diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzascm" "b/data_all_eng_slimpj/shuffled/split2/finalzzascm" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzascm" @@ -0,0 +1,5 @@ +{"text":"\\section{Sample}\n\\label{sec:sample}\n\nThe sample is composed of 70 archival {\\it Hubble Space Telescope}\n({\\it HST}) images of low-redshift QSOs. They have redshifts between $0.06 \\leq z \\leq 0.46$ and total\n(host plus nucleus) absolute magnitudes brighter than $M_V \\leq -23$.\nFurthermore, they must have been observed with the\n{\\it HST}'s Wide-Field Planetary Camera 2 (WFPC2), using broad-band\nfilters, and have images publicly available in the {\\it HST} archives\nas of 1999. This brings our sample to 70 QSOs.\nRather than restrict our study to a specific class of QSOs, we impose\nno physical criteria on the QSOs beyond those of magnitude and\nredshift. Thus we are able to study a broad range of properties and\ndraw general conclusions. The images are reduced and the physical parameters fitted \nas described by Hamilton et al.~(2002).\n\n\\section{The ``Fundamental Plane'' of QSOs}\n\\label{sec:fp}\n\nFor our Principal Components Analysis (PCA), we use a restricted sample of those QSOs for which we\nhave all of the following parameters: $M_V\\mathrm{(nuc)}$, $L_X$, \n$r_{1\/2}$, and $\\mu_e$, where $\\mu_e$ is the effective\nsurface magnitude of the galactic bulge. We further\nrequire that each QSO have a modeled, spheroidal bulge (the entire\ngalaxy, in the case of elliptical hosts). These qualifications\nrestrict the sample to 42 QSOs.\n\n\nWe can perform two PCAs, an optical one using $M_V\\mathrm{(nuc)}$, \n$\\log r_{1\/2}$, and $\\mu_e$ as the parameters, and an x-ray one that \nsubstitutes $\\log L_X$ for the nuclear luminosity.\nFrom the optical PCA performed on this sample of 42 objects, \nwe find that 96.1\\% of the variance can be explained with just the first two\nprincipal axes, and therefore the QSOs mostly lie in a plane within this parameter\nspace. This we consider to be a fundamental plane (FP) for QSOs. \nFor the corresponding x-ray results, the first two principal axes explain \n95.2\\% of the variance in the sample, and we find here an x-ray QSO fundamental plane.\nThe individual subsamples of QSOs (radio-loud or radio-quiet, with spiral or elliptical hosts, and all \ncombinations of these) \nare also examined in this way, and they show fundamental planes, as well.\n\n\nWe obtain the optical and x-ray formulae for the full sample's fundamental plane:\n\\begin{equation}\n\tM_V\\mathrm{(nuc)} = -77.5 + 3.14 \\mu_e - 14.2 \\log\n\tr_{1\/2}\n\t\\label{equ:oall-physical}\n\\end{equation}\n\\begin{equation}\n\t\\log L_X = 79.3 - 2.03 \\mu_e + 8.74 \\log\n\tr_{1\/2} \\mbox{ .} \n\t\\label{equ:xall-physical}\n\\end{equation}\nViews of the optical and x-ray fundamental planes, with the QSO data points superimposed, \nare displayed in Figure~\\ref{fig:fp-phys}. Note that the host properties describe the horizontal and the nuclear luminosity the vertical in these plots. Figure~\\ref{fig:fp-rms} illustrates the precision of the \nQSO fundamental plane in both forms, with the plane plotted against \nthe measured host sizes. Its highest precision is found when solving for $\\log r_{1\/2}$.\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\\subsection{Possible Derivation}\nThe fundamental plane for QSOs shows a relationship between the\nnuclear and host features that goes beyond the simple (and weak) correlation of nuclear and\nhost luminosities. This behavior may be connected to other, known\nrelations between the objects.\nFor example, there is already a well-studied\nfundamental plane for normal, elliptical galaxies (Djorgovski \\&\nDavis~1987; Dressler et al.~1987) that incorporates galaxy size, $r_{1\/2}$, central velocity dispersion,\n$\\sigma_c$, and effective surface magnitude,\n$\\mu_e$.\n\nLet us take a $V$-band measurement of the normal galaxy fundamental plane \n(Scodeggio et al.~1998), \n$\\log r_{1\/2} = 1.35 \\log \\sigma_c + 0.35 \\mu_e + \\mathit{Constant}$. \nThe ratio of the coefficients of $\\log r_{1\/2}$ to $\\mu_e$ differs by about 37\\% between the QSO\noptical fundamental plane and the normal galaxy FP, and the QSO x-ray FP\nshows a 34\\% difference. Still, there is a formal similarity between\nthe QSO and normal fundamental planes, which might point to a link between \nthe host galaxy's central velocity\ndispersion and the nuclear luminosity of the QSO. This\ncould derive from the fueling mechanism of QSOs, if the movement of gas\nto the center of the galaxy and the black hole is related to the\nvelocity dispersion. \n\n\nIt is therefore tempting to try to derive the\nQSO fundamental plane directly from the elliptical galaxy fundamental\nplane, but we find two problems with this approach, both arising from the relation of black hole mass to nuclear luminosity. Using the\nvelocity dispersion to black hole mass relation of Merritt \\& Ferrarese~(2001),\n$ \\mathcal{M}_{BH}=1.3 \\times 10^8 \n\t\\left( \\sigma_c \/ 200 \\mbox{ km s}^{-1} \\right)^{4.72} \n\t\\mathcal{M}_{\\odot} $\nwe can put the elliptical galaxy fundamental plane in terms of black hole mass. \nUsing the observed (but weak) correlation between black hole mass and \nnuclear luminosity in our sample, $M_V\\mathrm{(nuc)} = -1.98 \\log \\left( \\mathcal{M}_{\\mathrm{BH}} \/ \\mathcal{M}_{\\odot} \\right) -6.90$ and \n$\\log L_X = 2.77 \\log \\left( \\mathcal{M}_{\\mathrm{BH}} \/ \\mathcal{M}_{\\odot} \\right) + 19.8$,\nwe obtain\n\\begin{equation}\n\tM_V\\mathrm{(nuc)} = \\mathit{Constant} + 2.5 \\mu_e - 7.14 \\log r_{1\/2}\n\\end{equation}\n\\begin{equation}\n\t\\log L_X = \\mathit{Constant} - 3.5 \\mu_e + 10 \\log r_{1\/2} \\mbox{ .}\n\\end{equation}\nThese are our attempts to derive the QSO optical fundamental plane from the normal galaxy FP.\nThe optical form differs from the actual QSO FP, equation~(\\ref{equ:oall-physical}), \ncompletely outside the propagated errors, but \nthe x-ray form is within the errors of equation~(\\ref{equ:xall-physical}).\n\n\nBut any derivation of the QSO fundamental plane has an additional problem. \nAs mentioned before, the QSO fundamental plane for the full sample is \ncomposed of individual FPs of the several subsamples. Some subsample FPs \nactually slope in the opposite direction from the overall QSO FP.