bigl\\{ \\textcolor{gray}{P(s)}\n\t\t\t\t\\ar[r, \"\\mathit{ev}'\", color=gray] \n\t\t\t\t\\ar[d, shift right=0.56ex-.6ex, \"b\"', color=gray] \n\t\t\t\t\\ar[from=d, shift right=0.56ex+.6ex, \"a\"', color=gray] \n\t\t\t\t\\&\n\t\t\t\\overbrace{L(e)_{b^{(0)}} \\oplus L(e)\\hShift{1}_{i^{(\\bar{1})}} \\oplus L(e)\\hShift{1}_{x^{(\\bar{1})}} \\oplus L(e)\\hShift{2}_{b^{(\\bar{2})}}}^{\\Xi'_{L(e)} P(s)} \n\t\t\t\\mathrlap{\\Bigr\\}}\n\t\t\t\t\\ar[d, shift right, \"b_*=\\Mtrx{1\\\\&0&1\\\\&&&1}\"'] \n\t\t\t\\\\\n\t\t\tT'_{L(e)} P(e)\n\t\t\t\t\\ar[r, phantom, \"\\simeq\"]\n\t\t\t\t\\ar[u, shift right, \"T'_{L(e)} a\"']  \\&\n\t\t\t\\Bigl\\{ \\textcolor{gray}{P(e)}\n\t\t\t\t\\ar[r, \"\\mathit{ev}'\"', color=gray] \n\t\t\t\t\\&\n\t\t\t\\underbrace{L(e)_{i^{(0)}} \\oplus L(e)\\hShift{1}_{a^{(\\bar{1})}} \\oplus L(e)\\hShift{2}_{i^{(\\bar{2})}}}_{\\Xi'_{L(e)} P(e)}\n\t\t\t\\mathrlap{\\Bigr\\}.}\n\t\t\t\t\\ar[u, shift right, \"\\Mtrx{0\\\\&1\\\\&0\\\\&&0}=a_*\"'] \n\t\t\\end{tikzcd}\n\t\\]\n\tConsider again the left triple complex in \\eqref{eq:cotwist-of-P(e)-wrt-L(e)},\n\twhose black printed \"front layer\" represents $\\Xi'_{L(e)} P(e)$,\n\tand use the elimination lemma \\ref{lem:gauss-elimination}\n\tto eliminate only the bottom three identities\n\t$P(-)_{a^{(\\bar{1}})} \\to P(-)_{i^{(\\bar{2})}}$.\n\tSimilarly, a triple complex representing $T'_{L(e)} P(s)$ \n\tcan be obtained from \\eqref{eq:twisting-sl2-by-Le:Hom-Ps-Le}.\n\tWe obtain that $T'_{L(e)}a$ and $T'_{L(e)}b$ are represented by the maps\n\t\\begin{equation}\n\t\t\\label{eq:twisting-by-Le:action-on-a-b:triple-complexes}\n\t\t\\begin{tikzcd}[cramped, ampersand replacement=\\&, row sep={.7cm,between origins}, column sep={2cm,between origins}]\n\t\t\t%\n\t\t\t\\&[-0mm] \\mathmakebox[\\widthof{$P(e)_{i^{(0)}}$}]{P(e)_{b^{(0)}}}\n\t\t\t\t\\ar[rr]\n\t\t\t\t\\ar[dd, equal, cross line]\n\t\t\t\t\\ar[brace', -, dd, start anchor=north west, end anchor=south west, xshift=-1ex, \"T'_{L(e)} P(s) \\simeq{}\" {name=TPs, anchor=east, xshift=-.8ex, font=\\normalsize}]\n\t\t\t\t\\ar[dd, equal, cross line]\n\t\t\t\\&[-3mm]\\&[-0mm] P(s)_{b^{(0)}} \\ar[rr]\\ar[dd, equal] \n\t\t\t\\&[-3mm]\\&[-0mm] \\mathmakebox[\\widthof{$P(e)_{i^{(0)}}$}]{P(e)_{b^{(0)}}} \\ar[dd, equal]\n\t\t\t\\\\\n\t\t\t%\n\t\t\t\\&\\& |[gray]| P(s)\n\t\t\t\t\\ar[urrr, gray]\n\t\t\t\t\\ar[dr, gray, equal]\n\t\t\t\\\\\n\t\t\t%\n\t\t\t\\& P(e)_{i^{(\\bar{1})}} \\ar[rr]\t\n\t\t\t\\&\\& P(s)_{i^{(\\bar{1})}}\n\t\t\t\t\\ar[from=uu, equal, cross line]\n\t\t\t\t\\ar[rr] \n\t\t\t\\&\\& P(e)_{i^{(\\bar{1})}}\n\t\t\t\t\\ar[brace, -, from=uu, start anchor=north east, end anchor=south east]\n\t\t\t\\\\\n\t\t\t%\n\t\t\tP(e)_{i^{(0)}}\n\t\t\t\t\\ar[from=uuur, cross line, \"1\"'  near end, shift right]\n\t\t\t\t\\ar[to=uuur, cross line, \"0\"' very near end, shift right]\n\t\t\t\\&\\& P(s)_{i^{(0)}}\n\t\t\t\t\\ar[from=uuur, cross line, \"1\"', shift right]\n\t\t\t\t\\ar[to=uuur, cross line, \"0\"' very near end, shift right]\n\t\t\t\t\\ar[rr]\n\t\t\t\\&\\& P(e)_{i^{(0)}}\n\t\t\t\t\\ar[from=uuur, cross line, \"1\"', shift right]\n\t\t\t\t\\ar[to=uuur, cross line, \"0\"' very near end, shift right]\\\\\n\t\t\t\\phantom{P(e)_{i^{(0)}}} \n\t\t\t\t\\ar[brace', -, xshift=-1ex, from=u, start anchor=north west, end anchor=south west, \"T'_{L(e)} P(e) \\simeq{}\" {name=TPe, anchor=east, xshift=-.8ex, font=\\normalsize}]\n\t\t\t%\n\t\t\t\\& |[gray]| P(e) \n\t\t\t\t\\ar[urrr, gray, equal]\n\t\t\t\t\\ar[from=uuur, cross line, gray, \"b\"'  at end, shift right]\n\t\t\t\t\\ar[to=uuur, cross line, gray, \"a\"'  at end, shift right]\n\t\t\t\t\\ar[from=ul, to=ur, cross line]\n\t\t\t\\&\\&\\& \\phantom{P(e)_{i^{(0)}}} \n\t\t\t\t\\ar[brace, -, from=u, start anchor=north east, end anchor=south east]\n\t\t\t\t\\ar[from=TPs, to=TPe, \"T'_{L(e)}b\"', shift right, xshift=-2ex]\n\t\t\t\t\\ar[to=TPs, from=TPe, \"T'_{L(e)}a\"' near end, shift right, xshift=-2ex]\n\t\t\\end{tikzcd}\n\t\\end{equation}\n\tbetween the triple complexes representing $T'_{L(e)}P(s)$ and $T'_{L(e)}P(e)$.\n\tWe pass to the total complexes of \\eqref{eq:twisting-by-Le:action-on-a-b:triple-complexes}\n\tand use the elimination lemma \\ref{lem:gauss-elimination} \n\tto choose quasi-isomorphic replacements\n\t\\begin{equation*}\n\t\t\\scriptstyle\n\t\t\\begin{tikzcd}[ampersand replacement=\\&, cramped]\n\t\t\t\\&[-.6cm]\n\t\t\t\\& P(s) \\ar[from=d, \"\\Mtrx{0\\\\0\\\\1}\", xshift=-.5em] \\ar[d, \"\\Mtrx{0\\\\-1\\\\1}\", xshift=.5em] \\ar[d, phantom, \"\\simeq\" rotate=90]\n\t\t\t\\\\\n\t\t\tT'_{L(e)}P(s) \\ar[r, phantom, \"\\simeq\"] \\ar[d, \"T'_{L(e)}b\"', shift right] \\ar[from=d, \"T'_{L(e)}a\"', shift right]\\&\n\t\t\t\\smash{\\Bigl\\{}\n\t\t\tP(e)_{b^{(0)}} \n\t\t\t\t\\ar[d, \"1\"', shift right]\n\t\t\t\t\\ar[from=d, \"0\"', shift right]\n\t\t\t\t\\ar[r, \"\\Mtrx{-1\\\\a\\\\0}\"] \\& \n\t\t\t\\begin{matrix}P(e)_{i^{(\\bar{1})}} \u2295{}\\\\{} P(s)_{b^{(0)}} \u2295 \\textcolor{gray}{P(s)}\\end{matrix}\n\t\t\t\t\\ar[r, \"\\Mtrx{a&1&1\\\\0&b&\\textcolor{gray}{b}}\"] \n\t\t\t\t\\ar[d, \"\\Mtrx{0&1\\\\&0&\\textcolor{gray}{b}}\"', shift right]\n\t\t\t\t\\ar[from=d, \"\\Mtrx{0\\\\0&0\\\\&\\textcolor{gray}{a}}\"', shift right] \\&\n\t\t\t\\begin{matrix}\\phantom{{}\\oplus{}} P(s)_{i^{(\\bar{1})}} \\\\{} \u2295 P(e)_{b^{(0)}}\\end{matrix}\n\t\t\t\t\\ar[r, \"\\Mtrx{b & -1}\"] \n\t\t\t\t\\ar[d, \"\\Mtrx{0&1}\"', shift right]\n\t\t\t\t\\ar[from=d, \"0\"', shift right] \\& \n\t\t\tP(e)_{i^{(\\bar{1})}}\n\t\t\t\\mathrlap{\\Bigr\\}}\n\t\t\t\\\\\n\t\t\tT'_{L(e)}P(e) \\ar[r, phantom, \"\\simeq\"] \\&\n\t\t\t\\smash{\\Bigl\\{}\n\t\t\tP(e)_{i^{(0)}} \\ar[r, \"\\Mtrx{a\\\\0}\"] %\n\t\t\t\\&\n\t\t\tP(s)_{i^{(0)}} \u2295 \\textcolor{gray}{P(e)} \\ar[r, \"\\Mtrx{b & 1}\"'] \\& \n\t\t\tP(e)_{i^{(0)}}\n\t\t\t\\mathrlap{\\smash{\\Bigr\\}}}\n\t\t\t\\\\\n\t\t\t\\&\n\t\t\t\\& M(s)\\mathrlap{.} \\ar[from=u, \"\\Mtrx{\\can&0}\", xshift=.5em] \\ar[u, phantom, \"\\simeq\" rotate=90] \\ar[u, \"\\Mtrx{\\can\\\\-\\can}\", xshift=-.5em]\n\t\t\\end{tikzcd}\n\t\\end{equation*}\n\t%\n\tThis shows that $T'_{L(e)} b = \\LSh{s}b\\colon P(s)\\onto M(s)$\n\tand $T'_{L(e)}a = \\Sh{s} a\\colon M(s)\\into P(s)$ indeed.\n\tSince all morphisms in $D^\\mathrm{b}(\\mathcal{O}_0)$ are generated by $a$ and $b$,\n\tthis proves the statement.\n\\end{proof}\n\n\\section{\\texorpdfstring{$B_n$-actions for $\\mathfrak{sl}_3$ and $\\mathfrak{sl}_n$}{\ud835\udc35\u2099-actions for \ud835\udd30\ud835\udd29\u2083 and \ud835\udd30\ud835\udd29\u2099}}\n\\label{sec:sln-case}\nThe Lie algebra $\\mathfrak{sl}_3$ has as its Weyl group the symmetric group $S_3=\\{e, s, t, st, ts, w_0\\}$.\nA quiver $Q_{\\mathfrak{sl}_3}$,\nwhich has vertices indexed by $S_3$\nand unique edges $w \\leftrightarrows ws$ for all $w < ws$,\nand a homogeneous ideal $\\mathfrak a_{\\mathfrak{sl}_3}$ of $\\Complex Q_{\\mathfrak{sl}_3}$\nsuch that $\\mathcal{O}_{0,\\mathfrak{sl}_3}\\simeq\\Mod{}[A_{\\mathfrak{sl}_3}]$ \nfor the algebra $A_{\\mathfrak{sl}_3}=\\Complex Q_{\\mathfrak{sl}_3}\/\\mathfrak a_{\\mathfrak{sl}_3}$\nis provided in \\autocites{stroppel:quivers,Marko:quivers}.\n\nOne sees that $Q_{\\mathfrak{sl}_2}$ is a full subquiver of $Q_{\\mathfrak{sl}_3}$\nand $\\mathfrak a_{\\mathfrak{sl}_3}\\cap\\Complex Q_{\\mathfrak{sl}_2}=\\mathfrak a_{\\mathfrak{sl}_2}$.\nThe inclusion $A_{\\mathfrak{sl}_2}\\into A_{\\mathfrak{sl}_3}$ thus induced \ngives rise to an adjoint pair of functors\n\\begin{equation}\n\t\\label{eq:sl2-sl3:restriction-induction-category-O}\n\t\\begin{aligned}\n\t\t\\Res_{\\mathfrak{sl}_3}^{\\mathfrak{sl}_2}\\colon\\mathcal{O}_{0,\\mathfrak{sl}_3} \n\t\t\t&\\xtofrom{\\ \\dashv\\ } \\mathcal{O}_{0,\\mathfrak{sl}_2}\\noloc \\Ind_{\\mathfrak{sl}_2}^{\\mathfrak{sl}_3},\n\t\t\\\\\n\t\t\\begin{aligned}\n\t\t\tP(e), P(t) &\\mapsto P(e),\\\\\n\t\t\tP(s), P(st), P(ts), P(w_0) &\\mapsto P(s),\\\\\t\n\t\t\\end{aligned}\n\t\t&\n\t\t\\phantom{{}\\xtofrom{\\dashv}{}}\n\t\t\\begin{aligned}\n\t\t\tP(e) &\\mapsfrom P(e),\\\\\n\t\t\tP(s) &\\mapsfrom P(s),\n\t\t\\end{aligned}\n\t\\end{aligned}\n\\end{equation}\nwhich turns $\\mathcal{O}_{0,\\mathfrak{sl}_2}$ into a full subcategory of $\\mathcal{O}_{0,\\mathfrak{sl}_3}$.\nIn particular, $\\End_{\\mathcal{O}_{0,\\mathfrak{sl}_3}}(P(s)) \\cong \\Complex[x]\/(x^2)$ and\n$P(s)$ is $0$-spherelike also in $\\mathcal{O}_{0,\\mathfrak{sl}_3}$.\n\n\\begin{caveat}\n\tThe Calabi-Yau property from \\cref{def:spherical-object:calabi-yau} is not \"local\",\n\tin the sense that an object can lose this property in a larger ambient category.\n\tFor instance, there are non-trivial morphisms \n\t$P(s)\\to P(t)$ and $P(t)\\to P(s)$ in $\\mathcal{O}_{0,\\mathfrak{sl}_3}$\n\twhose composition is zero, so $P(s)$ cannot be spherical.\n\tWe shall present two possible remedies in this section.\n\\end{caveat}\n\n\\subsection{Spherical subcategories}\nConsider a $k$-linear triangulated category $\\mathcal T$.\n\\begin{definition}\n\tAn object $E\\in\\mathcal T$ is said to have a \\term{Serre dual} $\\mathrm{S} E$\n\tif the contravariant functor $\\Hom_\\mathcal T(E,-)^*$%\n\t---the star stands for vector space dual---%\n\tis represented by $\\mathrm{S} E$.\n\tIf a Serre dual can be chosen functorially\n\tvia an auto-equivalence $\\mathrm S$,\n\tthis functor $\\mathrm S$ is said to be a \\term{Serre functor} of $\\mathcal T$.\n\\end{definition}\n\n\\begin{remark}\n\tThe Calabi-Yau condition in \\cref{def:spherical-object:calabi-yau}\n\tfor a $d$-spherlike object $E$ to be spherical\n\tdemands that for all $F$, the composition pairing induce an isomorphism\n\t$\\Hom^d_{D^\\mathrm{b}}(E, F) \\cong \\Hom_{D^\\mathrm{b}}(F,E)^*$ or, equivalently, \n\tthat $E\\hShift{-d}$ be a Serre dual for $E$;\n\tsee also \\autocite[lem.\\ 2.15]{seidel-thomas:braid-group-actions}.\n\t%\n\t%\n\t%\n\\end{remark}\n\nLet $E\\in\\mathcal T$ be a $d$-spherelike (but not necessarily spherical) object\nthat has \\emph{some} Serre dual $\\mathrm SE$.\nSince in particular $\\End^d_\\mathcal T(E)^* = \\Hom_\\mathcal T(E, E\\hShift{-d})^*\n\\cong \\Hom_\\mathcal T(E\\hShift{-d}, \\mathrm SE)$,\nthere is a morphism $x^*\\colon E\\hShift{-d} \\to \\mathrm SE$\ndual to the non-trivial endomorphism $x \\in \\End^d_\\mathcal T(E)$.\n\n\\begin{definition}\n\tThe \\term{asphericality} of a spherelike object $E$ is $Q(E)\u2254\\cone(x^*)$.\n\tIts \\term{left complement} ${}^\\bot Q(E) \u2254 \\{X\\in\\mathcal T \\mid \\Hom_{\\mathcal T}(X, Q(E))=0 \\}$\n\tis a full triangulated subcategory of $\\mathcal T$.\n\\end{definition}\n\\begin{theorem}[(Hochenegger, Kalck, Ploog) {\\autocite[Thm.\\ 4.4, Appendix A]{HKP:spherical-subcategories}}]\n\tThe \\emph{spherical subcategory} $\\Sph(E)\u2254{}^\\bot Q(E)$ of $E$\n\tis the largest triangulated subcategory of $D^\\mathrm{b}(\\mathcal T)$\n\tin which $E$ is spherical.\n\\end{theorem}\n\\begin{example}\n\tFor $\\mathfrak g$ a semisimple complex Lie algebra and $\\lambda$ a regular weight\n\t(for instance, $\\lambda=0$),\n\tthe auto-equivalence $\\mathrm S\\coloneqq\\LSh{w_0}^2$ is a Serre functor of $D^\\mathrm{b}(\\mathcal{O}_\\lambda)$\n\t\\autocite[Prop.\\ 4.1]{MS:Serre-functors}.\n\\end{example}\n\\begin{proposition}\n\tThe $0$-spherelike module $P(s)\\inD^\\mathrm{b}(\\mathcal{O}_{0,\\mathfrak{sl}_3})$ has Serre dual $\\mathrm SP(s)\\cong P(s)^\\vee$.\n\\end{proposition}\n\\begin{proof}\n\tFor this proof, we take the graded structure on $\\mathcal{O}_0^{\\mathbf Z}$ into account.\n\tRecall from \\cref{def:graded-translation}\n\tthat $\\Theta_s^2 \u2245 \\Theta_s\u27e8-1\u27e9\u2295\\Theta_s\u27e81\u27e9$, and hence $\\Sh{s}\\Theta_s\\cong\\Theta_s\u27e8-1\u27e9$.\n\tRecall from \\cref{eqn:translation-defining-ses,eq:shuffling-on-Vermas} \n\tthat for $w<ws$, the functor $\\Sh{s}$ maps standard factors $M(w)$ to $M(ws)$\n\tand $M(ws)$ to $\\begin{psmallmatrix}M(ws)\\\\M(w)\/M(ws)\\end{psmallmatrix}\u27e8-1\u27e9$.\n\tApplying the factors of $\\mathrm S = \\LSh{w_0}^2$ successively\n\tshows that $P(s)$ is mapped under $\\mathrm S$ to\n\t\\begin{align}\n\t\t\\mathmakebox[1em][l]{P(s) = \\Theta_s M(s)}\n\t\t\\notag\\\\\n\t\t&\\xmapsto{\\Sh{s}} \n\t\t\tP(s)\\langle -1\\rangle = \\begin{psmallmatrix}M(s)\\\\M(e)\\end{psmallmatrix}\\langle -1\\rangle\n\t\t\\notag\\\\\n\t\t&\\xmapsto{\\Sh{t}} \n\t\t\t\\begin{psmallmatrix}M(ts)\\\\M(t)\\end{psmallmatrix}\\langle -1\\rangle \n\t\t\\notag\\\\\n\t\t&\\xmapsto{\\Sh{s}} \n\t\t\t\\begin{psmallmatrix}M(w_0)\\\\M(ts)\\end{psmallmatrix}\\langle -1\\rangle\n\t\t\t= \\Theta_t M(ts)\\langle -1\\rangle \n\t\t\\notag\\\\\n\t\t&\\xmapsto{\\Sh{t}} \n\t\t\t\\begin{psmallmatrix}M(w_0)\\\\M(ts)\\end{psmallmatrix}\\langle -2\\rangle \n\t\t\t&\n\t\t\\notag\\\\\n\t\t&\\xmapsto{\\Sh{s}}\n\t\t\t\\begin{psmallmatrix}\n\t\t\t\t\t           & M(w_0)\\\\\n\t\t\t\t\tM(ts)      & M(st)\/M(w_0)\\\\\n\t\t\t\t\tM(t)\/M(ts)\n\t\t\t\\end{psmallmatrix}\u27e8-3\u27e9 \n\t\t\\notag\\\\\n\t\t&\\xmapsto{\\Sh{t}}\n\t\t\t\\arraycolsep=3pt\n\t\t\t\\left(\n\t\t\t\t\\mkern-8mu\n\t\t\t\t\\begin{smallmatrix}\n\t\t\t\t\t&&& \\begin{smallmatrix} M(w_0)\\\\M(st)\/M(w_0) \\end{smallmatrix}\n\t\t\t\t\t\\\\[-0.9em]\n\t\t\t\t\t&\\begin{smallmatrix}\\phantom{()} \\\\ M(w_0) \\end{smallmatrix}\n\t\t\t\t\t& \\left( \\begin{smallmatrix} M(st)\\\\M(s)\/(st) \\end{smallmatrix} \\middle\/ \\begin{smallmatrix} M(w_0)\\\\M(ts)\/M(w_0) \\end{smallmatrix} \\right) \\\\[-1.225em]\n\t\t\t\t\t\\\\[0.125em]\n\t\t\t\t\t\\left(\n\t\t\t\t\t\t\\begin{smallmatrix} M(t)\\\\M(e)\/M(t) \\end{smallmatrix} \\middle\/ \\begin{smallmatrix} \\phantom{()}\\\\ M(w_0) \\end{smallmatrix}\n\t\t\t\t \t\\right)\n\t\t\t\t\\end{smallmatrix}\n\t\t\t\t\\mkern-8mu\n\t\t\t\\right)\n\t\t\t\\langle -4\\rangle\n\t\t\\notag\\\\\n\t\t&\\mathmakebox[\\widthof{${}\\xmapsto{\\Sh{t}}{}$}][r]{{}={}} \n\t\t\t\\begin{psmallmatrix}\n\t\t\t\t& L(w_0)\\\\\\\n\t\t\t\t& L(st)\\quad L(ts)\\\\\n\t\t\t\t L(w_0) &  L(s)\\quad L(t)\\\\\n\t\t\t\tL(st) \\quad L(ts) & L(e)\\\\\n\t\t\t\tL(s)\n\t\t\t\\end{psmallmatrix}\n\t\t\t\\langle -4\\rangle,\n\t\t\\label{eq:composition-series-of-P(s)-dual}\n\t\\end{align}\n\twhich is precisely the composition series\n\tof the dual module $P(s)^\\vee$.\n\\end{proof}\n\nThe space $\\Hom(P(s), P(s)^\\vee) = \\Hom(P(s), \\mathrm{S}P(s)) \\cong {\\End(P(s))}^*$\nis spanned by the morphisms $\\id^*$ and $x^*$\ndual to $\\id, x \\in \\End_\\mathcal{O}(P(s))$.\nAccording to the grading of $P(s)$ in $\\mathcal{O}_0^\\mathbf{Z}$,\nthe maps $\\id$ and $x$ respectively have degrees $0$ and $2$,\nso $\\id^*$ and $x^*$ are of degrees $0$ and $-2$.\nIn terms of composition series of $P(s)$ and $P(s)^\\vee$, the maps $\\id^*$ and $x^*$ are\n\\begin{equation}\n\t\\label{eq:P(s)-x-dual-made-explicit}\n\t\\begin{tikzpicture}[baseline=(Ps-1-5.base),\n\t\t,\n\t\tevery matrix\/.append style={\n\t\t\tcolumn sep={1.4em,between origins},\n\t\t\tanchor={base east},\n\t\t\trow sep={1em,between origins},\n\t\t},\n\t\tevery edge quotes\/.append style={\n\t\t\texecute at begin node={$},\n\t\t\texecute at end node={$}\n\t\t}\n\t]\n\t\t%\n\t\t\\matrix(Ps)[\n\t\t\tmatrix of math nodes, \n\t\t\tmatrix anchor=Ps-1-5.base,\n\t\t\tfont={\\scriptsize}\n\t\t]{\n\t\t\t&&&&L(s) &\\\\\n\t\t\t& L(e) && L(st) && L(ts)\\\\\n\t\t\tL(s) && |[gray]| L(t) &&|[gray]|L(w_0)\\\\\n\t\t\t|[gray]| L(st) && |[gray]| L(ts)\\\\\n\t\t\t&|[gray]| L(w_0)\\\\\n\t\t};\n\t\t%\n\t\t\\node[braced box'={Ps-label}, fit=(Ps-1-5) (Ps-3-1) (Ps-5-2)] {};\n\t\t\\node[left=-.25cm of Ps-label] {$P(s)={}$};\n\t\t%\n\t\t\\node [yshift=2pt, braced box''={T}, fit=(Ps-1-5) (Ps-3-1) (Ps-2-6)] {};\n\t\t%\n\t\t\\matrix(SPs)[\n\t\t\tright=9em of Ps-1-5.base,\n\t\t\tmatrix of math nodes, \n\t\t\tmatrix anchor=SPs-5-2.base, \n\t\t\tfont={\\scriptsize}\n\t\t]{\n\t\t\t&&&& |[gray]|L(w_0) &\\\\\n\t\t\t&&& |[gray]|L(st) && |[gray]|L(ts)\\\\\n\t\t\t& |[gray]|L(w_0) && L(s) && |[gray]|L(t)\\\\\n\t\t\tL(st) && L(ts) && L(e)\\\\\n\t\t\t& L(s)\\\\\n\t\t};\n\t\t%\n\t\t\\node[braced box={SPs-label}, fit=(SPs-1-5) (SPs-3-6) (SPs-5-2)] {};\n\t\t\\node[right=-.25cm of SPs-label] {$ {} = \\mathrm{S}P(s) = P(s)^\\vee$.};\n\t\t%\n\t\t\\node [braced box'''={T'}, fit=(SPs-5-2) (SPs-4-1) (SPs-4-5)] {};\n\t\t\n\t\t%\n\t\t\\draw[limit bb, ->] (T) to[out=80, in=-100, looseness=2, \"x^*\" at start] (T');\n\t\t%\n\t\t\\draw[cross line, ->] (Ps-1-5) to[\"\\id^*\"' {pos=.4, fill=white}] (SPs-5-2);\n\t\t\\matrix(G)[matrix of math nodes,matrix anchor=G-5-1.base,xshift=-2.5cm] at  (Ps-1-5.base -| Ps-3-1) {\n\t\t\t-4\\\\-3\\\\-2\\\\-1\\\\\\phantom{-}0\\\\\\phantom{-}1\\\\\\phantom{-}2\\\\\\phantom{-}3\\\\\\phantom{-}4\\\\\n\t\t};\n\t\\end{tikzpicture}\n\\end{equation}\nThe grading of the modules is as indicated on the left,\nsuch that the two factors $L(s)$ connected by $\\id^*$ are in degree zero\nand $x^*$ is a degree $-2$-map indeed.\nThe grey composition factors belong to the kernel and cokernel of $x^*$, respectively.\nUsing this explicit description for $x^*$, we can prove the following:\n\n\\begin{theorem}\n\tThe inclusion $D^\\mathrm{b}(\\mathcal{O}_{0,\\mathfrak{sl}_2})\\intoD^\\mathrm{b}(\\mathcal{O}_{0,\\mathfrak{sl}_3})$\n\tfrom \\cref{eq:sl2-sl3:restriction-induction-category-O}\n\tfactors through $\\Sph(P(s))$.\n\\end{theorem}\n\\begin{proof}\n\tFrom the composition series \\cref{eq:composition-series-of-P(s)-dual} of $P(s)^\\vee$,\n\twe can derive that $P(s)^\\vee$ has a projective resolution\n\t$\\{P(s)\\to P(w_0)\u27e8-2\u27e9\\to P(w_0)\u27e8-4\u27e9\\}$.\n\tExpressiong the map $x^*$ from \\cref{eq:P(s)-x-dual-made-explicit}\n\tin terms of the projective resolution of $P(s)^\\vee$ yields\n\tthat the asphericality $Q(P(s)) = \\cone(x^*)$ of $P(s)$ is the total complex\n\t\\begin{equation}\n\t\t\\label{eq:P(s):asphericality}\n\t\tQ \\simeq\n\t\t\\left\\{\n\t\t\t\\begin{tikzcd}[nodes={inner sep={2pt}},sep=small, slightly cramped, baseline=(x.base)]\n\t\t\t&& P(s)\u27e8-2\u27e9\\dar[\"\\strut\" name=x]\\\\\n\t\t\tP(s) \\rar & P(w_0)\u27e8-2\u27e9 \\rar & P(w_0)\u27e8-4\u27e9\n\t\t\t\\end{tikzcd}\n\t\t\\right\\}\n\t\\end{equation}\n\twith the canonical inclusions and\n\twith the bottom right $P(w_0)\u27e8-4\u27e9$ is in homological degree $0$.\n\tWe claim that \n\t\\[\n\t\t\\Hom_{D^\\mathrm{b}(\\mathcal{O}_{0,\\mathfrak{sl}_3})}(P(w), Q)\n\t\t\\begin{cases}\n\t\t\t= 0 & \\text{if $w\\in\\{e, s\\}$,}\\\\\n\t\t\t\\neq 0 & \\text{if $w\\in\\{t, st, ts, w_0\\}$.}\n\t\t\\end{cases}\n\t\\]\n\tConsider the composition series of the projective modules \n\tinvolved in \\cref{eq:P(s):asphericality}.\n\tThe $P(w_0)\u27e8-4\u27e9$ in degree $0$ has composition series\n\t\\begin{equation}\n\t\t\\label{eq:Ps-dual:resolution:composition-factors-of-Pw0}\n\t\tP(w_0)\u27e8-4\u27e9 = \n\t\t\\begin{psmallmatrix}\n\t\t\t& L(w_0)\\\\\n\t\t\t& L(st)\\quad L(ts)\\\\\n\t\t\t \\textcolor{gray}{L(s)} & \\textcolor{gray}{L(w_0)}\\quad L(w_0) & L(t)\\\\\n\t\t\t \\textcolor{gray}{L(st)}\\quad  \\textcolor{gray}{L(ts)} &  \\textcolor{gray}{L(e)} & \\textcolor{gray}{L(st)}\\quad \\textcolor{gray}{L(ts)}\\\\\n\t\t\t \\textcolor{gray}{L(w_0)} &  \\textcolor{gray}{L(s)}\\quad  \\textcolor{gray}{L(t)} & \\textcolor{gray}{L(w_0)}\\\\\n\t\t\t&  \\textcolor{gray}{L(st)}\\quad  \\textcolor{gray}{L(ts)}\\\\\n\t\t\t&  \\textcolor{gray}{L(w_0)}\n\t\t\\end{psmallmatrix}\u27e8-4\u27e9;\n\t\\end{equation}\n\tthe gray factors are a composition series of the image of\n\t\\[(P(s)\u27e8-2\u27e9\\oplus P(w_0)\u27e8-2\u27e9 \\to P(w_0)\u27e8-4\u27e9,\\]\n\twhich is the last non-trivial map of the total complex \\cref{eq:P(s):asphericality}.