\nFor example, in the optical form, the FP of radio-quiets in elliptical hosts \nslopes in the opposite direction. \nAnd in the x-ray form, the radio-quiet subsamples slope oppositely \nfrom the overall sample.\nYet these differences cannot be accounted for by different correlations of \nnuclear luminosity with black hole mass. \nThe poor correlation of black hole mass with nuclear luminosity lies in contrast with the \nrelatively thin QSO fundamental plane. \nFurthermore, Woo \\& Urry~(2002) suggest that the apparent correlations \nbetween black hole mass and nuclear luminosity are merely artifacts of \nsample selection.\nRegardless of how we take this interpretation, the relationship between \nblack hole mass and nuclear luminosity remains the missing link in any \nderivation of the QSO fundamental plane.\n\n\\subsection{Arrangement of Subsample Planes}\n\nThe insight into the origins of this new fundamental plane relationship might come from\nthe comparison of the QSO subsample FPs. The thickness of the overall QSO\nfundamental plane appears partly to be the result of the superposition of\nthe subsamples' planes.\nBecause the QSO FP mathematically describes a link between the host and the \nnucleus, it seems reasonable to suppose that the slope of the plane depends on the \nphysical nature of this link. The fueling mechanism at a QSO's core would seem to be the most directly related to this, depending on how we define ``fueling mechanism.'' \nWe could encompass within this term the details of the structure and dynamics of the accretion disk, as well as question of whether the QSO is efficiently or inefficiently fueled.\n\nIt is intriguing that as we change from one class to another, the fundamental plane \nessentially pivots about an axis, so the differences are mostly reduced to a single dimension, \nthe slope (or gradient) relative to the $\\mu_e$--$\\log r_{1\/2}$ plane.\nThe gradient directions, projected onto the $\\mu_e$--$\\log r_{1\/2}$ plane, are almost all \neither aligned (or anti-aligned, for those with opposite slope). \nThe optical subsample gradient directions are never more \nthan 3.8 degrees away from that of the full sample, and in the x-ray form, they never exceed \na 6.4 degree deviation.\n\nRadio-loudness has the strongest effect on the slopes. In the x-ray form, the subsample FPs \nare almost evenly divided between those aligned with the full sample and those anti-aligned. \nIn the optical form, only radio-quiets in elliptical hosts tilt opposite to the full sample.\nThis effect is interesting because we are seeing a stark difference between the radio-loud and \nradio-quiet nuclei in the hosts of the same morphology.\n\nIt would be interesting to find if the different QSO FP orientations described above come \nabout from different fueling mechanisms that might be found in the various subsamples.\nWe see, for instance, that radio-quiet and radio-loud QSOs are characterized by very \ndifferent slopes in their x-ray FPs, but the understanding of what makes these QSO types differ \nis still too limited to speculate further here. In our ongoing research, we are expanding the \nfundamental plane study to other types of AGN. \nWe can then compare their FP orientations with those of the different QSO\nsubsamples, which may teach us more about the physics underlying the QSO fundamental plane.\n\n\\section{Future Work}\n\nWe should ask if lower luminosity classes of AGN (such as Seyferts or LLAGN) also have \nfundamental planes of this sort. If they do, how do they compare with that of QSOs? We can \nimagine four possibilities:\n\\begin{enumerate}\n\\item{They share the same fundamental plane as QSOs. This would indicate that AGN power \nscales with the host properties, even across AGN types, and would support some form of \nunification.}\n\n\\item{The plane is parallel to that of QSOs, but shifted to lower nuclear luminosities. This would \nshow that these host properties don't determine the AGN class, and a given galaxy could \nhost different types.}\n\n\\item{The plane is tilted with respect to that of QSOs. Then the fundamental plane slope would \nbe characteristic of the AGN type, possibly supporting the idea that the slope is tied to the \naccretion mechanism.}\n\n\\item{There is no fundamental plane whatsoever. In that case, this type of fundamental plane \nwould be a unique property of QSOs. High-luminosity objects would be more closely \nconnected with their host properties.}\n\\end{enumerate}\n\nAny of these outcomes would teach us something useful.\n\n\\begin{figure}\n\\scalebox{0.675}{\\includegraphics{ofp2.eps}}\n\\scalebox{0.675}{\\includegraphics{xfp2.eps}}\n\\caption{\nViews of the optical (left) and x-ray (right) QSO\nfundamental planes, showing the individual QSOs (points) and the plane fitted to the overall sample. The host properties are the horizontal axes, while nuclear luminosity is vertical. \nThe shading of the planes is proportional to nuclear luminosity, ranging from black (faint) to white (bright). \nNote that only those points lying above the plane are visible here. \nIn the axis labels, ``M'' is $M_V\\mathrm{(nuc)}$, ``X'' is $\\log L_X$, \n``$\\mu$'' is $\\mu_e$, and ``r'' is $\\log r_{1\/2}$.\n}\n\\label{fig:fp-phys}\n\\end{figure}\n\n\n\\begin{figure}\n\\scalebox{0.34}{\\includegraphics{fp-errors-rx.eps}}\n\\scalebox{0.34}{\\includegraphics{fp-errors-ro.eps}}\n\\caption{Overall QSO fundamental plane (vertical\naxis), plotted against the measured host galaxy size ($\\log r_{1\/2}$, horizontal\naxis). Points on the diagonal line show perfect correspondence. \nThe left figure uses the QSO fundamental plane in its optical form, while the right\nfigure uses the x-ray form. The QSO fundamental plane is most precise when solved for the host size.}\n\\label{fig:fp-rms}\n\\end{figure}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThree types of fourth order Painlev\\'{e} type ordinary differential equations have been studied \\cite{FS,NY1,S}.\nThey are extensions of the Painlev\\'{e} equations $P_{\\rm{II}},\\ldots,P_{\\rm{VI}}$ and expressed as Hamiltonian systems\n\\[\n\t\\mathcal{H}^{X_n^{(1)}}:\\quad\n\t\\frac{dq_i}{dt} = \\frac{\\partial H^{X_n^{(1)}}}{\\partial p_i},\\quad\n\t\\frac{dp_i}{dt} = -\\frac{\\partial H^{X_n^{(1)}}}{\\partial q_i}\\quad\n\t(i=1,2),\n\\]\nwith the Coupled Hamiltonians\n\\[\\begin{split}\n\tH^{A_4^{(1)}} &= H_{\\rm{IV}}(q_1,p_1;\\alpha_2,\\alpha_1)\n\t+ H_{\\rm{IV}}(q_2,p_2;\\alpha_4,\\alpha_1+\\alpha_3) + 2q_1p_1p_2,\\\\\n\ttH^{A_5^{(1)}} &= H_{\\rm{V}}(q_1,p_1;\\alpha_2,\\alpha_1,\\alpha_1+\\alpha_3)\\\\\n\t&\\quad + H_{\\rm{V}}(q_2,p_2;\\alpha_4,\\alpha_1+\\alpha_3,\\alpha_1+\\alpha_3)\n\t+ 2q_1p_1(q_2-1)p_2,\\\\\n\tt(t-1)H^{D_6^{(1)}} &= H_{\\rm{VI}}(q_1,p_1;\\alpha_0,\\alpha_3+\\alpha_5,\n\t\\alpha_3+\\alpha_6,\\alpha_2(\\alpha_1+\\alpha_2))\\\\\n\t&\\quad + H_{\\rm{VI}}(q_2,p_2;\\alpha_0+\\alpha_3,\\alpha_5,\\alpha_6,\n\t\\alpha_4(\\alpha_1+2\\alpha_2+\\alpha_3+\\alpha_4))\\\\\n\t&\\quad + 2(q_1-t)p_1q_2\\{(q_2-1)p_2+\\alpha_4\\},\n\\end{split}\\]\nwhere\n\\[\\begin{split}\n\tH_{\\rm{IV}}(q,p;a,b) &= qp(p-q-t) - aq - bp,\\\\\n\tH_{\\rm{V}}(q,p;a,b,c) &= q(q-1)p(p+t) + atq + bp - cqp,\\\\\n\tH_{\\rm{VI}}(q,p;a,b,c,d) &= q(q-1)(q-t)p^2 - \\{(a-1)q(q-1)\\\\\n\t&\\quad +bq(q-t)+c(q-1)(q-t)\\}p + dq.\n\\end{split}\\]\nBut complete classification of fourth order Painlev\\'{e} systems is not achieved, so that the existence of unknown ones is expected.\nIn this article, we derive a class of fourth order Painlev\\'{e} systems from the Drinfeld-Sokolov hierarchies of type $A_n^{(1)}$ by similarity reductions.\n\nThe Drinfeld-Sokolov hierarchies are extensions of the KdV (or mKdV) hierarchy for the affine Lie algebras \\cite{DS}.\nFor type $A_n^{(1)}$, they imply several Painlev\\'{e} systems by similarity reductions \\cite{AS,KIK,KK1,KK2,NY1}; {\\it see Table 1}.\\DSPainleveKnown\nSuch fact clarifies the origines of several properties of the Painlev\\'{e} systems, Lax pairs, affine Weyl group symmetries and particular solutions in terms of the Schur polynomials.\n\nThe Drinfeld-Sokolov hierarchies are characterized by the Heisenberg subalgebras, that is maximal nilpotent subalgebras, of the affine Lie algebras.\nAnd the isomorphism classes of the Heisenberg subalgebras are in one-to-one correspondence with the conjugacy classes of the finite Weyl group {\\rm\\cite{KP}}.\nIn this article, we choose the {\\it regular} conjugacy classes of $W(A_n)$ and consider their associated hierarchies, called {\\it type I hierarchies} \\cite{GHM}.\nIn the notation of \\cite{DF}, the regular conjugacy classes of $W(A_n)$ correspond to the partitions $(p,\\ldots,p)$ and $(p,\\ldots,p,1)$.\nFor the derivation of fourth order Painlev\\'{e} systems, we investigate the partitions $(2,2)$, $(3,1)$, $(4,1)$, $(2,2,1)$ and $(3,3)$; {\\it see Table 2}.\\DSPainleveResult\n\nOne of impotant results in this article is the derivation of a new Painlev\\'{e} system.\nIt is expressed as a Hamiltonian system\n\\begin{equation}\\label{Eq:CP6}\n\t\\frac{dq_i}{dt} = \\frac{\\partial H_c}{\\partial p_i},\\quad\n\t\\frac{dp_i}{dt} = -\\frac{\\partial H_c}{\\partial q_i}\\quad (i=1,2),\n\\end{equation}\nwith a Coupled Hamiltonian\n\\begin{equation}\\begin{split}\\label{Eq:CP6_Ham}\n\tt(t-1)H_c &= H_{\\rm{VI}}(q_1,p_1;\\alpha_2,\\alpha_0+\\alpha_4,\n\t\\alpha_3+\\alpha_5-\\eta,\\eta\\alpha_1)\\\\\n\t&\\quad + H_{\\rm{VI}}(q_2,p_2;\\alpha_0+\\alpha_2,\\alpha_4,\n\t\\alpha_1+\\alpha_3-\\eta,\\eta\\alpha_5)\\\\\n\t&\\quad\n\t+ (q_1-t)(q_2-1)\\left\\{(q_1p_1+\\alpha_1)p_2+p_1(p_2q_2+\\alpha_5)\\right\\}.\n\\end{split}\\end{equation}\nThis system admits affine Weyl group symmetry of type $A_5^{(1)}$; see Appendix \\ref{Sec:AffWey}.\nOn the other hand, the system $\\mathcal{H}^{D_6^{(1)}}$ admits one of type $D_6^{(1)}$.\nThe relation between those two coupled Painlev\\'{e} VI systems is not clarified.\n\n\\begin{rem}\nFor the partition $(1,\\ldots,1)$ of $n+2$, we have the Garnier system in $n$-variables {\\rm\\cite{KK2}}.\nAlso for each partition $(5,1)$ and $(2,2,2)$, a system of sixth order is derived{\\rm;} we do not give the explicit formula here.\nThus we conjecture that any more fourth order Painlev\\'{e} system do not arise from the type I hierarchy.\n\\end{rem}\n\nThis article is organized as follows.\nIn Section \\ref{Sec:AffLie}, we recall the affine Lie algebra of type $A^{(1)}_n$ and realize it in a flamework of a central extension of the loop algebra $\\mathfrak{sl}_{n+1}[z,z^{-1}]$.\nIn Section \\ref{Sec:Heisenberg}, the Heisenberg subalgebra of $\\widehat{\\mathfrak{sl}}_{n+1}$ corresponding to the partition $\\mathbf{n}$ is introduced.\nIn Section \\ref{Sec:D-S}, we formulate the Drinfeld-Sokolov hierarchies and their similarity reductions.\nIn Section \\ref{Sec:Deri_CP6} and \\ref{Sec:Deri_Others}, the Painlev\\'{e} systems are derived from the Drinfeld-Sokolov hierarchies.\nIn Appendix \\ref{Sec:Lax}, we give explicit descriptions of Lax pairs by means of a bases of $\\widehat{\\mathfrak{sl}}_{n+1}$.\nIn Appendix \\ref{Sec:AffWey}, we discuss a group of symmetries for the system \\eqref{Eq:CP6} with \\eqref{Eq:CP6_Ham}.\n\n\n\\section{Affine Lie algebra}\\label{Sec:AffLie}\n\nIn this section, we recall the affine Lie algebra of type $A^{(1)}_n$ and realize it in a flamework of a central extension of the loop algebra $\\mathfrak{sl}_{n+1}[z,z^{-1}]$.\n\nIn the notation of \\cite{Kac}, the affine Lie algebra $\\mathfrak{g}=\\mathfrak{g}(A^{(1)}_n)$ is generated by the Chevalley generators $e_i,f_i,\\alpha_i^{\\vee}$ $(i=0,\\ldots,n)$ and the scaling element $d$ with the fundamental relations\n\\[\\begin{split}\n\t&(\\mathrm{ad}e_i)^{1-a_{i,j}}(e_j)=0,\\quad\n\t(\\mathrm{ad}f_i)^{1-a_{i,j}}(f_j)=0\\quad (i\\neq j),\\\\\n\t&[\\alpha_i^{\\vee},\\alpha_j^{\\vee}]=0,\\quad\n\t[\\alpha_i^{\\vee},e_j]=a_{i,j}e_j,\\quad\n\t[\\alpha_i^{\\vee},f_j]=-a_{i,j}f_j,\\quad\n\t[e_i,f_j]=\\delta_{i,j}\\alpha_i^{\\vee},\\\\\n\t&[d,\\alpha_i^{\\vee}]=0,\\quad [d,e_i]=\\delta_{i,0}e_0,\\quad\n\t[d,f_i]=-\\delta_{i,0}f_0,\n\\end{split}\\]\nfor $i,j=0,\\ldots,n$.\nThe generalized Cartan matrix $A=\\left[a_{i,j}\\right]_{i,j=0}^{n}$ for $\\mathfrak{g}$ is defined by\n\\[\\begin{array}{llll}\n\ta_{i,i}=2& (i=0,\\ldots,n),\\\\[4pt]\n\ta_{i,i+1}=a_{n,0}=a_{i+1,i}=a_{0,n}=-1& (i=0,\\ldots,n-1),\\\\[4pt]\n\ta_{i,j}=0& (\\text{otherwise}).