\n\tEvery map $P(e)\u27e8-\u27e9 \\to P(w_0)\u27e8-4\u27e9$ and $P(s)\u27e8-\u27e9 \\to P(w_0)\u27e8-4\u27e9$ factors through $P(s)\u27e8-2\u27e9\\oplus P(w_0)\u27e8-2\u27e9$\n\tand hence is null-homotopic.\n\tThis shows that $\\Hom_{D^\\mathrm{b}(\\mathcal{O}(\\mathfrak{sl}_3))}(P(w), Q) = 0$ if $w\\in\\{e, s\\}$.\n\t\n\tFor $w \\in \\{t, st, ts, w_0\\}$ on the other hand,\n\tthe black composition  factors $L(w)$ in \\cref{eq:Ps-dual:resolution:composition-factors-of-Pw0}  \n\tgenerate images of non-zero morphisms $P(w)\u27e8-\u27e9 \\to P(w_0)\u27e8-2\u27e9$\n\tthat cannot be factored through $P(s)\u27e8-2\u27e9\\oplus P(w_0)\u27e8-2\u27e9$\n\tand thus are not null-homotopic.\n\tThese morphisms therefore represent non-trivial morphisms\n\t$P(w)\u27e8-\u27e9 \\to Q$ in $D^\\mathrm{b}(\\mathcal{O}_{0,\\mathfrak{sl}_3})$\n\tand thus show that $\\Hom_{D^\\mathrm{b}(\\mathcal{O}(\\mathfrak{sl}_3))}(P(w), Q) \\neq 0$ if $w \\in \\{t, st, ts, w_0\\}$.\n\t\n\tWe see that $\\Sph(P(s))$  of $D^\\mathrm{b}(\\mathcal{O}_{0,\\mathfrak{sl}_3})$\n\tcontains the triangulated subcategory generated by $P(s)$ and $P(e)$,\n\twhich proves the claim.\n\\end{proof}\n\n\\subsection{Maximal parabolic subalgebras}\nWe shall now address parabolic subalgebras $\\mathfrak{p}$ of $\\mathfrak{sl}_3$ (\\cref{sec:parabolic-subalgebras})\nas another remedy for the failure of the Calabi-Yau property\nof $P(s)$ in $\\mathcal{O}_{0,\\mathfrak{sl}_3}$.\nConsider the category $\\mathcal{O}^\\mathfrak{p}_0$\ncorresponding to the parabolic subgroup $W_{\\mathfrak{p}} = \u27e8t\u27e9 \\cong S_1 \\times S_2$ of $W = S_3 = \u27e8s,t\u27e9$.\nThe minimal-length representatives of cosets in $W\/W_\\mathfrak{p}$ are $W^\\mathfrak{p}=\\{e, s, st\\}$.\nThere is an equivalence $\\mathcal{O}^\\mathfrak{p}_0\\simeq\\Mod{A_\\mathfrak{p}}$\nfor the path algebra quotient\n\\begin{equation}\n\t\\label{eq:quiver-sl3}\n\tA_\\mathfrak{p}\u2254A_{\\mathfrak{sl}_3}\/(\\trivpath{t}, \\trivpath{ts}, \\trivpath{w_0}) = \\Complex\\bigl[e \\rightleftarrows s \\rightleftarrows st\\bigr]\\Bigm\/ \n\t\t\\Bigl(\n\t\t\\begin{scriptaligned}\n\t\t\te\\from s\\from e &= 0, & \te\\from s\\from st &= 0,\\\\[-0.5em]\n\t\t\tst\\from s\\from e &= 0, &\ts\\from e \\from s &= s\\from st\\from s\n\t\t\\end{scriptaligned}\n\t\\Bigr)\n\\end{equation}\nof the path algebra $A_{\\mathfrak{sl}_3}$.\nThe parabolic Verma modules and projectives have the following composition series,\nwhose factors now are simple $A_\\mathfrak{p}$-modules:\n\\begin{equation}\n\t\\label{eq:parabolic-sl3-composition-factors}\n\t\\begin{array}{ccc|cc@{{}={}}cc@{{}={}}c}\n\t\t\\toprule\n\t\tM^\ud835\udd2d(e) &\n\t\tM^\ud835\udd2d(s)  &\n\t\tM^\ud835\udd2d(st) &\n\t\tP^\ud835\udd2d(e) & \n\t\t\\multicolumn{2}{c}{P^\ud835\udd2d(s)} &\n\t\t\\multicolumn{2}{c}{P^\ud835\udd2d(st)}\\\\\n\t\t\\midrule\n\t\t\\begin{smallmatrix}\\vphantom{L(e)}\\\\L(e)\\\\L(s)\\end{smallmatrix} &\n\t\t\\begin{smallmatrix}\\vphantom{L(e)}\\\\L(s)\\\\L(st)\\end{smallmatrix} &\n\t\t\\begin{smallmatrix}\\vphantom{L(e)}\\\\\\vphantom{L(e)}\\\\L(st)\\end{smallmatrix} &\n\t\t\\begin{smallmatrix}\\vphantom{L(e)}\\\\M(e)\\end{smallmatrix} &\n\t\t\\begin{smallmatrix}M(s)\\\\M(e)\\end{smallmatrix} &\n\t\t \\begin{smallmatrix}L(s)\\\\L(st)\\quad L(e)\\\\L(s)\\end{smallmatrix} &\n\t\t\\begin{smallmatrix}M(st)\\\\M(s)\\end{smallmatrix} &\n\t\t \\begin{smallmatrix}L(st)\\\\L(s)\\\\L(st)\\end{smallmatrix}\\\\\n\t\t\\bottomrule\n\t\\end{array}\n\\end{equation}\nAccording to \\cref{rmk:parabolic-translation-and-shuffling},\nthe defining short exact sequences \\cref{eqn:translation-defining-ses} of $\\Theta_s$ and $\\LSh{s}$\nrestrict to sequences \\cref{eq:ses-parabolic-wall-crossing} in $\\mathcal{O}_0^\\mathfrak{p}$\nso the indecomposable projective modules\nhave the following images under $\\Theta_s$ and $\\LSh{s}$:\n\\begin{equation}\n\t\\label{eq:parabolc-sl3:images-under-translation-and-shuffling}\n\t\\begin{array}{c|cr@{}>{{}}lcr@{}>{{}}l}\n\t\t\\toprule\n\t\tM                 & \\Theta_s M                        &              \\multicolumn{2}{c}{\\LSh{s} M}               & \\Theta_t M             &                                          \\multicolumn{2}{c}{\\LSh{t} M}                                           \\\\ \\midrule\n\t\tP^\\mathfrak{p}(e)  & P^\\mathfrak{p}(s)                  & \\{P^\\mathfrak{p}(e)  & \\to \\degZero{P^\\mathfrak{p}(s)}\\}   & 0                      & \\mathmakebox[\\widthof{$P^\\mathfrak{p}(e)$}][c]{P^\\mathfrak{p}(e)\\hShift{-1}} &                                     \\\\\n\t\tP^\\mathfrak{p}(s)  & P^\\mathfrak{p}(s)\u2295P^\\mathfrak{p}(s) &                     & \\phantom{{}\\to{}} P^\\mathfrak{p}(s) & P^\\mathfrak{p}(st)      & \\{P^\\mathfrak{p}(s)                                                         & \\to \\degZero{P^\\mathfrak{p}(st)}\\}   \\\\\n\t\tP^\\mathfrak{p}(st) & P^\\mathfrak{p}(s)                  & \\{P^\\mathfrak{p}(st) & \\to \\degZero{P^\\mathfrak{p}(s)}\\}   & P^\\mathfrak{p}(st)^{\u22952} &                                                                            & \\phantom{{}\\to{}} P^\\mathfrak{p}(st) \\\\ \\bottomrule\n\t\\end{array}\n\\end{equation}\n\n\\begin{remark}\n\t\\label{rmk:inclusion-of-parabolic-subalgebradoesnt-preserve-projectives}\n\tA module $M$ is $\\Sh{w}$-acyclic if and only if $\\LSh{w}$\n\tis quasi-isomorphic to a complex concentrated in degree zero.\n\tThe results form \\cref{eq:parabolc-sl3:images-under-translation-and-shuffling}\n\ttherefore are examples for \\cref{caveat:parabolic-projectives-not-projective}:\n\tthe objects $P^\\mathfrak{p}(-)$, \n\talbeit projective in the category $\\mathcal{O}_0^\\mathfrak{p}$,\n\tare not $\\Sh{s}$-acyclic and hence in particular not projective in $\\mathcal{O}_0$.\n\\end{remark}\n\\begin{lemma}\n\tThe set $\\{P^\\mathfrak{p}(s), P^\\mathfrak{p}(st)\\}$\n\tis an $\\mathrm{A}_2$-collection of $0$-spherical objects in $D^\\mathrm{b}(\\mathcal{O}^\\mathfrak{p}_0)$.\n\\end{lemma}\n\\begin{proof}\n\tFrom the composition series in \\cref{eq:parabolic-sl3-composition-factors},\n\twe see that $P^\\mathfrak{p}(s)$ and $P^\\mathfrak{p}(st)$ have endomorphism algebras\n\tisomorphic to $\\Complex[x]\/(x^2)$,\n\twith the non-trivial endomorphism\n\t$x\\colon P^\\mathfrak{p}(w) \\onto L(w) \\into P^\\mathfrak{p}(w)$ for $w \\in \\{s, st\\}$.\n\tAll Hom-spaces in the following are one-dimensional,\n\tand we see that for the indecomposable projectives, the composition pairings\n\t\\begin{align}\n\t\t\\label{eq:composition-pairing:SL3:Ps-Pe}\n\t\t\\Hom_\\mathcal O(P^\ud835\udd2d(e), P^\ud835\udd2d(s)) \\otimes \\Hom_\\mathcal O(P^\ud835\udd2d(s), P^\\mathfrak{p}(e)) \n\t\t&\\longto \\End(P^\\mathfrak{p}(s))\/\\langle\\id\\rangle\\\\* \\notag\n\t\t\\Biggl(\n\t\t\t\\begin{tikzpicture}[x=.4cm,y=.4cm, baseline=(Fb.base),every node\/.append style={font=\\scriptsize}]\n\t\t\t\t%\n\t\t\t\t\\node (Fa) at (0,1) {$L(s)$};\n\t\t\t\t\\node (Fb) at (1,0) {$L(e)$};\n\t\t\t\t\\draw[brace] ($(Fa)+(0.6,.7)$) to node[brace tip](Pa){} ($(Fb)+(1.1,0.1)$);\n\t\t\t\t\\node (Fc) at (-1,0) {$L(st)$};\n\t\t\t\t\\node (Fd) at (0,-1) {$L(s)$};\n\t\t\t\t\\node[font=\\normalsize] at (0,-2.5) {$P^\\mathfrak{p}(s)$};\n\t\t\t\t\\draw[brace'] (Fc.west |- Fd.south) to (Fd.south -| Fb.east);\n\t\t\t\t%\n\t\t\t\t\\node[font=\\normalsize] at (6,-2.5) {$P^\\mathfrak{p}(e)$};\n\t\t\t\t\\node (Ta) at (6,0.5) {$L(e)$};\n\t\t\t\t\\node (Tb) at (6,-0.5) {$L(s)$};\n\t\t\t\t\\draw[brace] ($(Tb)-(1,.3)$) to node[brace tip](Pb){} ($(Ta)-(1,-.3)$);\n\t\t\t\t\\draw[brace'] (Ta.west |- Fd.south) to (Fd.south -| Ta.east);\n\t\t\t\t%\n\t\t\t\t\\node[font=\\normalsize] at (12,-2.5) {$P^\\mathfrak{p}(s)$};\n\t\t\t\t\\node at (12,1) {$L(s)$};\n\t\t\t\t\\node (Sb) at (11,0) {$L(e)$};\n\t\t\t\t\\node (Sc) at (13,0) {$L(st)$};\n\t\t\t\t\\node (T') at (12,-1) {$L(s)$};\n\t\t\t\t\\draw[->, limit bb] (Pa.center) to[out=25,in=180] (Pb.center);\n\t\t\t\t\\draw[->, limit bb] (Ta) to[out=0,in=180] (T');\n\t\t\t\t\\draw[brace'] (Sb.west |- Fd.south) to (Fd.south -| Sc.east);\n\t\t\t\\end{tikzpicture}\n\t\t\\Biggr)&\\longmapsto \n\t\tx_{P^\\mathfrak{p}(s)},\\\\[3pt]\n\t\t\\label{eq:composition-pairing:SL3:Ps-Pst}\n\t\t\\Hom(P^\ud835\udd2d(st), P^\ud835\udd2d(s)) \\otimes \\Hom(P^\ud835\udd2d(s), P^\\mathfrak{p}(st))\n\t\t&\\longto \\End(P^\\mathfrak{p}(s))\/\\langle\\id\\rangle \\\\* \\notag\n\t\t\\Biggl(\n\t\t\t\\begin{tikzpicture}[x=.4cm,y=.4cm, baseline=(Fb.base),every node\/.append style={font=\\scriptsize}]\n\t\t\t\t\\node[font=\\normalsize] at (0,-2.5) {$P^\\mathfrak{p}(s)$};\n\t\t\t\t\\node (Fa) at (0,1) {$L(s)$};\n\t\t\t\t\\node at (-1,0) {$L(e)$};\n\t\t\t\t\\node (Fb) at (1,0) {$L(st)$};\n\t\t\t\t\\node at (0,-1) {$L(s)$};\n\t\t\t\t\\draw[brace] ($(Fa)+(0.6,.7)$) to node[brace tip](Pa){} ($(Fb)+(1.1,.1)$);\n\t\t\t\t\\draw[brace'] (Fc.west |- Fd.south) to (Fd.south -| Fb.east);\n\t\t\t\t%\n\t\t\t\t\\node[font=\\normalsize] at (6,-2.5) {$P^\\mathfrak{p}(st)$};\n\t\t\t\t\\node (Ta) at (6,1) {$L(st)$};\n\t\t\t\t\\node (Tb) at (6,0) {$L(s)$};\n\t\t\t\t\\node (Tc) at (6,-1) {$L(st)$};\n\t\t\t\t\\draw[brace] ($(Tc)-(1,.3)$) to node[brace tip](Pb){} ($(Tb)-(1,-.3)$);\n\t\t\t\t\\draw[brace] ($(Ta)+(1,.3)$) to node[brace tip](Pc){} ($(Tb)+(1,-.3)$);\n\t\t\t\t\\draw[brace'] (Ta.west |- Fd.south) to (Fd.south -| Ta.east);\n\t\t\t\t%\n\t\t\t\t\\node[font=\\normalsize] at (12,-2.5) {$P^\\mathfrak{p}(s)$};\n\t\t\t\t\\node (Sb) at (12,1) {$L(s)$};\n\t\t\t\t\\node (Za) at (11,0) {$L(st)$};\n\t\t\t\t\\node (Sc) at (13,0) {$L(e)$};\n\t\t\t\t\\node (Zb) at (12,-1) {$L(s)$};\n\t\t\t\t\\draw[brace] ($(Zb)-(0.8,.2)$) to node[brace tip](Pd){} ($(Za)-(1.0,0)$);\n\t\t\t\t\\draw[->,limit bb] (Pa.center) to[out=45,in=180] (Pb.center);\n\t\t\t\t\\draw[->,limit bb] (Pc.center) to[out=0,in=235] (Pd.center);\n\t\t\t\t\\draw[brace'] (Sb.west |- Fd.south) to (Fd.south -| Sc.east);\n\t\t\t\\end{tikzpicture}\n\t\t\\Biggr)\n\t\t&\\longmapsto x_{P^\\mathfrak{p}(s)},\\\\[3pt]\n\t\t%\n\t\t\\label{eq:composition-pairing:SL3:Pst-Pe}\n\t\t\\Hom_\\mathcal O(P^\ud835\udd2d(s), P^\ud835\udd2d(st)) \\otimes \\Hom_\\mathcal O(P^\ud835\udd2d(s), P^\\mathfrak{p}(st))\n\t\t&\\longto \\End(P^\\mathfrak{p}(ps))\/\\langle\\id\\rangle\\\\* \\notag\n\t\t\\Biggl(\n\t\t\t\\begin{tikzpicture}[x=.4cm,y=.4cm, baseline=(Fb.base),every node\/.append style={font=\\scriptsize, inner sep=2pt}]\n\t\t\t\t\\node[font=\\normalsize] at (0,-2.5) {$P^\\mathfrak{p}(st)$};\n\t\t\t\t\\node (Fa) at (0,1) {$L(st)$};\n\t\t\t\t\\node (Fb) at (0,0) {$L(s)$};\n\t\t\t\t\\node[braced box={Pa}, fit=(Fa) (Fb)]{};\n\t\t\t\t\\node at (0,-1) {$L(st)$};\n\t\t\t\t\\draw[brace'] (Fb.west |- Fd.south) to (Fd.south -| Fb.east);\n\t\t\t\t%\n\t\t\t\t\\node[font=\\normalsize] at (6,-2.5) {$P^\\mathfrak{p}(s)$};\n\t\t\t\t\\node (Ma) at (6,1) {$L(s)$};\n\t\t\t\t\\node (Mb) at (5,0) {$L(st)$};\n\t\t\t\t\\node (Md) at (7,0) {$L(e)$};\n\t\t\t\t\\node (Mc) at (6,-1) {$L(s)$};\n\t\t\t\t\\draw[brace'] (Mb.-170) to node[brace tip'](Pb){} (Mc.south west);\n\t\t\t\t\\draw[brace]  (Mb.170)  to node[brace tip] (Ta){} (Ma.north west);\n\t\t\t\t\\draw[brace'] (Mb.west |- Fd.south) to (Fd.south -| Md.east);\n\t\t\t\t%\n\t\t\t\t\\node[font=\\normalsize] at (12,-2.5) {$P^\\mathfrak{p}(st)$};\n\t\t\t\t\\node (Sb) at (12,1) {$L(st)$};\n\t\t\t\t\\node (Sc) at (12,0) {$L(s)$};\n\t\t\t\t\\node (Tb) at (12,-1) {$L(st)$};\n\t\t\t\t\\draw[->, limit bb] (Pa.center) to[out=0,in=-135] (Pb.center);\n\t\t\t\t\\draw[-, limit bb] (Ta) to[out=135, out looseness=1.5, in=135] (8,1.5);\n\t\t\t\t\\draw[->, limit bb] (8,1.5) to[out=-45, in=180] (Tb);\n\t\t\t\t\\draw[brace'] (Sb.west |- Fd.south) to (Fd.south -| Sc.east);\n\t\t\t\\end{tikzpicture}\n\t\t\\Biggr)\n\t\t&\\longmapsto x_{P^\\mathfrak{p}(st)}\n\t\\end{align}\n\tare non-degenerate and that\n\t$\\Hom_{\\mathcal{O}}(P^{\\mathfrak{p}}(e), P^{\\mathfrak{p}}(e)(s)) = 0 = \\Hom_{\\mathcal{O}}(P^{\\mathfrak{p}}(st), P^{\\mathfrak{p}}(e)(e))$.\n\tHence, $P^\\mathfrak{p}(s)$ and $P^\\mathfrak{p}(st)$ are $0$-spherical objects\n\tin $D^\\mathrm{b}(\\mathcal{O}_0^\\mathfrak{p})$.\n\tIn particular,\n\t$\\dim \\Hom(P^\\mathfrak{p}(s), P^\\mathfrak{p}(st)) = 1 = \\Hom(P^\\mathfrak{p}(st), P^\\mathfrak{p}(s))$\n\tas required;\n\tso the set $\\{P^\\mathfrak{p}(s), P^\\mathfrak{p}(st)\\}$ is an $\\mathrm A_2$-configuration.\n\\end{proof}\n\nWith the dimensions of the hom-spaces in\n\\crefrange{eq:composition-pairing:SL3:Ps-Pe}{eq:composition-pairing:SL3:Ps-Pst},\nit follows immediately from \\cref{def:twist-cotwist}\nthat the indecomposable projective modules under $T'_{P(s)}$ and $T'_{P(st)}$ are\n\\begin{equation}\n\t\\label{eq:parabolc-sl3:images-under-twist}\n\t\\begin{array}[t]{r @{} >{{}}c<{{}} @{} r @{} >{{}}c<{{}} @{} l}\n\t\tT'_{P^\\mathfrak{p}(s)}\\colon \n\t\tP^\\mathfrak{p}(e) & \\mapsto & \\{\\degZero{P^\\mathfrak{p}(e)}  & \\to & P^\\mathfrak{p}(s)\\}         \\\\\n\t\tP^\\mathfrak{p}(s)                                & \\mapsto &                               &     & P^\\mathfrak{p}(s)\\hShift{1} \\\\\n\t\tP^\\mathfrak{p}(st)                               & \\mapsto & \\{\\degZero{P^\\mathfrak{p}(st)} & \\to & P^\\mathfrak{p}(s)\\}\n\t\\end{array}\n\t\\qquad\n\t\\begin{array}[t]{r @{} >{{}}c<{{}} @{} r @{} >{{}}c<{{}} @{} l}\n\t\tT'_{P^\\mathfrak{p}(st)}\\colon \n\t\tP^\\mathfrak{p}(e) & \\mapsto & P^\\mathfrak{p}(e)             &     &  \\\\\n\t\tP^\\mathfrak{p}(s)                                 & \\mapsto & \\{\\degZero{P^\\mathfrak{p}(s)} & \\to & P^\\mathfrak{p}(st)\\}         \\\\\n\t\tP^\\mathfrak{p}(st)                                & \\mapsto &                              &     & P^\\mathfrak{p}(st)\\hShift{1}.\n\t\\end{array}\n\t\\hspace*{-2em}\n\\end{equation}\n\n\\begin{proposition}\n\t\\label{thm:parabolic-sl3:main-statement}\n\tFor $\\mathfrak{p}$ as above,\n\tthere are natural isomorphisms\n\t$T'_{P^\\mathfrak{p}(s)}  \\cong \\LSh{s}\\hShift{1}$ and\n\t$T'_{P^\\mathfrak{p}(st)} \\cong \\LSh{t}\\hShift{1}$ \n\tof autoequivalences of $\\mathcal{O}^\\mathfrak{p}_0$.\n\\end{proposition}\n\\begin{proof}\n\tWe see from \\cref{eq:parabolc-sl3:images-under-translation-and-shuffling,eq:parabolc-sl3:images-under-twist}\n\tthat $T'_{P^\\mathfrak{p}(s)}  P^\\mathfrak{p}(w) \\simeq \\LSh{s} P^\\mathfrak{p}(w)\\hShift{1}$\n\tand  $T'_{P^\\mathfrak{p}(st)} P^\\mathfrak{p}(w) \\simeq \\LSh{t} P^\\mathfrak{p}(w)\\hShift{1}$\n\tfor all $w \\in W^\\mathfrak{p}$.\n\tIt remains to show that the shuffling and spherical twist functors\n\talso map elements of $\\Hom_{\\mathcal{O}^\\mathfrak{p}}(P^\\mathfrak{p}(v), P^\\mathfrak{p}(w))$\n\t(for $v, w \\in W^\\mathfrak{p}$)\n\tto isomorphic maps.\n\tThis will be carried out analogously to \\cref{thm:shuffling-is-twisting:sl2}.\n\t\n\tThe proof for $\\Theta_t \\cong T'_{P^\\mathfrak{p}(st)}$ is done, mutatis mutandis, the same way as for $\\Theta_s \\cong T'_{P^\\mathfrak{p}(s)}$;\n\twe thus only show the latter.\n\tTo that end, we show that\n\tthe $A_\ud835\udd2d$-$A_\ud835\udd2d$-bimodules $\\FBim{\\Xi'_{P^\\mathfrak{p}(s)}}$ and $M_{\\Theta_s}$,\n\tfor which the functors $-\\otimes \\FBim{\\Xi'_{P^\\mathfrak{p}(s)}}$ and $-\\otimes M_{\\Theta_s}$\n\trespectively correspond to $\\Xi'_{P^\\mathfrak{p}(s)}$ and $\\Theta_s$\n\tunder $\\mathcal{O}_0^\\mathfrak{p} \\simeq \\Mod{}[A_\\mathfrak{p}]$,\n\tare isomorphic.\n\t\n\tSince as an $A_\\mathfrak{p}$-module, $P^\\mathfrak{p}(s) = \\trivpath{s} A_\\mathfrak{p}$\n\thas an explicit finite basis by paths of the quiver \\cref{eq:quiver-sl3} ending in the vertex $s$,\n\tthe module $\\FBim{\\Xi'_{P^\\mathfrak{p}(s)}} = P^\\mathfrak{p}(s)^* \u2297_\\Complex P^\\mathfrak{p}(s)$%\n\t---the star stands for vector space dual---%\n\thas a $\\Complex$-basis given by the sixteen pairwise tensor products in the schematic\n\t\\begin{equation}\n\t\t\\label{eq:sl3-p:M-Xi}\n\t\t\\begin{tikzpicture}[mth, ampersand replacement=\\&]\n\t\t\t\\matrix (A) [\n\t\t\t\tmatrix of math nodes, \n\t\t\t\trow sep=0mm, \n\t\t\t\tcolumn sep=2mm, \n\t\t\t\ttext height=2ex, \n\t\t\t\ttext depth=.5ex,\n\t\t\t\tcolumn 1\/.style={text width=width(\"$(s\u2190e\u2190s)^*.$\"), align=right},\n\t\t\t\tcolumn 3\/.style={text width=width(\"$(s\u2190e\u2190s)^*.$\")-1em, align=left},\n\t\t\t\tnodes={font=\\normalsize}\n\t\t\t] {\n\t\t\t\t(s\u2190e\u2190s)^*\t\\& \\phantom{\\otimes_\\Complex} \\& e    \t\\\\\n\t\t\t\t(s\u2190st)^*\t\\& \\& s\u2190e    \t\\\\\n\t\t\t\t(s\u2190e)^*\t\t\\& \\& s\u2190st \t\\\\\n\t\t\t\te^*\t\t\\& \\& s\u2190e\u2190s\t\\\\\n\t\t\t};\n\t\t\t\\node[braced box={T}, fit=(A-1-1) (A-4-1)] {} ;\n\t\t\t\\node[braced box'={T'},fit=(A-1-3) (A-4-3)] {} ;\n\t\t\t\\path[draw=none] (T) to node[anchor=center]{$\\displaystyle \u2297_\\Complex$} (T');\n\t\t\t%\n\t\t\t\\draw\n\t\t\t\t(A-1-1.west)\t\t\t\t\tedge[|->, out=195, in=165] node[swap] {$(e\u2190s)_*$} \t(A-2-1.west)\n\t\t\t\t(A-2-1.base west)\t\t \t\tedge[|->, out=195, in=165] node[swap] {$(e\u2190s)_*$} \t(A-4-1.west)\n\t\t\t\t([xshift=-5em]A-1-1.west) \t\tedge[|->, out=195, in=165] node[swap] {$(st\u2190s)_*$} \t([xshift=-5em]A-3-1.west)\n\t\t\t\t([xshift=-5em]A-3-1.base west) \tedge[|->, out=195, in=165] node[swap] {$(s\u2190st)_*$} \t([xshift=-5em]A-4-1.west)\n\t\t\t\t%\n\t\t\t\t(A-1-3.east)\t\t\t\t\tedge[|->, out=-15, in=15] node {$\u2218(s\u2190e)$} \t\t\t(A-2-3.east)\n\t\t\t\t(A-2-3.base east)\t\t\t\tedge[|->, out=-15, in=15] node {$\u2218(e\u2190s)$} \t\t\t(A-4-3.east)\n\t\t\t\t([xshift=4.5em]A-1-3.east)\t\tedge[|->, out=-15, in=15] node {$\u2218(s\u2190st)$} \t\t\t([xshift=4.5em]A-3-3.east)\n\t\t\t\t([xshift=4.5em]A-3-3.base east)\tedge[|->, out=-15, in=15] node {$\u2218(st\u2190s).$} \t\t([xshift=4.5em]A-4-3.east);\n\t\t\\end{tikzpicture}\n\t\\end{equation}\n\tThe right $A_\ud835\udd2d$-action is given by precomposition of paths;\n\tunder the left $A_\ud835\udd2d$-action, a\n\t$\\gamma \\in A_\ud835\udd2d$ acts on a $\\lambda \\in P^\\mathfrak{p}(s)^*$\n\tby $\\gamma_*(\\lambda) \\colon v \\mapsto \\lambda(v \\circ \\gamma)$.\n\tThe $A_\ud835\udd2d$-$A_\ud835\udd2d$-bimodule action on $\\FBim{\\Xi'_{P^\\mathfrak{p}(s)}}$ hence is\n\tsuch that the generating paths of $A_\ud835\udd2d$ act from the left and right as indicated.\n\t\n\tTo construct the module $\\FBim{\\Theta_s}$,\n\twe consider the endomorphism algebra $\\End_{\\mathcal{O}}(P^\\mathfrak{p})$ \n\tof $P^\\mathfrak{p} = P^\\mathfrak{p}(e) \\oplus P^\\mathfrak{p}(s) \\oplus P^\\mathfrak{p}(st)$,\n\twhich is generated by the elements\n\t\\begin{equation}\n\t\t\\label{eq:End-Ps-sl3:table}\n\t\t\\Mtrx{1\\\\&0\\\\&&0},\n\t\t\\Mtrx{0\\\\ &1\\\\&&0},\n\t\t\\Mtrx{0\\\\&0\\\\&&1},\n\t\t\\Mtrx{0 \\\\ (s \\from e) & 0 \\\\ && 0},\n\t\t\\Mtrx{0 \\\\ & 0 & (s \\from st) \\\\&& 0},\n\t\t\\Mtrx{0 & (e \\from s) \\\\ & 0 \\\\ && 0},\n\t\t\\Mtrx{0 \\\\ & 0 \\\\ & (st \\from s) & 0}.\n\t\\end{equation}\n\tSince according to \\cref{eq:parabolc-sl3:images-under-translation-and-shuffling}\n\twe have $\\Theta_s P^\\mathfrak{p} = P^\\mathfrak{p}(s)^{\\oplus 4}$,\n\twe endow these $P^\\mathfrak{p}(s)$'s with indices to make them distinguishable,\n\tidentifying $P^\\mathfrak{p}(s)_1$ with $\\Theta_s P^\\mathfrak{p}(e)$,\n\t$P^\\mathfrak{p}(s)_2 \\oplus P^\\mathfrak{p}(s)_3$ with $\\Theta_s P^\\mathfrak{p}(s)$\n\tand $P^\\mathfrak{p}(s)_4$ with $\\Theta_s P^\\mathfrak{p}(st)$.