\n\\end{array}\\]\nWe denote the Cartan subalgebra of $\\mathfrak{g}$ by\n\\[\n\t\\mathfrak{h} = \\mathbb{C}\\alpha_0^{\\vee}\\oplus\\mathbb{C}\\alpha_1^{\\vee}\n\t\\oplus\\cdots\\oplus\\mathbb{C}\\alpha_n^{\\vee}\\oplus\\mathbb{C}d\n\t= \\mathfrak{h}'\\oplus\\mathbb{C}d.\n\\]\nThe normalized invariant form $(\\cdot|\\cdot):\\mathfrak{g}\\times\\mathfrak{g}\\to\\mathbb{C}$ is determined by the conditions\n\\[\\begin{array}{lll}\n\t(\\alpha_i^{\\vee}|\\alpha_j^{\\vee}) = a_{i,j},& (e_i|f_j) = \\delta_{i,j},&\n\t(\\alpha_i^{\\vee}|e_j) = (\\alpha_i^{\\vee}|f_j) = 0,\\\\[4pt]\n\t(d|d) = 0,& (d|\\alpha_j^{\\vee}) = \\delta_{0,j},& (d|e_j) = (d|f_j) = 0,\n\\end{array}\\]\nfor $i,j=0,\\ldots,n$.\n\nLet $\\mathfrak{n}_{+}$ and $\\mathfrak{n}_{-}$ be the subalgebras of $\\mathfrak{g}$ generated by $e_i$ and $f_i$ $(i=0,\\ldots,n)$ respectively.\nThen the Borel subalgebra $\\mathfrak{b}_{+}$ of $\\mathfrak{g}$ is defined by $\\mathfrak{b}_{+}=\\mathfrak{h}\\oplus\\mathfrak{n}_{+}$.\nNote that we have the triangular decomposition\n\\[\n\t\\mathfrak{g} = \\mathfrak{n}_{-}\\oplus\\mathfrak{h}\\oplus\\mathfrak{n}_{+}\n\t= \\mathfrak{n}_{-}\\oplus\\mathfrak{b}_{+}.\n\\]\nThe corresponding infinite demensional groups are defined by\n\\[\n\tN_{\\pm} = \\exp(\\mathfrak{n}_{\\pm}^*),\\quad H = \\exp(\\mathfrak{h}'),\\quad\n\tB_{+} = HN_{+},\n\\]\nwhere $\\mathfrak{n}_{\\pm}^*$ are completions of $\\mathfrak{n}_{\\pm}$ respectively.\n\nLet $\\mathbf{s}=(s_0,\\ldots,s_n)$ be a vector of non-negative integers.\nWe consider a gradation $\\mathfrak{g}=\\bigoplus_{k\\in\\mathbb{Z}}\\mathfrak{g}_k(\\mathbf{s})$ of type $\\mathbf{s}$ by setting\n\\[\n\t\\deg\\mathfrak{h}=0,\\quad \\deg e_i=s_i,\\quad \\deg f_i=-s_i\\quad\n\t(i=0,\\ldots,n).\n\\]\nWith an element $\\vartheta(\\mathbf{s})\\in\\mathfrak{h}$ such that\n\\[\n\t(\\vartheta(\\mathbf{s})|\\alpha_i^{\\vee}) = s_i\\quad (i=0,\\ldots,n),\n\\]\nthis gradation is defined by\n\\[\n\t\\mathfrak{g}_k(\\mathbf{s}) = \\left\\{x\\in\\mathfrak{g}\\bigm|\n\t[\\vartheta(\\mathbf{s}),x]=kx\\right\\}\\quad (k\\in\\mathbb{Z}).\n\\]\nWe denote by\n\\[\n\t\\mathfrak{g}_{n_{r+1}=\\ldots=n_s=1$.\nConsider a partition of matrix corresponding to $\\mathbf{n}$\n\\[\n\t\\begin{bmatrix}\n\t\tB_{11}& B_{12}& \\cdots& B_{1s}\\\\\n\t\tB_{21}& B_{22}& \\cdots& B_{2s}\\\\\n\t\t\\vdots& \\vdots& \\ddots& \\vdots\\\\\n\t\tB_{s1}& B_{s2}& \\cdots& B_{ss}\n\t\\end{bmatrix},\n\\]\nwhere each block $B_{ij}$ is an $n_i\\times n_j$-matrix.\nWith this blockform, we define matricies $\\Lambda_i'\\in\\widehat{\\mathfrak{sl}}_{n+1}$ $(i=1,\\ldots,r)$ by\n\\[\n\t\\Lambda_i' = \\begin{bmatrix}\n\t\tO& & \\cdots& & O\\\\ & & & & \\\\ \\vdots& & B_{ii}& & \\vdots\\\\\n\t\t& & & & \\\\ O& & \\cdots& & O\n\t\\end{bmatrix},\\quad\n\tB_{ii} = \\begin{bmatrix}\n\t\t0& 1& 0& \\cdots& 0\\\\ 0& 0& 1& & 0\\\\ \\vdots& \\vdots& & \\ddots& \\\\\n\t\t0& 0& 0& & 1\\\\ z& 0& 0& \\cdots& 0\n\t\\end{bmatrix},\n\\]\ndiagonal matricies $H_j'\\in\\widehat{\\mathfrak{sl}}_{n+1}$ $(i=j,\\ldots,s-1)$ by\n\\[\n\tH_j' = n_{j+1}z^{-1}(\\Lambda_j')^{n_j}\n\t- n_jz^{-1}(\\Lambda_{j+1}')^{n_{j+1}},\n\\]\nand a diagonal matrix $\\eta_{\\mathbf{n}}'\\in\\widehat{\\mathfrak{sl}}_{n+1}$ by\n\\[\n\tB_{ii} = \\frac{1}{2n_i}\\mathrm{diag}(n_i-1,n_i-3,\\ldots,-n_i+1)\\quad\n\t(i=1,\\ldots,r).\n\\]\n\nDenoting the matrix $\\eta_{\\mathbf{n}}'$ by $\\mathrm{diag}(\\eta_1',\\eta_2',\\ldots,\\eta_{n+1}')$, we consider a permutation\n\\[\n\t\\sigma = \\left(\\begin{array}{llll}\n\t\t\\eta_1'& \\eta_2'& \\ldots& \\eta_{n+1}'\\\\[4pt]\n\t\t\\eta_1& \\eta_2& \\ldots& \\eta_{n+1}\n\t\\end{array}\\right),\n\\]\nsuch that $\\eta_1\\geq\\eta_2\\geq\\ldots\\geq\\eta_{n+1}$.\nThis permutation can be lifted to the transformation $\\sigma$ acting on the matricies $\\Lambda_i'$ and $H_j'$.\nWe set\n\\[\n\t\\Lambda_i = \\sigma(\\Lambda_i')\\quad (i=1,\\ldots,r),\\quad\n\tH_j = \\sigma(H_j')\\quad (j=1,\\ldots,s-1).\n\\]\nThen the Heisenberg subalgebra of $\\widehat{\\mathfrak{sl}}_{n+1}$ corresponding to the partition $\\mathbf{n}$ is defined by\n\\[\n\t\\mathfrak{s}_\\mathbf{n} = \\bigoplus_{i=1}^{r}\n\t\\bigoplus_{k\\in\\mathbb{Z}\\setminus n_i\\mathbb{Z}}\\mathbb{C}\\Lambda_i^k\n\t\\oplus\\bigoplus_{j=1}^{s-1}\\bigoplus_{k\\in\\mathbb{Z}\\setminus\\{0\\}}\n\t\\mathbb{C}z^kH_j\\oplus\\mathbb{C}K.\n\\]\n\nLet $N_{\\mathbf{n}}'$ be the least common multiple of $n_1,\\ldots,n_s$.\nAlso let\n\\[\n\tN_{\\mathbf{n}} = \\left\\{\\begin{array}{ll}\n\t\tN_{\\mathbf{n}}'&\n\t\t\\text{if $\\displaystyle N_{\\mathbf{n}}'\\left(\\frac{1}{n_i}+\\frac{1}{n_j}\\right)\n\t\t\\in2\\mathbb{Z}$ for $\\forall(i,j)$}\\\\[12pt]\n\t\t2N_{\\mathbf{n}}'& \\text{otherwise}\n\t\\end{array}\\right..\n\\]\nWe consider a operator corresponding to $\\mathbf{n}$\n\\[\n\t\\vartheta_{\\mathbf{n}}\n\t= N_{\\mathbf{n}}\\left(z\\frac{d}{dz}+\\mathrm{ad}\\eta_{\\mathbf{n}}\\right),\n\\]\nwhere $\\eta_{\\mathbf{n}}=\\sigma(\\eta_{\\mathbf{n}}')$.\nThen the operator $\\vartheta_{\\mathbf{n}}$ implies a gradation $\\mathbf{s}=(s_0,\\ldots,s_n)$ as follows:\n\\[\n\t\\vartheta_{\\mathbf{n}}(e_i) = s_ie_i\\quad (i=0,\\ldots,n).\n\\]\nNote that the Heisenberg subalgebra $\\mathfrak{s}_\\mathbf{n}$ admits the gradation $\\mathbf{s}$ defined by $\\vartheta_{\\mathbf{n}}$.\n\n\n\\section{Drinfeld-Sokolov hierarchy}\\label{Sec:D-S}\n\nIn this section, we formulate the Drinfeld-Sokolov hierarchy associated with the Heisenberg subalgebra $\\mathfrak{s}_\\mathbf{n}$.\nIts similarity reduction is also formulated.\n\nLet $\\Lambda_i$ and $H_j$ be the generators for $\\mathfrak{s}_\\mathbf{n}$ given in Section \\ref{Sec:Heisenberg}.\nIntroducing time variables $t_{i,k}$ $(i=1,\\ldots,r;k\\in\\mathbb{N})$, we consider an $N_{-}B_{+}$-valued function $G=G(t_{1,1},t_{1,2},\\ldots)$ defined by\n\\[\n\tG = \\exp\\left(\\sum_{i=1}^r\\sum_{k=1}^{\\infty}t_{i,k}\\Lambda_i^k\\right)G(0).\n\\]\nHere we assume the $\\mathbf{n}$-reduced condition\n\\[\n\tt_{i,l}=0\\quad (i=1,\\ldots,r;l\\in n_i\\mathbb{N}).\n\\]\nThen we have a system of partial differential equations\n\\begin{equation}\\label{Eq:DS_exp}\n\t\\partial_{i,k}(G) = \\Lambda_i^kG\\quad (i=1,\\ldots,r;k\\in\\mathbb{N}),\n\\end{equation}\nwhere $\\partial_{i,k}=\\partial\/\\partial t_{i,k}$\nVia the trianglar decomposition\n\\[\n\tG = W^{-1}Z,\\quad W\\in N_{-},\\quad Z\\in B_{+},\n\\]\nthe system \\eqref{Eq:DS_exp} implies {\\it a Sato equation}\n\\begin{equation}\\label{Eq:Sato}\n\t\\partial_{i,k}(W) = B_{i,k}W - W\\Lambda_i^k\\quad\n\t(i=1,\\ldots,r;k\\in\\mathbb{N}),\n\\end{equation}\nwhere $B_{i,k}$ stands for the $b_{+}$-component of $W\\Lambda_i^kW^{-1}$.\nThe compatibility condition of \\eqref{Eq:Sato} gives the Drinfeld-Sokolov hierarchy\n\\begin{equation}\\label{Eq:DS}\n\t\\left[\\partial_{i,k}-B_{i,k},\\partial_{j,l}-B_{j,l}\\right] = 0\\quad\n\t(i,j=1,\\ldots,r;k,l\\in\\mathbb{N}).\n\\end{equation}\n\nUnder the system \\eqref{Eq:Sato}, we consider an equation\n\\begin{equation}\\label{Eq:Sato_SR}\n\t(\\vartheta_{\\mathbf{n}}-\\mathrm{ad}\\rho)(W)\n\t= \\sum_{i=1}^r\\sum_{k=1}^{\\infty}d_ikt_{i,k}\\partial_{i,k}(W),\n\\end{equation}\nwhere $d_i=\\deg\\Lambda_i$ $(i=1,\\ldots,r)$ and $\\rho=\\sum_{j=1}^{s-1}\\rho_jH_j$.\nNote that each $\\rho_j$ is independent of time vatiables $t_{i,k}$.\nThe compatibility condition of \\eqref{Eq:Sato} and \\eqref{Eq:Sato_SR} gives\n\\begin{equation}\\label{Eq:DS_SR}\n\t\\left[\\vartheta_{\\mathbf{n}}-M,\\partial_{i,k}-B_{i,k}\\right] = 0\\quad\n\t(i=1,\\ldots,r;k\\in\\mathbb{N}),\n\\end{equation}\nwhere\n\\[\n\tM = \\rho + \\sum_{i=1}^r\\sum_{k=1}^{\\infty}d_ikt_{i,k}B_{i,k}.\n\\]\nWe call the systems \\eqref{Eq:DS} and \\eqref{Eq:DS_SR} a similarity reduction of the Drinfeld-Sokolov hierarchy.\n\n\\begin{rem}\\label{Rem:Lax}\nThe similarity reduction can be regarded as the compatibility condition of a Lax form\n\\[\n\t\\partial_{i,k}(\\Psi) = B_{i,k}\\Psi\\quad (i=1,\\ldots,r;k\\in\\mathbb{N}),\\quad\n\t\\vartheta_{\\mathbf{n}}(\\Psi) = M\\Psi.\n\\]\nHere an $N_{-}B_{+}$-valued function $\\Psi$ is given by\n\\[\n\t\\Psi = W\\exp\\left(\\sum_{i=1}^r\\sum_{k=1}^{\\infty}t_{i,k}\\Lambda_i^k\\right).\n\\]\n\\end{rem}\n\n\n\\section{Derivation of Coupled $P_{\\rm{VI}}$}\\label{Sec:Deri_CP6}\n\nIn this section, we derive the Painlev\\'{e} system \\eqref{Eq:CP6} with \\eqref{Eq:CP6_Ham} from the Drinfeld-Sokolov hierarchies for $\\mathfrak{s}_{(3,3)}$ and $\\mathfrak{s}_{(2,2,1)}$ by similarity reductions.\n\n\n\\subsection{For the partition $(3,3)$}\\label{Sec:System33}\n\nAt first, we define the Heisenberg subalgebra $\\mathfrak{s}_{(3,3)}$ of $\\mathfrak{g}(A^{(1)}_5)$.\nLet\n\\[\n\t\\Lambda_1 = e_{1,2} + e_{3,4} + e_{5,0},\\quad\n\t\\Lambda_2 = e_{0,1} + e_{2,3} + e_{4,5},\\quad\n\tH_1 = \\alpha^{\\vee}_1 + \\alpha^{\\vee}_3 + \\alpha^{\\vee}_5,\n\\]\nwhere\n\\[\n\te_{i_1,i_2,\\ldots,i_{n-1},i_n} = \\mathrm{ad}e_{i_1}\\mathrm{ad}e_{i_2}\n\t\\ldots\\mathrm{ad}e_{i_{n-1}}(e_{i_n}).\n\\]\nThen we have\n\\[\n\t\\mathfrak{s}_{(3,3)} = \\bigoplus_{k\\in\\mathbb{Z}\\setminus3\\mathbb{Z}}\n\t\\mathbb{C}\\Lambda_1^k\\oplus\\bigoplus_{k\\in\\mathbb{Z}\\setminus3\\mathbb{Z}}\n\t\\mathbb{C}\\Lambda_2^k\\oplus\\bigoplus_{k\\in\\mathbb{Z}\\setminus\\{0\\}}\n\t\\mathbb{C}z^kH_1\\oplus\\mathbb{C}K.\n\\]\nThe grade operator for $\\mathfrak{s}_{(3,3)}$ is given by\n\\[\n\t\\vartheta_{(3,3)} = 3\\left(z\\frac{d}{dz}+\\mathrm{ad}\\eta_{(3,3)}\\right),\n\\]\nwhere\n\\[\n\t\\eta_{(3,3)} = \\frac{1}{3}(\\alpha^{\\vee}_1+2\\alpha^{\\vee}_2+2\\alpha^{\\vee}_3\n\t+2\\alpha^{\\vee}_4+\\alpha^{\\vee}_5).\n\\]\nIt follows that $\\mathfrak{s}_{(3,3)}$ admits the gradation of type $\\mathbf{s}=(1,0,1,0,1,0)$, namely\n\\[\n\t\\vartheta_{(3,3)}(e_i) = e_i\\quad (i=0,2,4),\\quad\n\t\\vartheta_{(3,3)}(e_j) = 0\\quad (j=1,3,5).\n\\]\nNote that\n\\[\n\t\\mathfrak{g}_{\\geq0}(1,0,1,0,1,0) = \\mathbb{C}f_1\\oplus\\mathbb{C}f_3\n\t\\oplus\\mathbb{C}f_5\\oplus\\mathfrak{b}_{+}.\n\\]\n\nWe now assume $t_{2,1}=1$ and $t_{1,k}=t_{2,k}=0$ $(k\\geq2)$.\nThen the similarity reduction \\eqref{Eq:DS} and \\eqref{Eq:DS_SR} for $\\mathfrak{s}_{(3,3)}$ is expressed as\n\\begin{equation}\\label{Eq:DS_SR_33_b}\n\t\\left[\\vartheta_{(3,3)}-M,\\partial_{1,1}-B_{1,1}\\right] = 0.\n\\end{equation}\nHere the $\\mathfrak{b}_{+}$-valued functions $M$ and $B_{1,1}$ are defined by\n\\begin{equation}\\begin{split}\\label{Eq:DS_SR_33_b_BM}\n\tM &= \\vartheta_{(3,3)}(W)W^{-1}\n\t+ W(\\rho_1H_1+t_{1,1}\\Lambda_1+\\Lambda_2)W^{-1},\\\\\n\tB_{1,1} &= \\partial_{1,1}(W)W^{-1} + W\\Lambda_1W^{-1},\n\\end{split}\\end{equation}\nwhere $W$ is an $N_{-}$-valued function; its explicit formula is given below.\nIn the following, we derive the Painlev\\'{e} system from the system \\eqref{Eq:DS_SR_33_b} with \\eqref{Eq:DS_SR_33_b_BM}.\n\nWe denote by\n\\[\n\tW = \\exp(\\omega_0)\\exp(\\omega_{-1})\\exp(\\omega_{<-1}),\n\\]\nwhere\n\\[\\begin{split}\n\t\\omega_0 &= -w_1f_1 - w_3f_3 - w_5f_5,\\\\\n\t\\omega_{-1} &= -w_0f_0 - w_2f_2 - w_4f_4 - w_{0,1}f_{0,1} - w_{1,2}f_{1,2}\n\t- w_{2,3}f_{2,3} - w_{3,4}f_{3,4}\\\\\n\t&\\quad - w_{4,5}f_{4,5} - w_{5,0}f_{5,0} - w_{1,2,3}f_{1,2,3}\n\t- w_{3,4,5}f_{3,4,5} - w_{5,0,1}f_{5,0,1},\n\\end{split}\\]\nand $\\omega_{<-1}\\in\\mathfrak{g}_{<-1}(1,0,1,0,1,0)$.\nThen the $\\mathfrak{b}_{+}$-valued function $M$ is described as\n\\[\\begin{split}\n\tM &= \\kappa_0\\alpha^{\\vee}_0 + \\kappa_1\\alpha^{\\vee}_1\n\t+ \\kappa_2\\alpha^{\\vee}_2 + \\kappa_3\\alpha^{\\vee}_3\n\t+ \\kappa_4\\alpha^{\\vee}_4 + \\kappa_5\\alpha^{\\vee}_5 - (t_{1,1}w_5-w_1)e_0\n\t+ \\varphi_1e_1\\\\\n\t&\\quad - (t_{1,1}w_1-w_3)e_2 + \\varphi_3e_3 - (t_{1,1}w_3-w_5)e_4\n\t+ \\varphi_5e_5 + t_{1,1}\\Lambda_1 + \\Lambda_2,\\\\\n\\end{split}\\]\nwith dependent variables\n\\[\n\t\\varphi_1 = t_{1,1}w_2 - w_0,\\quad \\varphi_3= t_{1,1}w_4 - w_2,\\quad\n\t\\varphi_5 = t_{1,1}w_0 - w_4,\n\\]\nand parameters\n\\[\\begin{split}\n\t&\\kappa_0 = -t_{1,1}w_{5,0} - w_{0,1},\\quad\n\t\\kappa_1 = t_{1,1}(w_1w_2-w_{1,2}) - (w_0w_1+w_{0,1}) + \\rho_1,\\\\\n\t&\\kappa_2 = -t_{1,1}w_{1,2} - w_{2,3},\\quad\n\t\\kappa_3 = t_{1,1}(w_3w_4-w_{3,4}) - (w_2w_3+w_{2,3}) + \\rho_1,\\\\\n\t&\\kappa_4 = -t_{1,1}w_{3,4} - w_{4,5},\\quad\n\t\\kappa_5 = t_{1,1}(w_0w_5-w_{5,0}) - (w_4w_5+w_{4,5}) + \\rho_1.\n\\end{split}\\]\nNote that\n\\[\n\t\\partial_{1,1}(\\kappa_i) = 0\\quad (i=0,\\ldots,5).\n\\]\nWe also remark that\n\\[\n\tw_1\\varphi_1 + w_3\\varphi_3 + w_5\\varphi_5 + \\kappa_0 - \\kappa_1\n\t+ \\kappa_2 - \\kappa_3 + \\kappa_4 - \\kappa_5 + 3\\rho_1 = 0.