\n\tA diagram chase of morphisms through the relevant naturality diagrams\n\tshows that $\\Theta_s$ respectively maps \n\tthe above generators of $\\End_{\\mathcal{O}}(P^\\mathfrak{p})$ from \\cref{eq:End-Ps-sl3:table}\n\tto the endomorphisms\n\t\\begin{equation}\n\t\t\\label{eq:Theta-s-action-on-End-Ps-sl3:table}\n\t\t\\Mtrx{1\\\\&0\\\\&&0\\\\&&&0}, \n\t\t\\Mtrx{0\\\\&1\\\\&&1\\\\&&&0}, \n\t\t\\Mtrx{0\\\\&0\\\\&&0\\\\&&&1} , \n\t\t\\Mtrx{0&& 1\\\\&0\\\\&&0\\\\&&&0}, \n\t\t\\Mtrx{0&&\\\\&0\\\\&&0\\\\&&1&0}, \n\t\t\\Mtrx{0\\\\1&0\\\\&&0\\\\&&&0}, \n\t\t\\Mtrx{0\\\\&0&&1\\\\&&0\\\\&&&0}.\n\t\\end{equation}\n\tof $\\Theta_s P^\\mathfrak{p} = P^\\mathfrak{p}(s)^{\\oplus 4}$.\n\tSomewhat more suggestively, \n\twe may write these morphisms $f$ as arrows connecting the two $P^\\mathfrak{p}(-)$'s\n\ton which they have non-zero kernel or cokernel.\n\t$\\Theta_s$ then maps the last four of the generators $f$ from the table,\n\twhich we depict by the following morphisms on the left,\n\tto the respective solid or dashed morphism on the right:\n\t\\settowidth{\\algnRef}{${}\u2295{}$}\n\t\\begin{equation}\n\t\t\\label{eq:Theta-s-action-on-End-Ps-sl3}\n\t\t\\begin{tikzcd}[column sep={\\the\\algnRef}, nodes={inner xsep=0pt}, row sep=small, ampersand replacement=\\&]\n\t\t\tP^\\mathfrak{p} = P(e)\\rar[phantom, \"\u2295\"] \n\t\t\t\\& P(s)\\ar[dl]\\ar[dr, dashed] \\rar[phantom, \"\u2295\"]\n\t\t\t\\& P(st) \\ar[d, phantom, \"\" name=C]\n\t\t\t\\&[4em] P(s)_1 \\rar[phantom, \"\u2295\"]  \\ar[d, phantom, \"\" name=C']\n\t\t\t\\& P(s)_2 \\rar[phantom, \"\u2295\"] \n\t\t\t\\& P(s)_3 \\rar[phantom, \"\u2295\"]\\ar[dll, \"\\id\"' very near end]\\ar[dr, dashed, \"\\id\" very near end] \n\t\t\t\\& P(s)_4 =\\Theta_s P^\\mathfrak{p}\n\t\t\t\\\\\n\t\t\t\\phantom{P^\\mathfrak{p} = {}} P(e)\\rar[phantom, \"\u2295\"] \\ar[dr] \n\t\t\t\\& P(s) \\rar[phantom, \"\u2295\"]\n\t\t\t\\& P(st) \\ar[dl, dashed] \\ar[d, phantom, \"\" name=D]\n\t\t\t\\& P(s)_1 \\rar[phantom, \"\u2295\"] \\ar[dr, \"\\id\"' very near start] \\ar[d, phantom, \"\" name=D']\n\t\t\t\\& P(s)_2 \\rar[phantom, \"\u2295\"] \n\t\t\t\\& P(s)_3 \\rar[phantom, \"\u2295\"]\n\t\t\t\\& P(s)_4 \\ar[dll, \"\\id\" very near start, dashed] \\phantom{{}=\\Theta_s P^\\mathfrak{p}}\n\t\t\t\\\\\n\t\t\t\\phantom{P^\\mathfrak{p} = {}} P(e)\\rar[phantom, \"\u2295\"] \n\t\t\t\\& P(s) \\rar[phantom, \"\u2295\"]\n\t\t\t\\& P(st)\n\t\t\t\\& P(s)_1 \\rar[phantom, \"\u2295\"] \n\t\t\t\\& P(s)_2 \\rar[phantom, \"\u2295\"] \n\t\t\t\\& P(s)_3 \\rar[phantom, \"\u2295\"]\n\t\t\t\\& P(s)_4 \\mathrlap{.}\\phantom{{}=\\Theta_s P^\\mathfrak{p}}  \\ar[phantom, from=C, to=C', \"\\xmapsto{\\Theta_s}\"]\\ar[phantom, from=D, to=D', \"\\xmapsto{\\Theta_s}\"]\n\t\t\\end{tikzcd}\n\t\\end{equation}\n\tA vector space basis of $M_{\\Theta_s} = \\Hom(P^\\mathfrak{p}, \\Theta_s P^\\mathfrak{p})$\n\tis given by morphisms\n\tsending $P^\\mathfrak{p}$ to one of its summands $P^\\mathfrak{p}(w)$ (where $w \\in \\{e, s, st\\}$)\n\tand further to a summand $P^\\mathfrak{p}(s)_i$ of $\\Theta_s P^\\mathfrak{p}$.\n\tIn other words, this basis consists of the 16 possible ways\n\tto map a summand $P^\\mathfrak{p}(w)$ of $P^\\mathfrak{p}$ on the right\n\tto the $P^\\mathfrak{p}(s)$ in the middle \n\tand embed this into $\\Theta_s P^\\mathfrak{p}$\n\tas one of the summands on the left:\n\t\\begin{equation}\n\t\t\\label{eq:sl3-p:M-Theta}\n\t\t\\begin{tikzpicture}[mth, ampersand replacement=\\&]\n\t\t\t\\matrix (A) [\n\t\t\tmatrix of math nodes, \n\t\t\trow sep={3.5ex,between origins},\n\t\t\tcolumn sep=10mm,\n\t\t\tcolumn 1\/.style={anchor=base east, text width=width(\"$P^\\mathfrak{p}(s)_4$\")},\n\t\t\tcolumn 3\/.style={anchor=base west, text width=width(\"$P^\\mathfrak{p}(st)$\")}\n\t\t\t] {\n\t\t\t\tP^\ud835\udd2d(s)_3 \\&  \\& P^\ud835\udd2d(s)\\\\\n\t\t\t\tP^\ud835\udd2d(s)_1 \\&  \\& P^\ud835\udd2d(e)\\\\[-1.75ex]\n\t\t\t\t\\& P^\ud835\udd2d(s) \\\\[-1.75ex]\n\t\t\t\tP^\ud835\udd2d(s)_4 \\&  \\& P^\ud835\udd2d(st)\\\\\n\t\t\t\tP^\ud835\udd2d(s)_2 \\&  \\& P^\ud835\udd2d(s)\\\\\n\t\t\t};\n\t\t\t%\n\t\t\t%\n\t\t\t%\n\t\t\t%\n\t\t\t%\n\t\t\t%\n\t\t\t\\draw[->] (A-3-2)\n\t\t\t\tedge[->, out=180, in=0] (A-1-1)\n\t\t\t\tedge[->, out=180, in=0] (A-2-1)\n\t\t\t\tedge[->, out=180, in=0] (A-4-1)\n\t\t\t\tedge[->, out=180, in=0] (A-5-1)\n\t\t\t\tedge[<-, in=180, out=0, \"$\\id$\" very near end] (A-1-3)\n\t\t\t\tedge[-, in=180, out=0] (A-2-3)\n\t\t\t\tedge[-, in=180, out=0] (A-4-3)\n\t\t\t\tedge[-, in=180, out=0, \"$x$\"' very near end] (A-5-3);\n\t\t\t\\draw[looseness=1]\n\t\t\t\t%\n\t\t\t\t(A-1-1.west)                     edge[|->, out=180+15, in=180-15] node[swap] {$\ud835\udee9_s(e\u2190s)\u2218$}             (A-2-1.west)\n\t\t\t\t(A-2-1.base west)                edge[|->, out=180+15, in=180-15] node[swap, pos=.4] {$\ud835\udee9_s(e\u2190s)\u2218$}     (A-5-1.west)\n\t\t\t\t([xshift=-5em]A-1-1.west)        edge[|->, out=180+15, in=180-15] node[swap] {$\ud835\udee9_s(st\u2190s)\u2218$}            ([xshift=-5em]A-4-1.west)\n\t\t\t\t([xshift=-5em]A-4-1.base west)   edge[|->, out=180+15, in=180-15] node[swap] {$\ud835\udee9_s(s\u2190st)\u2218$}            ([xshift=-5em]A-5-1.west)\n\t\t\t\t%\n\t\t\t\t(A-1-3.east)                     edge[|->, out=   -15, in=    15] node {$\u2218(s\u2190e)$}                      (A-2-3.east)\n\t\t\t\t(A-2-3.base east)                edge[|->, out=   -15, in=    15] node[pos=.4] {$\u2218(e\u2190s)$}              (A-5-3.east)\n\t\t\t\t([xshift=4.5em]A-1-3.east)       edge[|->, out=   -15, in=    15] node {$\u2218(s\u2190st)$}                     ([xshift=4.5em]A-4-3.east)\n\t\t\t\t([xshift=4.5em]A-4-3.base east)  edge[|->, out=   -15, in=    15] node {$\u2218(st\u2190s).$}                    ([xshift=4.5em]A-5-3.east);\n\t\t\\end{tikzpicture}\n\t\\end{equation}\n\tTo understand the $A^\\mathfrak{p}$-$A^\\mathfrak{p}$-bimodule action on $\\FBim{\\Theta_s}$,\n\trecall from \\cref{cor:morita-equivalence-and-induced-module}\n\tthat $\\phi, \\psi \\in A^{\\mathfrak{p}}$ act on $m\\in \\FBim{\\Theta_s}$\n\tby $\\phi \\mathbin{.} m \\mathbin{.} \\psi = \\Theta_s(\\phi) \\circ m \\circ \\psi$,\n\twith the images $\\Theta_s(\\phi)$ from\n\t\\cref{eq:Theta-s-action-on-End-Ps-sl3:table,eq:Theta-s-action-on-End-Ps-sl3}.\n\tOn the vector space basis of $\\FBim{\\Theta_s}$ from \\cref{eq:sl3-p:M-Theta},\n\tthe generating paths of the algebra $A^\\mathfrak{p}$ therefore\n\tact from the left and right as indicated.\n\t\n\tComparing \\cref{eq:sl3-p:M-Xi} and \\cref{eq:sl3-p:M-Theta}\n\tshows that the obvious isomorphism $M_{\\Xi'_{P^\\mathfrak{p}(s)}} \\cong M_{\\Theta_s}$\n\tof vector spaces is an isomorphism of $A^\\mathfrak{p}$-$A^\\mathfrak{p}$-bimodules.\n\tIt follows that $T'_{P^\\mathfrak{p}(s)} \\simeq \\{\\id \\Rightarrow -\\otimes M_{\\Xi'_{P^\\mathfrak{p}(s)}}\\}$\n\tand $\\Sh{s} = \\{\\id \\Rightarrow -\\} \\cong M_{\\Theta_s}$ are naturally isomorphic functors.\n\\end{proof}\n\n\\subsection*{Parabolic subalgebras of $\\mathfrak{sl}_n$}\nWe now transfer results for $\\mathcal{O}^\\mathfrak{p}_0$ from $\\mathfrak{sl}_3$ to $\\mathfrak{sl}_n$.\nConsider the parabolic subalgebra $\\mathfrak{p}$ of $\\mathfrak{sl}_n$\ncorresponding to the subgroup $W_\\mathfrak{p}=\u27e8s_2, \\dotsc, s_{n-1}\u27e9 = S_{1}\\times S_{n-1}$\nof $S_n = \u27e8s_1,\\dotsc,s_{n-1}\u27e9$.\nWe let $\\sigma_0 \u2254 e$ and $\\sigma_i \u2254 s_1\\dotsm s_i$ for $i \\geq 1$;\nthe minimal length coset representatives then are $W^\\mathfrak{p}=\\{\\sigma_0,\\dotsc,\\sigma_{n-1}\\}$. \n\n\\begin{lemma}\n\t\\label{lem:parabolic-sln:path-algebra}\n\tThe category $\\mathcal{O}^\\mathfrak{p}_0(\\mathfrak{sl}_n)$ is equivalent to $\\Mod{}[A^\\mathfrak{p}(\\mathfrak{sl}_n)]$ for\n\t\\begin{equation*}\n\t\t\\label{eq:parabolic-sln:path-algebra}\n\t\tA^\\mathfrak{p}=\\Complex\\bigl[e\\rightleftarrows\\sigma_1\\rightleftarrows\\dotsb\\rightleftarrows\\sigma_{n-1}\\bigr]\n\t\t\\Bigm\/\n\t\t\\Bigl(\n\t\t\\setlength\\jot{-.3ex}\n\t\t\\begin{scriptaligned}\n\t\t\te\\from\\sigma_1\\from e &= 0,\\\\\n\t\t\t\\sigma_i\\from\\sigma_{i+1}\\from\\sigma_i &= \\sigma_i\\from\\sigma_{i-1}\\from\\sigma_i,\\\\\n\t\t\t\\sigma_i\\from\\sigma_{i\\pm 1}\\from\\sigma_{i\\pm 2} &= 0\n\t\t\\end{scriptaligned}\n\t\t\\Bigr)_{1 \\leq i \\leq n-2.}\n\t\\end{equation*}\n\tWe shall denote this quiver by $Q^\\mathfrak{p}$.\n\\end{lemma}\n\n\\begin{proof}\n\tWe compute the composition series of Verma modules and indecomposable projectives in $\\mathcal{O}^\\mathfrak{p}_0$\n\tusing the generalised Kazhdan-Lusztig theorem\n\t\\autocites[Cor.\\ 7.1.3]{Irving:filtered-category-OS}[Thm.\\ 1.3]{CC:parabolic-KL}.\n\tFor parabolic subgroups of the form $W_\\mathfrak{p}=S_{k}\\times S_{n-k}\\leq S_n$\n\tthere is a graphical calculus for computing parabolic Kazhdan-Lusztig polynomials\n\t\\autocites[\u00a75]{Brundan-Stroppel:Highest-weight-categories-I}{JS:Graphical-Kazhdan-Lusztig}.\n\tThe composition series thus obtained are listed in \\cref{eq:composition-series-in-Op-Sn}.\n\t\\begin{table}\n\t\t\\caption{Composition series of Verma modules and indecomposable projectives\n\t\t\tindexed by the $\\sigma_i \\in W^\\mathfrak{p}$.}\n\t\t\\label{eq:composition-series-in-Op-Sn}\n\t\t\\centering\n\t\t$\\begin{array}{c|ccccc}\n\t\t\t\\toprule\n\t\t\t& \\sigma_0 & \\sigma_1 & \\sigma_2 &\\cdots & \\sigma_{n-1}\\\\\n\t\t\t\\midrule\n\t\t\tM^\ud835\udd2d(-)\n\t\t\t& \\begin{smallmatrix}L(e)\\\\L(s_1)\\end{smallmatrix}\n\t\t\t& \\begin{smallmatrix}L(\\sigma_1)\\\\L(\\sigma_2)\\end{smallmatrix}\n\t\t\t& \\begin{smallmatrix}L(\\sigma_2)\\\\L(\\sigma_3)\\end{smallmatrix}\n\t\t\t& \\cdots\n\t\t\t& L(\\sigma_{n-1})\\\\[1em]\n\t\t\tP^\ud835\udd2d(-)\n\t\t\t& \\begin{smallmatrix}L(e)\\\\L(s_1)\\end{smallmatrix}\n\t\t\t& \\begin{smallmatrix}L(\\sigma_1)\\\\L(e)\\quad L(\\sigma_2)\\\\L(\\sigma_1)\\end{smallmatrix}\n\t\t\t& \\begin{smallmatrix}L(\\sigma_2)\\\\L(s_1)\\quad L(\\sigma_3)\\\\L(\\sigma_2)\\end{smallmatrix}\n\t\t\t& \\cdots\n\t\t\t& \\begin{smallmatrix}L(\\sigma_{n-1})\\\\L(\\sigma_{n-2})\\\\L(\\sigma_{n-1})\\end{smallmatrix}\\\\\n\t\t\t\\bottomrule\n\t\t\\end{array}$\n\t\\end{table}\n\tThese composition series show that the unique (up to scalar)\n\tmorphisms $P^\\mathfrak{p}(\\sigma_i) \\to P^\\mathfrak{p}(\\sigma_{i \\pm 1})$\n\tare irreducible,\n\tthat they generate $\\End_{\\mathcal{O}^\\mathfrak{p}} \\bigoplus_{i=0}^{n-1} P(\\sigma_i)$,\n\tand that their relations generate precisely\n\tthe ideal quotiented out in the statement of the lemma.\n\\end{proof}\n\nEvery Verma module $M^\\mathfrak{p}(\\sigma_i)$\nfits uniquely into a short exact sequence\n\\[\n\tM^\\mathfrak{p}(\\sigma_i)\\into P^\\mathfrak{p}(\\sigma_{i+1})\\onto M^\\mathfrak{p}(\\sigma_{i+1});\n\\]\nin particular, $\\Theta_{s_i} M^\\mathfrak{p}(\\sigma_i)=P^\\mathfrak{p}(\\sigma_i)$\nis always projective,\nwhich is not true in the non-parabolic category $\\mathcal{O}_0$.\nFrom these sequences we obtain\nthe images of the indecomposable projective and Verma modules $M$\nunder translation and shuffling functors\nlisted in \\crefrange{eq:parabolic-sln:translation-of-vermas-and-projectives}{eq:parabolic-sln:shuffling-of-projectives}\n(two-term complexes are understood to have the right entry in degree zero).\n\\bgroup\n\\newcommand{\\M}[1]{M^\\mathfrak{p}(\\sigma_{#1})}\n\\renewcommand{\\P}[1]{P^\\mathfrak{p}(\\sigma_{#1})}\n\\begin{table}\n\t\\caption{Images of Verma modules and indecomposable projectives under translation functors.}\n\t\\label{eq:parabolic-sln:translation-of-vermas-and-projectives}\n\t$\\begin{array}{@{} c|cccc@{}}\n\t\t\\toprule\n\t\tM     & \\Theta_{s_1} M & \\Theta_{s_2} M & \\Theta_{s_3} M & \\cdots         \\\\ \\midrule\n\t\t\\M{0} & \\P{1}          &                &  \\\\\n\t\t\\M{1} & \\P{1}          & \\P{2}          &                &  \\\\\n\t\t\\M{2} & \\P{1}          & \\P{2}          & \\P{3}          &                \\\\\n\t\t\\M{3} &                & \\P{2}          & \\P{3}          &                \\\\\n\t\t\\M{4} &                &                & \\P{3}          & \\smash{\\ddots} \\\\ \\bottomrule\n\t\\end{array}$\n\t\\hfill\n\t$\\begin{array}{@{} c|cccc @{}}\n\t\t\\toprule\n\t\tM     & \\Theta_{s_1} M & \\Theta_{s_2} M & \\Theta_{s_3} M & \\cdots         \\\\ \\midrule\n\t\t\\P{0} & \\P{1}          &                &  \\\\\n\t\t\\P{1} & \\P{1}^2        & \\P{2}          &                &  \\\\\n\t\t\\P{2} & \\P{1}          & \\P{2}^2        & \\P{3}          &                \\\\\n\t\t\\P{3} &                & \\P{2}          & \\P{3}^2        &                \\\\\n\t\t\\P{4} &                &                & \\P{3}          & \\smash{\\ddots} \\\\ \\bottomrule\n\t\\end{array}$\n\\end{table}\n\\begin{table}\n\t\\caption{Images of Verma modules under derived shuffling functors.}\n\t\\label{eq:parabolic-sln:shuffling-of-vermas}\n\t\\centering\n\t$\\begin{array}{@{} c | *{3}{r@{}>{{}}l<{{}}@{}l} l @{} }\n\t\t\\toprule\n\t\t%\n\t\tM & \\multicolumn{3}{c}{\\LSh{s_1} M}  &  \\multicolumn{3}{c}{\\LSh{s_2} M}  & \\multicolumn{3}{c}{\\LSh{s_3} M}  & \\cdots         \\\\ \\midrule\n\t\t\\M{0}                   &         &             & \\M{1}    & \\M{0}    & \\hShift{-1} &          & \\M{0}    & \\hShift{-1} &         &                \\\\\n\t\t\\M{1}                   & \\{\\M{1} & \\longto     & \\P{1})\\} &          &             & \\M{2}    & \\M{1}    & \\hShift{-1} &         &                \\\\\n\t\t\\M{2}                   & \\M{2}   & \\hShift{-1} &          & \\{ \\M{2} & \\longto     & \\P{2} \\} &          &             & \\M{3}   &                \\\\\n\t\t\\M{3}                   & \\M{3}   & \\hShift{-1} &          & \\M{3}    & \\hShift{-1} &          & \\{ \\M{3} & \\to         & \\P{3}\\} &                \\\\\n\t\t\\M{4}                   & \\M{4}   & \\hShift{-1} &          & \\M{4}    & \\hShift{-1} &          & \\M{4}    &             &         &  \\\\ \n\t\t\\M{5}                   & \\M{5}   & \\hShift{-1} &          & \\M{5}    & \\hShift{-1} &          & \\M{5}    &             &         & \\smash{\\ddots} \\\\ \\bottomrule\n\t\\end{array}$\n\\end{table}\n\\begin{table}\n\t\\caption{Images of indecomposable projectives under derived shuffling functors.}\n\t\\label{eq:parabolic-sln:shuffling-of-projectives}\n\t\\centering\n\t$\\begin{array}{@{} c | *{3}{r@{}>{{}}l<{{}}@{}l} l @{} }\n\t\t\\toprule\n\t\t%\n\t\tM & \\multicolumn{3}{c}{\\LSh{s_1} M}  & \\multicolumn{3}{c}{\\LSh{s_2} M}  & \\multicolumn{3}{c}{\\LSh{s_3} M} & \\cdots         \\\\ \\midrule\n\t\t\\P{0}                   &         &             & \\M{1}    & \\P{0}   & \\hShift{-1} &          & \\P{0}   & \\hShift{-1} &         &  \\\\\n\t\t\\P{1}                   &         &             & \\P{1})   & \\{\\P{1} & \\longto     & \\P{2}\\}  & \\P{1}   & \\hShift{-1} &         &  \\\\\n\t\t\\P{2}                   & \\{\\P{2} & \\longto     & \\P{1})\\} &         &             & \\P{2}    & \\{\\P{2} & \\longto     & \\P{3}\\} &                \\\\\n\t\t\\P{3}                   & \\P{3}   & \\hShift{-1} &          & \\{\\P{3} & \\longto     & \\P{2} \\} &         &             & \\P{3}   &                \\\\\n\t\t\\P{4}                   & \\P{4}   & \\hShift{-1} &          & \\P{4}   & \\hShift{-1} &          & \\{\\P{4} & \\longto     & \\P{2}\\} & \\\\\n\t\t\\P{5}                   & \\P{5}   & \\hShift{-1} &          & \\P{5}   & \\hShift{-1} &          & \\P{5}   & \\hShift{-1} &         &  \\smash{\\ddots} \\\\ \\bottomrule\n\t\\end{array}$\n\\end{table}\n\\egroup\n\n\\begin{lemma}\n\t$\\{P^\\mathfrak{p}(\\sigma_1),\\dotsc,P^\\mathfrak{p}(\\sigma_{n-1})\\}$\n\tis an $\\mathrm A_{n-2}$-configuration of $0$-spherical objects.\n\\end{lemma}\n\\begin{proof}\n\tThe composition series from \\cref{eq:composition-series-in-Op-Sn}\n\texhibit that $\\Hom_{\\mathcal{O}_0^\\mathfrak{p}}(P^\\mathfrak{p}(\\sigma_i), P^\\mathfrak{p}(\\sigma_i)) \\cong \\Complex[x]\/(x^2)$\n\tfor all $1 \\leq i \\leq n-1$,\n\tthat the nontrivial endomorphism $x$ is the degree $2$-map\n\t\\[\n\t\tx\\colon P^\\mathfrak{p}(\\sigma_i) \\onto L^\\mathfrak{p}(\\sigma_i) \\into P^\\mathfrak{p}(\\sigma_i),\n\t\\]\n\tand that\n\t\\[\n\t\t\\dim\\Hom_{\\mathcal{O}_0^\\mathfrak{p}}(P^\\mathfrak{p}(\\sigma_j), P^\\mathfrak{p}(\\sigma_i)) =\n\t\t\\begin{smallcases}\n\t\t\t2 & \\text{if $i=j$},\\\\\n\t\t\t1 & \\text{if $|i-j|=1$}\\\\\n\t\t\t0 & \\text{otherwise.}\n\t\t\\end{smallcases}\n\t\\]\n\tIt is sufficient to check non-degeneracy of the composition pairing\n\t\\[\n\t\t\\Hom_{\\mathcal{O}_0^\\mathfrak{p}}(-, P^\\mathfrak{p}(\\sigma_i)) \n\t\t\\otimes \\Hom_{\\mathcal{O}_0^\\mathfrak{p}}(P^\\mathfrak{p}(\\sigma_i), -)\n\t\t\\to \\langle x_{P^\\mathfrak{p}(\\sigma_i)} \\rangle\n\t\\]\n\tonly for the indecomposable projectives $P^\\mathfrak{p}(\\sigma_{i\\pm 1})$\n\tthat are connected to $P^\\mathfrak{p}(\\sigma_{i})$ by an arrow in $Q^\\mathfrak{p}$\n\tbecause for $j \\notin \\{i-1, i, i+1\\}$,\n\tthe composition pairing\n\t\\[\n\t\t\\underbrace{\\Hom_{\\mathcal{O}_0^\\mathfrak{p}}(P^\\mathfrak{p}(\\sigma_j), P^\\mathfrak{p}(\\sigma_i))}_{0}\n\t\t\\otimes \n\t\t\\underbrace{\\Hom_{\\mathcal{O}_0^\\mathfrak{p}}(P^\\mathfrak{p}(\\sigma_i), P^\\mathfrak{p}(\\sigma_j))}_{0}\n\t\t\\to \\langle x_{P^\\mathfrak{p}(\\sigma_i)} \\rangle\n\t\\]\n\tis non-degenerate trivially.\n\t\n\tFor $P^\\mathfrak{p}(\\sigma_{i\\pm 1})$, we see,\n\tanaloguously to \\crefrange{eq:composition-pairing:SL3:Ps-Pe}{eq:composition-pairing:SL3:Pst-Pe},\n\tthat the composition of the only (up to scalars) non-zero morphisms to and from $P^\\mathfrak{p}(\\sigma_{i})$,\n\twhich are written down in terms of composition series in the following,\n\tis $x_{P^\\mathfrak{p}(\\sigma_i)}$.\n\tThis shows that for all $1\\leq i<n-2$\n\t(and mutatis mutandis also for $i = n-1$),\n\tthe composition pairing\n\t\\begin{align}\n\t\t\\Hom(P^\\mathfrak{p}(\\sigma_{i\\pm 1}), P^\\mathfrak{p}(\\sigma_{i}))\n\t\t\t\\otimes\\Hom(P^\\mathfrak{p}(\\sigma_i), P^\\mathfrak{p}(\\sigma_{i\\pm 1}))\n\t\t\t& \\longto \\End(P^\\mathfrak{p}(\\sigma_i))\/\\langle \\id\\rangle\\\\\n\t\t\t\\Biggl(\n\t\t\t\\tikzset{x=.4cm,y=.4cm, baseline=(Fb.base),every node\/.append style={font=\\scriptsize, inner sep=0pt}}\n\t\t\t\\underbrace{\n\t\t\t\t\\begin{tikzpicture}[remember picture]\n\t\t\t\t\t\\node (Fa) at (0,1) {$L(\\sigma_i)$};\n\t\t\t\t\t\\node at (-1.25,0) {$L(\\sigma_{i+1})$};\n\t\t\t\t\t\\node (Fb) at (1.75,0) {$L(\\sigma_{i-1})$};\n\t\t\t\t\t\\node (Fc) at (0,-1) {$L(\\sigma_i)$};\n\t\t\t\t\t\\draw[brace] (Fa.50) to node[brace tip](Pa){} (Fb.10);\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}_{\\textstyle P^\\mathfrak{p}(\\sigma_{i})}\n\t\t\t\\qquad\n\t\t\t\\underbrace{\n\t\t\t\t\\begin{tikzpicture}[remember picture]\n\t\t\t\t\t\\node (Ma) at (0,1) {$L(\\sigma_{i+1})$};\n\t\t\t\t\t\\node (Mb) at (-1.25,0) {$L(\\sigma_{i\\pm 2})$};\n\t\t\t\t\t\\node at (1.25,0) {$L(\\sigma_{i})$};\n\t\t\t\t\t\\node (Mc) at (0,-1) {$L(\\sigma_{i+1})$};\n\t\t\t\t\t\\draw[brace'] (Mb.