\n\\]\nThe $\\mathfrak{b}_{+}$-valued function $B_{1,1}$ is described as\n\\[\\begin{split}\n\tB_{1,1} &= u_0K + (u_1+w_1x_1)\\alpha^{\\vee}_1 + u_2\\alpha^{\\vee}_2\n\t+ (u_3+w_3x_3)\\alpha^{\\vee}_3 + u_4\\alpha^{\\vee}_4\\\\\n\t&\\quad + w_5x_5\\alpha^{\\vee}_5 - w_5e_0 + x_1e_1 - w_1e_2 + x_3e_3\n\t- w_3e_4 + x_5e_5 + \\Lambda_1,\n\\end{split}\\]\nwhere\n\\[\\begin{split}\n\t&u_1 = \\frac{-2w_1\\varphi_1+w_3\\varphi_3+w_5\\varphi_5\n\t-2\\kappa_0+2\\kappa_1+\\kappa_2-\\kappa_3+\\kappa_4-\\kappa_5}{3t_{1,1}},\\\\\n\t&u_2 = -\\frac{w_1\\varphi_1+\\kappa_0-\\kappa_1+\\rho_1}{t_{1,1}},\\\\\n\t&u_3 = \\frac{-w_1\\varphi_1-w_3\\varphi_3+2w_5\\varphi_5\n\t-\\kappa_0+\\kappa_1-\\kappa_2+\\kappa_3+2\\kappa_4-2\\kappa_5}{3t_{1,1}},\\\\\n\t&u_4 = \\frac{w_5\\varphi_5+\\kappa_4-\\kappa_5+\\rho_1}{t_{1,1}},\\quad\n\tx_1 = \\frac{t_{1,1}^2\\varphi_1+t_{1,1}\\varphi_5+\\varphi_3}{t_{1,1}^3-1},\\\\\n\t&x_3 = \\frac{t_{1,1}^2\\varphi_3+t_{1,1}\\varphi_1+\\varphi_5}\n\t{t_{1,1}^3-1},\\quad\n\tx_5 = \\frac{t_{1,1}^2\\varphi_5+t_{1,1}\\varphi_3+\\varphi_1}{t_{1,1}^3-1}.\n\\end{split}\\]\nHence the system \\eqref{Eq:DS_SR_33_b} with \\eqref{Eq:DS_SR_33_b_BM} can be expressed as a system of ordinary differential equations in terms of the variabes $\\varphi_1,\\varphi_5,w_1,w_3,w_5$; we do not give its explicit formula.\n\nLet\n\\[\n\tq_1 = \\frac{w_1}{t_{1,1}^2w_3},\\quad\n\tp_1 = \\frac{t_{1,1}^2w_3\\varphi_1}{3},\\quad\n\tq_2 = \\frac{w_5}{t_{1,1}w_3},\\quad\n\tp_2 = \\frac{t_{1,1}w_3\\varphi_5}{3},\\quad t = \\frac{1}{t_{1,1}^3}.\n\\]\nWe also set\n\\[\\begin{split}\n\t&\\alpha_0 = \\frac{1}{3}(1-2\\kappa_0+\\kappa_1+\\kappa_5),\\quad\n\t\\alpha_1 = \\frac{1}{3}(\\kappa_0-2\\kappa_1+\\kappa_2),\\\\\n\t&\\alpha_2 = \\frac{1}{3}(1+\\kappa_1-2\\kappa_2+\\kappa_3),\\quad\n\t\\alpha_3 = \\frac{1}{3}(\\kappa_2-2\\kappa_3+\\kappa_4),\\\\\n\t&\\alpha_4 = \\frac{1}{3}(1+\\kappa_3-2\\kappa_4+\\kappa_5),\\quad\n\t\\alpha_5 = \\frac{1}{3}(\\kappa_0+\\kappa_4-2\\kappa_5),\n\\end{split}\\]\nand\n\\[\n\t\\eta = \\rho_1 + \\frac{1}{2}(\\alpha_1+\\alpha_3+\\alpha_5).\n\\]\nThen we have\n\n\\begin{thm}\nThe system \\eqref{Eq:DS_SR_33_b} with \\eqref{Eq:DS_SR_33_b_BM} gives the Painlev\\'{e} system \\eqref{Eq:CP6} with \\eqref{Eq:CP6_Ham}.\nFurthermore, $w_3$ satisfies the completely integrable Pfaffian equation\n\\[\\begin{split}\n\tt(t-1)\\frac{d}{dt}\\log w_3 &= -(q_1-1)(q_1-t)p_1 - (q_2-1)(q_2-t)p_2\\\\\n\t&\\quad - \\alpha_1q_1 - \\alpha_5q_2\n\t+ \\frac{1}{3}(\\alpha_1+\\alpha_2-\\alpha_3-\\alpha_4+2\\eta)t\\\\\n\t&\\quad - \\frac{1}{3}(\\alpha_1+\\alpha_2+2\\alpha_3-\\alpha_4-4\\eta).\n\\end{split}\\]\n\\end{thm}\n\n\n\\subsection{For the partition $(2,2,1)$}\\label{Sec:System221}\n\nThe Heisenberg subalgebra $\\mathfrak{s}_{(2,2,1)}$ of $\\mathfrak{g}(A^{(1)}_4)$ is defined by\n\\[\n\t\\mathfrak{s}_{(2,2,1)} = \\bigoplus_{k\\in\\mathbb{Z}\\setminus2\\mathbb{Z}}\n\t\\mathbb{C}\\Lambda_1^k\\oplus\\bigoplus_{k\\in\\mathbb{Z}\\setminus2\\mathbb{Z}}\n\t\\mathbb{C}\\Lambda_2^k\\oplus\\bigoplus_{k\\in\\mathbb{Z}\\setminus\\{0\\}}\n\t\\mathbb{C}z^kH_1\\oplus\\bigoplus_{k\\in\\mathbb{Z}\\setminus\\{0\\}}\n\t\\mathbb{C}z^kH_2\\oplus\\mathbb{C}K,\n\\]\nwith\n\\[\\begin{array}{ll}\n\t\\Lambda_1 = e_{4,0} + e_{1,2,3},& \\Lambda_2 = e_{0,1} + e_{2,3,4},\\\\[4pt]\n\tH_1 = \\alpha^{\\vee}_1 + \\alpha^{\\vee}_2 - \\alpha^{\\vee}_3,&\n\tH_2 = -\\alpha^{\\vee}_2 + \\alpha^{\\vee}_3 + \\alpha^{\\vee}_4.\n\\end{array}\\]\nThe subalgebra $\\mathfrak{s}_{(2,2,1)}$ admits the gradation of type $\\mathbf{s}=(2,0,1,1,0)$ with the grade operator\n\\[\n\t\\vartheta_{(2,2,1)}\n\t= 4\\left(z\\frac{d}{dz}+\\mathrm{ad}\\eta_{(2,2,1)}\\right),\\quad\n\t\\eta_{(2,2,1)} = \\frac{1}{4}\n\t(\\alpha^{\\vee}_1+2\\alpha^{\\vee}_2+2\\alpha^{\\vee}_3+\\alpha^{\\vee}_4).\n\\]\nNote that\n\\[\n\t\\mathfrak{g}_{\\geq0}(2,0,1,1,0)\n\t= \\mathbb{C}f_1\\oplus\\mathbb{C}f_4\\oplus\\mathfrak{b}_{+}.\n\\]\n\nWe now assume $t_{1,2}=1$ and $t_{1,k}=t_{2,k}=0$ $(k\\geq3)$.\nThen the similarity reduction \\eqref{Eq:DS_SR} for $\\mathfrak{s}_{(2,2,1)}$ is expressed as\n\\begin{equation}\\label{Eq:DS_SR_221_b}\n\t\\left[\\vartheta_{(2,2,1)}-M,\\partial_{1,1}-B_{1,1}\\right] = 0,\n\\end{equation}\nwith\n\\begin{equation}\\begin{split}\\label{Eq:DS_SR_221_b_BM}\n\tM &= \\vartheta_{(2,2,1)}(W)W^{-1}\n\t+ W(\\rho_1H_1+\\rho_2H_2+2t_{1,1}\\Lambda_1+2\\Lambda_2)W^{-1},\\\\\n\tB_{1,1} &= \\partial_{1,1}(W)W^{-1} + W\\Lambda_1W^{-1}.\n\\end{split}\\end{equation}\n\nLet\n\\[\n\tW = \\exp(\\omega_0)\\exp(\\omega_{-1})\\exp(\\omega_{-2})\\exp(\\omega_{<-2}),\n\\]\nwhere\n\\[\\begin{split}\n\t\\omega_0 &= -w_1f_1 - w_4f_4,\\\\\n\t\\omega_{-1} &= -w_2f_2 - w_3f_3 - w_{1,2}f_{1,2} - w_{3,4}f_{3,4},\\\\\n\t\\omega_{-2} &= -w_0f_0 - w_{0,1}f_{0,1} - w_{2,3}f_{2,3} - w_{4,0}f_{4,0}\\\\\n\t&\\quad - w_{1,2,3}f_{1,2,3} - w_{2,3,4}f_{2,3,4} - w_{4,0,1}f_{4,0,1}\n\t- w_{1,2,3,4}f_{1,2,3,4},\n\\end{split}\\]\nand $\\omega_{<-2}\\in\\mathfrak{g}_{<-2}(2,0,1,1,0)$.\nThen the system \\eqref{Eq:DS_SR_221_b_BM} gives explicit formulas of $M,B_{1,1}$ as follows:\n\\[\\begin{split}\n\tM &= \\kappa_0\\alpha^{\\vee}_0 + \\kappa_1\\alpha^{\\vee}_1\n\t+ \\kappa_2\\alpha^{\\vee}_2 + \\kappa_3\\alpha^{\\vee}_3\n\t+ \\kappa_4\\alpha^{\\vee}_4 + 2(w_1-t_{1,1}w_4)e_0\\\\\n\t&\\quad + \\varphi_1e_1 + (\\varphi_2-w_1\\varphi_{1,2})e_2\n\t+ (\\varphi_3+w_4\\varphi_{3,4})e_3 + \\varphi_4e_4\\\\\n\t&\\quad + \\varphi_{1,2}e_{1,2} + 2(t_{1,1}w_1-w_4)e_{2,3}\n\t- \\varphi_{3,4}e_{3,4} + 2t_{1,1}\\Lambda_1 + 2\\Lambda_2,\\\\\n\tB_{1,1} &= u_0K + (u_2+w_1x_1)\\alpha^{\\vee}_1 + u_2\\alpha^{\\vee}_2\n\t+ u_3\\alpha^{\\vee}_3 + w_4x_4\\alpha^{\\vee}_4 - w_4e_0\\\\\n\t&\\quad + x_1e_1 - w_1x_{1,2}e_2 + \\frac{\\varphi_3}{2t_{1,1}}e_3 + x_4e_4\n\t+ x_{1,2}e_{1,2} - w_1e_{2,3} + \\Lambda_1,\n\\end{split}\\]\nwhere\n\\[\\begin{split}\n\t&\\varphi_1 = -2w_0 + t_{1,1}w_2w_3 - 2t_{1,1}w_{2,3},\\quad\n\t\\varphi_2 = -2w_{3,4},\\quad \\varphi_3 = 2t_{1,1}w_{1,2},\\\\\n\t&\\varphi_4 = 2t_{1,1}w_0 + w_2w_3 + 2w_{2,3},\\quad\n\t\\varphi_{1,2} = 2t_{1,1}w_3,\\quad \\varphi_{3,4} = -2w_2,\n\\end{split}\\]\nand\n\\[\\begin{split}\n\t&u_2 = -\\frac{w_1\\varphi_1+\\kappa_0-\\kappa_1+\\rho_1}{2t_{1,1}},\\quad\n\tu_3 = \\frac{w_4\\varphi_4+\\kappa_3-\\kappa_4+\\rho_1}{2t_{1,1}},\\\\\n\t&x_1 = \\frac{(t_{1,1}\\varphi_1+\\varphi_4)\\varphi_3\n\t+(w_1\\varphi_1+w_4\\varphi_4+\\kappa_0-\\kappa_1+\\kappa_3-\\kappa_4+2\\rho_1)\n\t\\varphi_{3,4}}{2(t_{1,1}^2-1)\\varphi_3},\\\\\n\t&x_4 = \\frac{(\\varphi_1+t_{1,1}\\varphi_4)\\varphi_3+t_{1,1}\n\t(w_1\\varphi_1+w_4\\varphi_4+\\kappa_0-\\kappa_1+\\kappa_3-\\kappa_4+2\\rho_1)\n\t\\varphi_{3,4}}{2(t_{1,1}^2-1)\\varphi_3},\\\\\n\t&x_{1,2} = \\frac{w_1\\varphi_1+w_4\\varphi_4+\\kappa_0-\\kappa_1+\\kappa_3\n\t-\\kappa_4+2\\rho_1}{\\varphi_3}.