-170) to node[brace tip'](Pb){} (Mc.south west);\n\t\t\t\t\t\\draw[brace]  (Mb.170)  to node[brace tip] (Ta){} (Ma.north west);\n\t\t\t\t\t\\begin{scope}[overlay]\n\t\t\t\t\t\t\\draw[->] (Pa.center) to[out=70, out looseness=1.6, in=-135] (Pb.center);\n\t\t\t\t\t\\end{scope} \n\t\t\t\t\\end{tikzpicture}\n\t\t\t}_{\\textstyle P^\\mathfrak{p}(\\sigma_{i\\pm 1})}\n\t\t\t\\qquad\n\t\t\t\\underbrace{\n\t\t\t\t\\begin{tikzpicture}[remember picture, trim left=(Gd.west)]\n\t\t\t\t\t\\node (Ga) at (0,1) {$L(\\sigma_i)$};\n\t\t\t\t\t\\node (Gd) at (-1.75,0) {$L(\\sigma_{i+1})$};\n\t\t\t\t\t\\node (Gb) at (1.25,0) {$L(\\sigma_{i-1})$};\n\t\t\t\t\t\\node (Gc) at (0,-1) {$L(\\sigma_i)$};\n\t\t\t\t\t\\draw[brace']  (Gc.-80)to node[brace tip'](Tb){} (Gb.-10) ;\n\t\t\t\t\t\t\\draw[-, limit bb] (Ta) to[out=135, in looseness=1.5, in=115] (-4, 0);\n\t\t\t\t\t\t\\draw[->, limit bb] (-4, 0) to[out=-65, out looseness=1.5, in=-70] (Tb);\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}_{\\textstyle P^\\mathfrak{p}(\\sigma_{i})}\n\t\t\t\\Biggr)\n\t\t\t& \\longmapsto x_{P^\\mathfrak{p}(\\sigma_i)} \\notag\n\t\t\\end{align}\n\t\tis non-degenerate.\n\\end{proof}\n\\begin{theorem}\n\t\\label{thm:main-result}\n\tFor the parabolic subalgebra $\\mathfrak{p}\\subseteq\\mathfrak{sl}_n$\n\tcorresponding to the subgroup $W_\\mathfrak{p} = S_{n-1} \\times S_1 < S_n$,\n\tthe auto-equivalences $\\LSh{s_i} \\hShift{1}$ and $T'_{P^\\mathfrak{p}(\\sigma_i)}$ of $D^\\mathrm{b}(\\mathcal{O}^\\mathfrak{p}_0)$\n\tare naturally isomorphic\n\tfor every $1 \\leq i \\leq n-1$.\n\\end{theorem}\n\\begin{proof}\n\tLet $A^\\mathfrak{p}_n$ and $Q^\\mathfrak{p}_n$\n\trespectively be the path algebra quotient and the quiver from \\cref{lem:parabolic-sln:path-algebra}\n\tfor $\\mathfrak{sl}_n$.\n\tThe assignment $p\\colon A^\\mathfrak{p}_n \\to A^\\mathfrak{p}_n\/(\\trivpath{\\sigma_0}) \\xto{\\cong} A^\\mathfrak{p}_{n-1}$\n\tinduces fully faithful functor\n\t\\begin{equation*}\n\t\tp^* \\colon \\mathcal{O}^\\mathfrak{p}_0(\\mathfrak{sl}_{n-1})\\to\\mathcal{O}^\\mathfrak{p}_0(\\mathfrak{sl}_{n}),\\quad\n\t\tP^ \\mathfrak{p}(e) \\mapsto M^\\mathfrak{p}(\\sigma_1),\\quad\n\t\tP^\\mathfrak{p}(\\sigma_{i-1})\\mapsto P^\\mathfrak{p}(\\sigma_{i})\\ \\text{for $1\\leq k \\leq n-2$,}\n\t\\end{equation*}\n\twhich exhibits $\\mathcal{O}^\\mathfrak{p}_0(\\mathfrak{sl}_{n-1})$ as a full subcategory of $\\mathcal{O}^\\mathfrak{p}_0(\\mathfrak{sl}_{n})$;\n\tby induction, it follows that on the triangulated category of $D^\\mathrm{b}(\\mathfrak{sl}_{n})$\n\tgenerated by $P^\\mathfrak{p}(\\sigma_2), \\dotsc, P^\\mathfrak{p}(\\sigma_{n-1})$\n\tthere are natural isomorphisms $\\LSh{s_i} \\hShift{1} \\cong T'_{P^\\mathfrak{p}(\\sigma_i)}$\n\tfor $2 \\leq i \\leq n-1$.\n\tWe shall see, however, that it is easier to show the statement\n\talso for $i = 1$ and all of $D^\\mathrm{b}(\\mathfrak{sl}_{n})$ directly,\n\twithout resorting to induction,\n\temploying the proof of \\cref{thm:parabolic-sl3:main-statement}\n\twith the following alterations:\n\t\n\tSince\n\t\\begin{equation}\n\t\t\\label{eq:parabolic-sln:proof-of-main-thm:zero-hom}\n\t\t\\Hom_\\mathcal{O}(P^\\mathfrak{p}(\\sigma_i), P^\\mathfrak{p}(\\sigma_j)) = 0\n\t\t\\qquad\n\t\t\\text{for $0 \\leq i,j \\leq n-1$ with $\\lvert i-j\\rvert \\geq 2$}\n\t\\end{equation}\n\tit follows from the definition of $T'$\n\tthat $T'_{P^\\mathfrak{p}(\\sigma_i)}$ (for $1 \\leq i \\leq n-1$)\n\tacts as identity on all $P^\\mathfrak{p}(\\sigma_j)$\n\tfor $0 \\leq j \\leq n-1$ with $\\lvert i-j\\rvert \\geq 2$,\n\tas does, according to \\cref{eq:parabolic-sln:shuffling-of-projectives},\n\tthe functor $\\LSh{s_i}\\hShift{1}$.\n\tOne checks that both functors act by identity also on morphisms\n\tbetween these modules;\n\tthis shows that $T'_{P(\\sigma_i)} \\cong \\LSh{s_i} \\hShift{1}$ for all $1 \\leq i \\leq n-1$\n\tas functors on the triangulated subcategory of $D^\\mathrm{b}(\\mathcal{O}^\\mathfrak{p}_0(\\mathfrak{sl}_n))$\n\tgenerated by these $P^\\mathfrak{p}(\\sigma_j)$'s.\n\t\n\tThe category $\\mathcal{O}(\\mathfrak{sl}_n)$ has a projective generator\n\t$P^\\mathfrak{p}_n \\coloneqq \\bigoplus_{k=0}^{n-1} P^\\mathfrak{p}(\\sigma_k)$,\n\twith image $\\Theta_{s_i} P^\\mathfrak{p}_n = P^\\mathfrak{p}(\\sigma_i)^4$\n\tfor all $1 \\leq i \\leq n-1$.\n\tAgain due to \\cref{eq:parabolic-sln:shuffling-of-projectives},\n\twe obtain that\n\t\\[\\FBim{\\Theta_{s_i}} = \\Hom_{\\mathcal{O}_0}( P^\\mathfrak{p}_n, \\Theta_{s_i} P^\\mathfrak{p}_n)\n\t\t= \\Hom_{\\mathcal{O}_0}\\bigl(\n\t\t\tP^\\mathfrak{p}_n(\\sigma_i-1)\\oplus P^\\mathfrak{p}_n(\\sigma_i) \\oplus P^\\mathfrak{p}_n(\\sigma_i+1), \n\t\t\tP^\\mathfrak{p}_n(\\sigma_i)^4\\bigr\n\t\t)\n\t\\]\n\tReplacing $e$ by $\\sigma_{i-1}$, $s$ by $\\sigma_i$ and $st$ by $\\sigma_{i+1}$\n\tin the proof of \\cref{thm:parabolic-sl3:main-statement}\n\tthen gives a proof of $T'_{P(\\sigma_i)} \\cong \\LSh{s_i}\\hShift{1}$ for all $1 \\leq i \\leq n-1$\n\tas functors on $D^\\mathrm{b}(\\mathcal{O}^\\mathfrak{p}_0(\\mathfrak{sl}_n))$.\n\\end{proof}\n\n\\Needspace{4\\baselineskip}\n\\section{Further observations and final remarks}\n\\subsection{\\texorpdfstring{$\\mathcal{O}^\\mathfrak{p}_0$}{O} as a spherical subcategory}\nWe know that the object $P^\\mathfrak{p}(s)\\in\\mathcal{O}_0^\\mathfrak{p}(\\mathfrak{sl}_3)$ is spherical, so\none might ask whether $\\mathcal{O}^\\mathfrak{p}_0(\\mathfrak{sl}_3)$ arises\nas the spherical subcategory $\\Sph(P^\\mathfrak{p}(s))$\nof $P^\\mathfrak{p}(s)\\in\\mathcal{O}_{0,\\mathfrak{sl}_3}$.\nHowever, $P^\\mathfrak{p}(s)$ is not spherelike in $D^\\mathrm{b}(\\mathcal{O}(\\mathfrak{sl}_3))$,\n\\ie, we cannot assign a meaningful spherical subcategory to it.\n\nTo see this, consider the projective resolution $P^\\mathfrak{p}(s) \\simeq \\{P(s)\\to P(ts)\\to P(s)\\}$\nin $D^\\mathrm{b}(\\mathcal{O}(\\mathfrak{sl}_3))$.\nUsing this resolution, we obtain the chain complex\n(see \\cref{sec:twisting-by-Le} for an explanation of the notation)\n\\bgroup\n\\tikzset{\n\tampersand replacement=\\&,\n\tcommutative diagrams\/diagrams={\n\t\trow sep=small,\n\t\tcolumn sep=tiny,\n\t\tnodes={inner sep=1pt, font=\\scriptsize}\n\t}\n}%\n\\begin{multline*}\n\t\\hom^\\bullet_{D^\\mathrm{b}(\\mathcal{O}_0)}(P^\\mathfrak{p}(s), P^\\mathfrak{p}(s))\n\t\\\\\n\t\\simeq\\left\\{\n\t\\left\\langle\n\t\\begin{tikzcd}[baseline=(R.base), ampersand replacement=\\&]\n\t\tP(s) \\rar\\dar \\& P(ts)\\rar\\dar \\& P(s)\\ar[d, \"{\\id, x}\" name=R]\\\\\n\t\tP(s) \\rar \\& P(ts)\\rar \\& P(s)\n\t\\end{tikzcd}\n\t\\right\\rangle\n\t\\to\n\t0\n\t\\to\n\t\\left\\langle\n\t\\begin{tikzcd}[baseline=(R.base), ampersand replacement=\\&]\n\t\t\\&\\& P(s) \\rar\\dar[\"{\\id}\"']\\dar[phantom, \"\\phantom{\\id, x}\" name=R] \\& P(ts) \\rar \\&  P(s) \\\\\n\t\tP(s) \\rar \\& P(ts) \\rar \\&  P(s)\n\t\\end{tikzcd}\n\t\\right\\rangle\n\t\\right\\},\n\\end{multline*}\n\\egroup\nwhose leftmost bracket is in degree zero.\nThe complex $\\hom^\\bullet_{D^\\mathrm{b}(\\mathcal{O}_0)}(P^\\mathfrak{p}(s), P^\\mathfrak{p}(s))$ \nthus has total dimension $3$, so $P^\\mathfrak{p}(s)$ is not spherelike.\nAs a side note, we notice that the inclusion\n$D^\\mathrm{b}(\\mathcal{O}_0^\\mathfrak{p}) \\subset D^\\mathrm{b}(\\mathcal{O}_0)$,\ngiven by mapping projectives to projectives,\nis not full.\n\n\\subsection{Necessity of extremal partitions}\nIs it necessary to choose a parabolic subalgebra $\ud835\udd2d$ which corresponds to the \"extremal\" partition $(n-1, 1)$,\n\\ie\\ to the parabolic subgroup $S_{n-1}\\times S_1<S_n$?\nThe maximal parabolic subalgebra\n\\begin{equation}\n\t\\label{eq:sl4:subalgebra-without-spherical-objects}\n\t\ud835\udd2d = \\begin{psmallmatrix}* & * & * &*\\\\ *&*&*&*\\\\0&0&*&*\\\\0&0&*&*\\end{psmallmatrix} \u2282 \\mathfrak{sl}_4,\n\\end{equation}\ncorresponding to the parabolic subgroup $W_\ud835\udd2d=\u27e8s\u27e9\u00d7\u27e8u\u27e9=S_2\\times S_2<S_4=\u27e8s,t,u\u27e9$\nwith minimal length coset representatives $W^\ud835\udd2d=\\{e, t, tu, ts, tsu, tsut\\}$\nhas parabolic Verma modules and indecomposable projectives\nwith composition series listed in \\cref{tab:sl4:subalgebra-without-spherical-objects:composition-series};\nthese can be obtained from parabolic Kazhdan\\-\/Lusztig polynomials\n(see also \\autocite{Brundan-Stroppel:Highest-weight-categories-III}).\nOne checks from the composition series \nthat $P^\ud835\udd2d(t)$, $P^\ud835\udd2d(ts)$ and $P^\ud835\udd2d(tsu)$ are the only spherelike indecomposable projectives,\nand there no possible $\\mathrm A_3$ configuration of indecomposable projectives.\nOne might wonder if one can find at least an $\\mathrm A_2$-configuration or a single spherical object.\n\nUnfortunately, we encounter the same problem of the present spherelike objects\nas in the non\\-\/parabolic $\\mathfrak{sl}_3$-case;\nthis can be seen from the composition series as follows.