\n\\end{split}\\]\nNote that $\\kappa_0,\\ldots,\\kappa_4$ are constants.\nWe also remark that\n\\[\\begin{split}\n\t&\\varphi_2\\varphi_{3,4} + 2(w_1\\varphi_1+w_4\\varphi_4+\\kappa_0-\\kappa_1\n\t+\\kappa_2-\\kappa_4+2\\rho_2) = 0,\\\\\n\t&\\varphi_3\\varphi_{1,2} - 2t_{1,1}(w_1\\varphi_1+w_4\\varphi_4+\\kappa_0\n\t-\\kappa_1+\\kappa_3-\\kappa_4+2\\rho_1) = 0.\n\\end{split}\\]\nHence the system \\eqref{Eq:DS_SR_221_b} can be expressed as a system of ordinary differential equations in terms of the variables $\\varphi_1,\\varphi_3,\\varphi_4,\\varphi_{3,4},w_1,w_4$.\n\nLet\n\\[\\begin{split}\n\t&q_1 = -\\frac{t_{1,1}^2\\varphi_{3,4}w_4}{\\varphi_3},\\quad\n\tp_1 = -\\frac{\\varphi_3\\varphi_4}{4t_{1,1}^2\\varphi_{3,4}},\\\\\n\t&q_2 = -\\frac{t_{1,1}\\varphi_{3,4}w_1}{\\varphi_3},\\quad\n\tp_2 = -\\frac{\\varphi_3\\varphi_1}{4t_{1,1}\\varphi_{3,4}},\\quad\n\tt = t_{1,1}^2.\n\\end{split}\\]\nWe also set\n\\[\\begin{split}\n\t&\\alpha_0 = \\frac{1}{4}(2-2\\kappa_0+\\kappa_1+\\kappa_4),\\quad\n\t\\alpha_1 = \\frac{1}{4}(\\kappa_0+\\kappa_3-2\\kappa_4),\\\\\n\t&\\alpha_2 = \\frac{1}{4}(1+\\kappa_2-2\\kappa_3+\\kappa_4),\\quad\n\t\\alpha_3 = \\frac{1}{4}(-\\kappa_2+\\kappa_3+2\\rho_1-2\\rho_2),\\\\\n\t&\\alpha_4 = \\frac{1}{4}(1+\\kappa_1-\\kappa_2-2\\rho_1+2\\rho_2),\\quad\n\t\\alpha_5 = \\frac{1}{4}(\\kappa_0-2\\kappa_1+\\kappa_2),\\\\\n\t&\\eta\n\t= \\frac{1}{4}(2\\kappa_0-2\\kappa_1+2\\kappa_3-2\\kappa_4+3\\rho_1-\\rho_2).\n\\end{split}\\]\nThen we have\n\n\\begin{thm}\nThe system \\eqref{Eq:DS_SR_221_b} with \\eqref{Eq:DS_SR_221_b_BM} gives the Painlev\\'{e} system \\eqref{Eq:CP6} with \\eqref{Eq:CP6_Ham}.\nFurthermore, $\\varphi_3$ and $\\varphi_{3,4}$ satisfy the completely integrable Pfaffian equations\n\\[\\begin{split}\n\tt(t-1)\\frac{d}{dt}\\log\\varphi_3 &= -q_1(q_1-t)p_1 - q_2(q_2-t)p_2\n\t- \\alpha_1q_1 - \\alpha_5q_2\\\\\n\t&\\quad + \\frac{1}{4}\n\t(1+2\\alpha_2-2\\alpha_3-2\\alpha_4-2\\alpha_5+6\\eta)t\\\\\n\t&\\quad - \\frac{1}{4}\n\t(1+2\\alpha_2+2\\alpha_3-2\\alpha_4-2\\alpha_5+2\\eta),\\\\\n\tt(t-1)\\frac{d}{dt}\\log\\varphi_{3,4} &= -(q_1-t)p_1 - (q_2-t)p_2 - \\eta.\n\\end{split}\\]\n\\end{thm}\n\n\n\\section{Derivation of other systems}\\label{Sec:Deri_Others}\n\nIn this section, we discuss the derivation of the Painlev\\'{e} systems for $\\mathfrak{s}_{(2,2)}$, $\\mathfrak{s}_{(3,1)}$ and $\\mathfrak{s}_{(4,1)}$ by a similar manner as in Section \\ref{Sec:Deri_CP6}.\n\n\n\\subsection{For the partition $(2,2)$}\\label{Sec:System22}\n\nThe Heisenberg subalgebra $\\mathfrak{s}_{(2,2)}$ of $\\mathfrak{g}(A^{(1)}_3)$ is defined by\n\\[\n\t\\mathfrak{s}_{(2,2)} = \\bigoplus_{k\\in\\mathbb{Z}\\setminus2\\mathbb{Z}}\n\t\\mathbb{C}\\Lambda_1^k\\oplus\\bigoplus_{k\\in\\mathbb{Z}\\setminus2\\mathbb{Z}}\n\t\\mathbb{C}\\Lambda_2^k\\oplus\\bigoplus_{k\\in\\mathbb{Z}\\setminus\\{0\\}}\n\t\\mathbb{C}z^kH_1\\oplus\\mathbb{C}K,\n\\]\nwith\n\\[\n\t\\Lambda_1 = e_{1,2} + e_{3,0},\\quad\n\t\\Lambda_2 = e_{0,1} + e_{2,3},\\quad\n\tH_1 = \\alpha^{\\vee}_1 + \\alpha^{\\vee}_3.\n\\]\nThe subalgebra $\\mathfrak{s}_{(2,2)}$ admits the gradation of type $\\mathbf{s}=(1,0,1,0)$ with the grade operator\n\\[\n\t\\vartheta_{(2,2)}\n\t= 2\\left(z\\frac{d}{dz}+\\mathrm{ad}\\eta_{(2,2)}\\right),\\quad\n\t\\eta_{(2,2)}\n\t= \\frac{1}{2}(\\alpha^{\\vee}_1+2\\alpha^{\\vee}_2+\\alpha^{\\vee}_3).\n\\]\nNote that\n\\[\n\t\\mathfrak{g}_{\\geq0}(1,0,1,0) = \\mathbb{C}f_1\\oplus\\mathbb{C}f_3\\oplus\\mathfrak{b}_{+}.\n\\]\n\nWe now assume $t_{1,2}=1$ and $t_{1,k}=t_{2,k}=0$ $(k\\geq3)$.\nThen the similarity reduction \\eqref{Eq:DS_SR} for $\\mathfrak{s}_{(2,2)}$ is expressed as\n\\begin{equation}\\label{Eq:DS_SR_22_b}\n\t\\left[\\vartheta_{(2,2)}-M,\\partial_{1,1}-B_{1,1}\\right] = 0,\n\\end{equation}\nwith\n\\begin{equation}\\begin{split}\\label{Eq:DS_SR_22_b_BM}\n\tM &= \\vartheta_{(2,2)}(W)W^{-1}\n\t+ W(\\rho_1H_1+t_{1,1}\\Lambda_1+\\Lambda_2)W^{-1},\\\\\n\tB_{1,1} &= \\partial_{1,1}(W)W^{-1} + W\\Lambda_1W^{-1}.\n\\end{split}\\end{equation}\n\nLet\n\\[\n\tW = \\exp(\\omega_0)\\exp(\\omega_{-1})\\exp(\\omega_{<-1}),\n\\]\nwhere\n\\[\\begin{split}\n\t\\omega_{0} &= -w_1f_1 - w_3f_3,\\\\\n\t\\omega_{-1} &= -w_0f_0 - w_2f_2 - w_{0,2}f_{0,2} - w_{1,2}f_{1,2}\\\\\n\t&\\quad - w_{2,3}f_{2,3} - w_{3,0}f_{3,0} - w_{1,2,3}f_{1,2,3}\n\t- w_{3,0,1}f_{3,0,1},\n\\end{split}\\]\nand $\\omega_{<-1}\\in\\mathfrak{g}_{<-1}(1,0,1,0)$.\nThen the system \\eqref{Eq:DS_SR_22_b_BM} gives explicit formulas of $M,B_{1,1}$ as follows:\n\\[\\begin{split}\n\tM &= \\kappa_0\\alpha^{\\vee}_0 + \\kappa_1\\alpha^{\\vee}_1\n\t+ \\kappa_2\\alpha^{\\vee}_2 + \\kappa_3\\alpha^{\\vee}_3 + (w_1-t_{1,1}w_3)e_0\\\\\n\t&\\quad + \\varphi_1e_1 + (w_3-t_{1,1}w_1)e_2 + \\varphi_3e_3\n\t+ t_{1,1}\\Lambda_1 + \\Lambda_2,\\\\\n\tB_{1,1} &= u_0K + u_1\\alpha^{\\vee}_1 + u_2\\alpha^{\\vee}_2\n\t+ w_3x_3\\alpha^{\\vee}_3 + w_1e_0 + x_1e_1 + w_3e_2 + x_3e_3 + \\Lambda_1,\n\\end{split}\\]\nwhere\n\\[\n\t\\varphi_1 = t_{1,1}w_2 - w_0,\\quad \\varphi_3 = t_{1,1}w_0 - w_2,\n\\]\nand\n\\[\\begin{split}\n\tu_1 &= \\frac{w_1}{t_{1,1}}x_3\n\t- \\frac{\\kappa_0-\\kappa_1+\\rho_1}{t_{1,1}},\\quad\n\tu_2 = \\frac{w_3\\varphi_3+\\kappa_2-\\kappa_3+\\rho_1}{t_{1,1}},\\\\\n\tx_1 &= \\frac{(w_1-t_{1,1}w_3)\\varphi_3\n\t-(\\kappa_0-\\kappa_1+\\kappa_2-\\kappa_3+2\\rho_1)t_{1,1}}{(t_{1,1}^2-1)w_1},\\\\\n\tx_3 &= \\frac{(t_{1,1}w_1-w_3)\\varphi_3\n\t-(\\kappa_0-\\kappa_1+\\kappa_2-\\kappa_3+2\\rho_1)}{(t_{1,1}^2-1)w_1}.\n\\end{split}\\]\nNote that $\\kappa_0,\\ldots,\\kappa_3$ are constants.\nWe also remark that\n\\[\n\tw_1\\varphi_1 + w_3\\varphi_3 + \\kappa_0 - \\kappa_1 + \\kappa_2 - \\kappa_3\n\t+ 2\\rho_1 = 0.\n\\]\nHence the system \\eqref{Eq:DS_SR_22_b} can be expressed as a system of ordinary differential equations in terms of the variables $\\varphi_3,w_1,w_3$.\n\nLet\n\\[\n\tp = \\frac{w_1\\varphi_3}{2t_{1,1}},\\quad q = \\frac{t_{1,1}w_3}{w_1},\\quad\n\tt = t_{1,1}^2.