\nThe projectives $P^\ud835\udd2d(ts)$, $P^\ud835\udd2d(tu)$ corresponding to the two incomparable weights $ts\\not\\gtrless tu$\nhave non-trivial morphisms $P^\ud835\udd2d(ts)\u2192P^\ud835\udd2d(tu)$ and $P^\ud835\udd2d(tu)\u2192P^\ud835\udd2d(ts)$ whose composition\n\\begin{equation}\n\t%\n\t\\label{eq:sl4:P(ts)-P(tu)-not-spherical}\n\t\\begin{tikzpicture}[x=.4cm,y=.4cm, node distance=1mm, baseline=(current bounding box.center),every node\/.append style={inner sep=0mm}]\n\t\t\\begin{scope}[every node\/.append style={font=\\scriptsize}]\n\t\t\t%\n\t\t\t%\n\t\t\t\\node (I-ts) at (0,0) {$L(ts)$};\n\t\t\t\\node[right=of I-ts] (I-tsut) {$L(tsut)$};\n\t\t\t\\node[right=of I-tsut] (I-tu) {$L(tu)$};\n\t\t\t\\node[above=of I-tsut] (I-t) {$L(t)$};\n\t\t\t\\node[below=of I-tsut] (I-tsu) {$L(tsu)$};\n\t\t\t\\node[above=of I-tu] (I-tsu') {$L(tsu)$};\n\t\t\t\\node[above=of I-tsu'] (I-ts') {$L(ts)$};\n\t\t\t\\node[braced box={Pa}, fit=(I-tsu') (I-ts')] {} ;\n\t\t\t%\n\t\t\t%\n\t\t\t%\n\t\t\t\\node[right=4 of I-tu] (II-ts) {$L(ts)$};\n\t\t\t\\node[right=of II-ts] (II-tsut) {$L(tsut)$};\n\t\t\t\\node[right=of II-tsut] (II-tu) {$L(tu)$};\n\t\t\t\\node[above=of II-tsut] (II-t) {$L(t)$};\n\t\t\t\\node[below=of II-tsut] (II-tsu) {$L(tsu)$};\n\t\t\t\\node[above=of II-tu] (II-tsu') {$L(tsu)$};\n\t\t\t\\node[above=of II-tsu'] (II-tu') {$L(tu)$};\n\t\t\t\\draw[brace] ([xshift=1mm]II-tsu.south west) coordinate (c1) to node[brace tip] (Pb){} ([xshift=-1mm]II-ts.base west) coordinate (c2);\n\t\t\t\\draw[->, limit bb] \n\t\t\tlet \\p1=(c1),\\p2=(c2),\\n1={atan2(\\y2-\\y1,\\x2-\\x1)}\n\t\t\tin (Pa) to[out=0,in=\\n1+90] (Pb.center);\n\t\t\t\\node[braced box={Pc}, fit=(II-tsu') (II-tu')] {};\n\t\t\t%\n\t\t\t%\n\t\t\t%\n\t\t\t\\node[right=4 of II-tu] (III-ts) {$L(ts)$};\n\t\t\t\\node[right=of III-ts] (III-tsut) {$L(tsut)$};\n\t\t\t\\node[right=of III-tsut] (III-tu) {$L(tu)$};\n\t\t\t\\node[above=of III-tsut] (III-t) {$L(t)$};\n\t\t\t\\node[below=of III-tsut] (III-tsu) {$L(tsu)$};\n\t\t\t\\node[above=of III-tu] (III-tsu') {$L(tsu)$};\n\t\t\t\\node[above=of III-tsu'] (III-ts' ){$L(ts)$};\n\t\t\t\\draw[brace] ([xshift=1mm]III-tsu.south west) coordinate (c1) to node[brace tip] (Pd){} ([xshift=-1mm]III-ts.base west) coordinate (c2);\n\t\t\t\\draw[->, limit bb] \n\t\t\tlet \\p1=(c1),\\p2=(c2),\\n1={atan2(\\y2-\\y1,\\x2-\\x1)}\n\t\t\tin (Pc) to[out=0,in=\\n1+90] (Pd.center);\n\t\t\\end{scope}\n\t\t%\n\t\t\\node[above=1 of I-t] (Pts) {$P^\ud835\udd2d(ts)$};\n\t\t\\node[above=1 of II-t] (Ptu) {$P^\ud835\udd2d(tu)$};\n\t\t\\node[above=1 of III-t] (Pts') {$P^\ud835\udd2d(ts)$};\n\t\t\\draw[->] (Pts) edge (Ptu) (Ptu) edge (Pts');\n\t\\end{tikzpicture}\n\t%\n\\end{equation}\nis the zero morphism.\nThe same holds true for $\\bigl(P^\ud835\udd2d(tu)\u2192P^\ud835\udd2d(t)\u2192P^\ud835\udd2d(tu)\\bigr) = 0$.\nHence neither $P^\ud835\udd2d(tu)$ nor $P^\ud835\udd2d(ts)$ is spherical.\nFor $P^\ud835\udd2d(t)$ the composition\n\\begin{equation}\n\t%\n\t\\label{eq:sl4:P(t)-not-spherical}\n\t\\begin{tikzpicture}[x=.4cm,y=.4cm, node distance=1mm, baseline=(current bounding box.center),every node\/.append style={inner sep=0mm}]\n\t\t\\begin{scope}[every node\/.append style={font=\\scriptsize}]\n\t\t\t\\path\n\t\t\t%\n\t\t\t%\n\t\t\t%\n\t\t\tnode(I-ts) at (0,0) {$L(ts)$}\n\t\t\tnode[right=of I-ts] (I-tsut) {$L(tsut)$}\n\t\t\tnode[right=of I-tsut] (I-tu) {$L(tu)$}\n\t\t\tnode[above=of I-tsut] (I-t) {$L(t)$}\n\t\t\tnode[below=of I-tsut] (I-tsu) {$L(tsu)$}\n\t\t\t%\n\t\t\tnode[left=of I-ts] (I-e') {$L(e)$}\n\t\t\tnode[below=of I-e'] (I-t') {$L(t)$}\n\t\t\tnode[below=of I-t'] (I-tsut') {$L(tsut)$}\n\t\t\t%\n\t\t\tnode[braced box={P-I}, fit=(I-t) (I-tsu) (I-tu)] {}\n\t\t\t%\n\t\t\t%\n\t\t\t%\n\t\t\t%\n\t\t\tnode[right=6.5 of I-tsu] (II-t) {$L(t)$}\n\t\t\tnode[below=of II-t] (II-tsut) {$L(tsut)$}\n\t\t\tnode[left=of II-tsut] (II-ts)  {$L(ts)$}\n\t\t\tnode[right=of II-tsut] (II-tu) {$L(tu)$}\n\t\t\tnode[below=of II-tsut] (II-tsu) {$L(tsu)$}\n\t\t\t%\n\t\t\tnode[above=of II-tu] (II-tsu') {$L(tsu)$}\n\t\t\tnode[above=of II-tsu'] (II-ts') {$L(ts)$}\n\t\t\t%\n\t\t\tnode[right=of II-tsu']  (II-tsu'') {$L(tsu)$}\n\t\t\tnode[above=of II-tsu''] (II-tu'') {$L(tu)$}\n\t\t\t%\n\t\t\tnode[right=of II-tu''] (II-tsut''') {$L(tsut)$}\n\t\t\tnode[above=of II-tsut'''] (II-tsu''') {$L(tsu)$}\n\t\t\t%\n\t\t\tnode[braced box'={P-II}, fit=(II-t) (II-tsu) (II-ts)] {}\n\t\t\tnode[braced box={P-II'}, fit=(II-tsu''') (II-tsut''') ] {}\n\t\t\t%\n\t\t\t%\n\t\t\t%\n\t\t\t%\n\t\t\tnode[right=6 of II-tu''](III-ts) {$L(ts)$}\n\t\t\tnode[right=of III-ts] (III-tsut) {$L(tsut)$}\n\t\t\tnode[right=of III-tsut] (III-tu) {$L(tu)$}\n\t\t\tnode[above=of III-tsut] (III-t) {$L(t)$}\n\t\t\tnode[below=of III-tsut] (III-tsu) {$L(tsu)$}\n\t\t\t%\n\t\t\tnode[left=of III-ts] (III-e') {$L(e)$}\n\t\t\tnode[below=of III-e'] (III-t') {$L(t)$}\n\t\t\tnode[below=of III-t'] (III-tsut') {$L(tsut)$};\n\t\t\t%\n\t\t\t\\draw[brace] ([yshift=-1mm]III-tsu.base east) coordinate (C-III') to node[brace tip] (P-III){} ([yshift=-1mm]III-tsut'.south west) coordinate (C-III);\n\t\t\t\\draw[->]\n\t\t\t(P-I) to[out=0, in=180,looseness=1.5] (P-II);\n\t\t\t\\draw[limit bb] \n\t\t\t(P-II') to[out=0, in=135] ($(III-tsut')-(2,1.2)$) coordinate (H1);\n\t\t\t\\draw[->, limit bb]\n\t\t\tlet \\p1=(C-III),\\p2=(C-III'),\\n1={atan2(\\y2-\\y1,\\x2-\\x1)}\n\t\t\tin (H1) to[out=-45,in=\\n1-90, looseness=1] (P-III.center);\n\t\t\\end{scope}\n\t\t\\path\n\t\tnode[above=1 of I-t] (H2) {\\strut}\n\t\tnode (Pt) at ($(H2) !.5! (I-ts |- H2)$) {$P^\ud835\udd2d(t)$}\n\t\tnode (Ptsu) at (Pt -| II-ts') {$P^\ud835\udd2d(tsu)$}\n\t\tnode (Pt') at ($(III-t |- H2) !.5! (III-ts |- H2)$) {$P^\ud835\udd2d(t)$};\n\t\t\\draw[->] (Pt) -- (Ptsu);\n\t\t\\draw[->] (Ptsu) -- (Pt');\n\t\\end{tikzpicture}\n\t%\n\\end{equation}\nshows that $P^\ud835\udd2d(t)$ it is not spherical either.\n\nTherefore, for $\ud835\udd2d\u2286\\mathfrak{sl}_4$ the parabolic subalgebra\ncorresponding to the parabolic subgroup $S_2\\times S_2\\leq S_4$,\nthe modules $P^\ud835\udd2d(t)$, $P^\ud835\udd2d(ts)$ and $P^\ud835\udd2d(tu)$ \nare the only spherelike indecomposable projective modules,\nand none of them is spherical.\n\n\\begin{table}\n\t\\caption{%\n\t\tComposition series of parabolic Verma modules and indecomposable projectives\n\t\tin $\\mathcal{O}^\\mathfrak{p}_0$ for $\\mathfrak{p}\u2286\\mathfrak{sl}_4$ the parabolic subalgebra\n\t\tcorresponding to the parabolic subgroup $W_\\mathfrak{p} \u2254 S_2\\times S_2 \\leq S_4$\n\t\t(see \\cref{eq:sl4:subalgebra-without-spherical-objects}).\n\t}\n\t\\label{tab:sl4:subalgebra-without-spherical-objects:composition-series}\n\t\\tikzset{node distance=1mm, x=3mm, every node\/.append style={inner sep=0mm, font=\\scriptsize}, baseline}\n\t\\centering\n\t\\begin{tabular}{c|c|c}\n\t\t\\toprule\n\t\t$w\u2208W^\ud835\udd2d$ & $M^\ud835\udd2d(w)$ & $P^\ud835\udd2d(w)$\\\\\n\t\t\\midrule\n\t\t$e$\n\t\t& \n\t\t%\n\t\t\\tikz\\draw\n\t\tnode(I-e')  at (0,0) {$L(e)$}\n\t\tnode[below=of I-e'] (I-t') {$L(t)$}\n\t\tnode[below=of I-t'] (I-tsut') {$L(tsut)$};\n\t\t&\n\t\t\\scriptsize dto.\n\t\t\\\\\n\t\t\\midrule\n\t\t$t$\n\t\t&\n\t\t%\n\t\t\\tikz\\draw\n\t\tnode (I-ts) at (0,0) {$L(ts)$}\n\t\tnode[right=of I-ts] (I-tsut) {$L(tsut)$}\n\t\tnode[right=of I-tsut] (I-tu) {$L(tu)$}\n\t\tnode[above=of I-tsut] (I-t) {$L(t)$}\n\t\tnode[below=of I-tsut] (I-tsu) {$L(tsu)$};\n\t\t&\n\t\t%\n\t\t\\tikz\\draw\n\t\t%\n\t\tnode(I-ts) at (0,0) {$L(ts)$}\n\t\tnode[right=of I-ts] (I-tsut) {$L(tsut)$}\n\t\tnode[right=of I-tsut] (I-tu) {$L(tu)$}\n\t\tnode[above=of I-tsut] (I-t) {$L(t)$}\n\t\tnode[below=of I-tsut] (I-tsu) {$L(tsu)$}\n\t\t%\n\t\tnode[left=1 of I-ts] (I-e') {$L(e)$}\n\t\tnode[below=of I-e'] (I-t') {$L(t)$}\n\t\tnode[below=of I-t'] (I-tsut') {$L(tsut)$};\n\t\t\\\\\\midrule\n\t\t$ts$\n\t\t&\t\n\t\t%\n\t\t\\tikz\\draw\n\t\tnode (ts'') at (0,0) {$L(ts)$}\n\t\tnode[below=of ts''] (tsu'') {$L(tsu)$};\n\t\t&\n\t\t%\n\t\t\\tikz\\path\n\t\t%\n\t\tnode (ts'') at (0,0) {$L(ts)$}\n\t\tnode[below=of ts''] (tsu'') {$L(tsu)$}\t\t\n\t\t%\n\t\tnode[below left=of tsu''] (tsu''') {$L(tsu)$}\n\t\tnode[left=of tsu'''](tsut''') {$L(tsut)$}\n\t\tnode[left=of tsut'''] {$L(ts)$}\n\t\tnode[above=of tsut'''] {$L(t)$}\n\t\tnode[below=of tsut'''] {$L(tsu)$};\t\n\t\t\\\\\t\\midrule\n\t\t$tu$\n\t\t&\n\t\t%\n\t\t\\tikz\\path\n\t\t%\n\t\tnode (tu'') at (0,0) {$L(tu)$}\n\t\tnode[below=of tu''] (tsu'') {$L(tsu)$};\n\t\t&\n\t\t%\n\t\t\\tikz\\path\n\t\t%\n\t\tnode (tu'') at (0,0) {$L(tu)$}\n\t\tnode[below=of tu''] (tsu'') {$L(tsu)$}\t\t\n\t\t%\n\t\tnode[below left=of tsu''] (tsu''') {$L(tsu)$}\n\t\tnode[left=of tsu'''](tsut''') {$L(tsut)$}\n\t\tnode[left=of tsut'''] {$L(ts)$}\n\t\tnode[above=of tsut'''] {$L(t)$}\n\t\tnode[below=of tsut'''] {$L(tsu)$};\n\t\t\\\\\\midrule\n\t\t$tsu$\n\t\t&\n\t\t%\n\t\t\\tikz\\draw\n\t\tnode (tsu) at(0,0) {$L(tsu)$}\n\t\tnode[below=of tsu] (tsut) {$L(tsut)$};\n\t\t&\n\t\t%\n\t\t\\tikz\\path\n\t\t%\n\t\tnode (tsu) at(0,0) {$L(tsu)$}\n\t\tnode[below=of tsu] (tsut) {$L(tsut)$}\n\t\t%\n\t\tnode[left=1 of tsut]  (tu') {$L(tu)$}\n\t\tnode[below=of tu'] (tsu') {$L(tsu)$}\t\n\t\t%\n\t\tnode[left=1 of tu'] (ts'') {$L(ts)$}\n\t\tnode[below=of ts''] (tsu'') {$L(tsu)$}\t\t\n\t\t%\n\t\tnode[below left=of tsu''] (tsu''') {$L(tsu)$}\n\t\tnode[left=of tsu'''](tsut''') {$L(tsut)$}\n\t\tnode[left=of tsut'''] {$L(ts)$}\n\t\tnode[above=of tsut'''] {$L(t)$}\n\t\tnode[below=of tsut'''] {$L(tsu)$};\n\t\t\\\\\\midrule\n\t\t$tsut$\n\t\t&\n\t\t%\n\t\t\\tikz\\draw\n\t\tnode at (0,0)  {$L(tsut)$};\n\t\t&\n\t\t%\n\t\t\\tikz\\path\n\t\t%\n\t\tnode (tsut) at (0,0) {$L(tsut)$}\n\t\t%\n\t\tnode[below left=of tsut] (tsu') {$L(tsu)$}\n\t\tnode[below=of tsu'] (tsut') {$L(tsut)$}\n\t\t%\n\t\tnode[below left=1mm and 1 of tsut'] (tu'') {$L(tu)$}\n\t\tnode[left=of tu''] (tsut'') {$L(tsut)$}\n\t\tnode[left=of tsut''] (ts'') {$L(ts)$}\n\t\tnode[above=of tsut''] {$L(t)$}\n\t\tnode[below=of tsut''] {$L(tsu)$}\n\t\t%\n\t\tnode[left=1 of ts''] (e''') {$L(e)$}\n\t\tnode[below=of e'''] (t''') {$L(t)$}\n\t\tnode[below=of t'''] {$L(tsut)$};\n\t\t\\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\\end{table}\n\n\\subsection*{Acknowledgements}\nThis article compiles the results of a Master's thesis\nwritten with the advice of Catharina Stroppel,\nwhom the author wants to thank for her continuous support\nas well as the opportunity provided to dive into this topic.\n\n\\printbibliography\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}