\n\\]\nWe also set\n\\[\\begin{split}\n\t&\\alpha_0 = \\displaystyle\\frac{1}{2}(1+\\kappa_1-2\\kappa_2+\\kappa_3),\\quad\n\t\\alpha_1 = \\displaystyle\\frac{1}{2}(-\\kappa_1+\\kappa_3+2\\rho_1),\\\\\n\t&\\alpha_2 = \\kappa_0 + \\kappa_2 - 2\\kappa_3,\\quad\n\t\\alpha_3 = \\displaystyle\\frac{1}{2}(1-2\\kappa_0+\\kappa_1+\\kappa_3),\\\\\n\t&\\alpha_4 = \\displaystyle\\frac{1}{2}(-\\kappa_1+\\kappa_3-2\\rho_1),\n\\end{split}\\]\nand\n\\[\n\ta = \\alpha_0,\\quad b = \\alpha_3,\\quad c = \\alpha_4,\\quad\n\td = \\alpha_2(\\alpha_1+\\alpha_2).\n\\]\nThen we have\n\n\\begin{thm}\nThe system \\eqref{Eq:DS_SR_22_b} with \\eqref{Eq:DS_SR_22_b_BM} gives the sixth Painlev\\'{e} equation.\nFurthermore, $w_1$ satisfies the completely integrable Pfaffian equation\n\\[\\begin{split}\n\tt(t-1)\\frac{d}{dt}\\log w_1 &= -(q-1)(q-t)p - \\alpha_2q\\\\\n\t&\\quad + \\frac{1}{4}(1+2\\alpha_1-2\\alpha_3-4\\alpha_4)t\n\t- \\frac{1}{4}(1-2\\alpha_1-4\\alpha_2-2\\alpha_3).\n\\end{split}\\]\n\\end{thm}\n\n\n\\subsection{For the partition $(3,1)$}\\label{Sec:System31}\n\nThe Heisenberg subalgebra $\\mathfrak{s}_{(3,1)}$ of $\\mathfrak{g}(A^{(1)}_3)$ is defined by\n\\[\n\t\\mathfrak{s}_{(3,1)} = \\bigoplus_{k\\in\\mathbb{Z}\\setminus3\\mathbb{Z}}\n\t\\mathbb{C}\\Lambda_1^k\\oplus\\bigoplus_{k\\in\\mathbb{Z}\\setminus\\{0\\}}\n\t\\mathbb{C}z^kH_1\\oplus\\mathbb{C}K,\n\\]\nwith\n\\[\n\t\\Lambda_1 = e_0 + e_1 + e_{2,3},\\quad\n\tH_1 = \\alpha^{\\vee}_1 + 2\\alpha^{\\vee}_2 - \\alpha^{\\vee}_3.\n\\]\nThe subalgebra $\\mathfrak{s}_{(3,1)}$ admits the gradation of type $\\mathbf{s}=(1,1,0,1)$ with the grade operator\n\\[\n\t\\vartheta_{(3,1)}\n\t= 3z\\left(\\frac{d}{dz}+\\mathrm{ad}\\eta_{(3,1)}\\right),\\quad\n\t\\eta_{(3,1)} = \\frac{1}{3}(\\alpha^{\\vee}_1+\\alpha^{\\vee}_2+\\alpha^{\\vee}_3).\n\\]\nNote that\n\\[\n\t\\mathfrak{g}_{\\geq0}(1,1,0,1) = \\mathbb{C}f_2\\oplus\\mathfrak{b}_{+}.\n\\]\n\nWe now assume $t_{1,2}=1$ and $t_{1,k}=0$ $(k\\geq3)$.\nThen the similarity reduction \\eqref{Eq:DS_SR} for $\\mathfrak{s}_{(3,1)}$ is expressed as\n\\begin{equation}\\label{Eq:DS_SR_31_b}\n\t\\left[\\vartheta_{(3,1)}-M,\\partial_{1,1}-B_{1,1}\\right] = 0,\n\\end{equation}\nwith\n\\begin{equation}\\begin{split}\\label{Eq:DS_SR_31_b_BM}\n\tM &= \\vartheta_{(3,1)}(W)W^{-1}\n\t+ W(\\rho_1H_1+t_{1,1}\\Lambda_1+2\\Lambda_1^2)W^{-1},\\\\\n\tB_{1,1} &= \\partial_{1,1}(W)W^{-1} + W\\Lambda_1W^{-1}.\n\\end{split}\\end{equation}\n\nLet\n\\[\n\tW = \\exp(-w_2f_2)\\exp(\\omega_{-1})\\exp(\\omega_{-2})\\exp(\\omega_{<-2}),\n\\]\nwhere\n\\[\\begin{split}\n\t\\omega_{-1} &= -w_0f_0 - w_1f_1 - w_3f_3 - w_{1,2}f_{1,2}\n\t- w_{2,3}f_{2,3},\\\\\n\t\\omega_{-2} &= -w_{0,1}f_{0,1} - w_{3,0}f_{3,0} - w_{0,1,2}f_{0,1,2} \n\t- w_{1,2,3}f_{1,2,3} - w_{2,3,0}f_{2,3,0},\n\\end{split}\\]\nand $\\omega_{<-2}\\in\\mathfrak{g}_{<-2}(1,1,0,1)$.\nThen the system \\eqref{Eq:DS_SR_31_b_BM} gives explicit formulas of $M,B_{1,1}$ as follows:\n\\[\\begin{split}\n\tM &= \\kappa_0\\alpha^{\\vee}_0 + \\kappa_1\\alpha^{\\vee}_1\n\t+ \\kappa_2\\alpha^{\\vee}_2 + \\kappa_3\\alpha^{\\vee}_3 + \\varphi_0e_0\n\t+ (\\varphi_1+w_2\\varphi_{1,2})e_1\\\\\n\t&\\quad + \\varphi_2e_2 + (\\varphi_3-w_2\\varphi_{2,3})e_3\n\t+ \\varphi_{1,2}e_{1,2} + \\varphi_{2,3}e_{2,3} - 2w_2e_{3,0}\n\t+ 2\\Lambda_1^2,\\\\\n\tB_{1,1} &= u_3K - \\frac{\\varphi_1-t_{1,1}}{2}\\alpha^{\\vee}_0\n\t+ \\frac{\\varphi_0-t_{1,1}}{2}\\alpha^{\\vee}_1\n\t+ \\frac{w_2\\varphi_{1,2}}{2}\\alpha^{\\vee}_2\n\t+ \\frac{\\varphi_{1,2}}{2}e_2 - w_2e_3 + \\Lambda_1,\n\\end{split}\\]\nwhere\n\\[\\begin{split}\n\t&\\varphi_0 = 2w_1 + 2w_{2,3} + t_{1,1},\\quad\n\t\\varphi_1 = -2w_0 - 2w_{2,3} + t_{1,1},\\\\\n\t&\\varphi_2 = (w_0-2w_1+t_{1,1})w_3 - 2w_{3,0},\\quad \\varphi_3 = 2w_{1,2},\\\\\n\t&\\varphi_{1,2} = 2w_3,\\quad \\varphi_{2,3} = 2w_0 - 2w_1 + t_{1,1}.\n\\end{split}\\]\nNote that $\\kappa_0,\\ldots,\\kappa_4$ are constants.\nWe also remark that\n\\[\n\t2w_2\\varphi_2 - \\varphi_3\\varphi_{1,2} = 2(\\kappa_2-\\kappa_3-3\\rho_1),\\quad\n\t\\varphi_0 + \\varphi_1 + \\varphi_{2,3} = 3t_{1,1}.\n\\]\nHence the system \\eqref{Eq:DS_SR_31_b} can be expressed as a system of ordinary differential equations in terms of the variables $\\varphi_0,\\varphi_1,\\varphi_2,\\varphi_{1,2},w_2$.\n\nLet\n\\[\n\tq_1 = -\\frac{w_2\\varphi_{1,2}}{\\sqrt{6}},\\quad\n\tp_1 = -\\frac{2\\varphi_2}{\\sqrt{6}\\varphi_{1,2}},\\quad\n\tq_2 = \\frac{\\varphi_1}{\\sqrt{6}},\\quad\n\tp_2 = -\\frac{\\varphi_0}{\\sqrt{6}},\\quad t = -\\frac{\\sqrt{6}t_{1,1}}{2}.\n\\]\nWe also set\n\\[\\begin{split}\n\t&\\alpha_1 = \\displaystyle\\frac{1}{3}(\\kappa_2-\\kappa_3-3\\rho_1),\\quad\n\t\\alpha_2 = \\displaystyle\\frac{1}{3}(\\kappa_1-2\\kappa_2+\\kappa_3),\\\\\n\t&\\alpha_3 = \\displaystyle\\frac{1}{3}(1+\\kappa_0-2\\kappa_1+\\kappa_2),\\quad\n\t\\alpha_4 = \\displaystyle\\frac{1}{3}(1-2\\kappa_0+\\kappa_1+\\kappa_3).\n\\end{split}\\]\nThen we have\n\n\\begin{thm}\nThe system \\eqref{Eq:DS_SR_31_b} with \\eqref{Eq:DS_SR_31_b_BM} gives the Painlev\\'{e} system $\\mathcal{H}^{A_4^{(1)}}$.\nFurthermore, $\\varphi_{1,2}$ satisfies the completely integrable Pfaffian equation\n\\[\n\t\\frac{d}{dt}\\log\\varphi_{1,2} = p_1 + p_2 - \\frac{2}{3}t.\n\\]\n\\end{thm}\n\n\n\\subsection{For the partition $(4,1)$}\\label{Sec:System41}\n\nThe Heisenberg subalgebra $\\mathfrak{s}_{(4,1)}$ of $\\mathfrak{g}(A^{(1)}_4)$ is defined by\n\\[\n\t\\mathfrak{s}_{(4,1)} = \\bigoplus_{k\\in\\mathbb{Z}\\setminus4\\mathbb{Z}}\n\t\\mathbb{C}\\Lambda_1^k\\oplus\\bigoplus_{k\\in\\mathbb{Z}\\setminus\\{0\\}}\n\t\\mathbb{C}z^kH_1\\oplus\\mathbb{C}K,\n\\]\nwith\n\\[\n\t\\Lambda_1 = e_0 + e_1 + e_4 + e_{2,3},\\quad\n\tH_1 = \\alpha^{\\vee}_1 + 2\\alpha^{\\vee}_2 - 2\\alpha^{\\vee}_3\n\t- \\alpha^{\\vee}_4.\n\\]\nThe subalgebra $\\mathfrak{s}_{(4,1)}$ admits the gradation of type $\\mathbf{s}=(2,2,1,1,2)$ with the grade operator\n\\[\n\t\\vartheta_{(4,1)}\n\t= 8\\left(z\\frac{d}{dz}+\\mathrm{ad}\\eta_{(4,1)}\\right),\\quad\n\t\\eta_{(4,1)} = \\frac{1}{8}\n\t(3\\alpha^{\\vee}_1+4\\alpha^{\\vee}_2+4\\alpha^{\\vee}_3+3\\alpha^{\\vee}_4).\n\\]\nNote that\n\\[\n\t\\mathfrak{g}_{\\geq0}(2,2,1,1,2) = \\mathfrak{b}_{+}.\n\\]\n\nWe now assume $t_{1,2}=1$ and $t_{1,k}=0$ $(k\\geq3)$.\nThen the similarity reduction \\eqref{Eq:DS_SR} for $\\mathfrak{s}_{(4,1)}$ is expressed as\n\\begin{equation}\\label{Eq:DS_SR_41_b}\n\t\\left[\\vartheta_{(4,1)}-M,\\partial_{1,1}-B_{1,1}\\right] = 0,\n\\end{equation}\nwith\n\\begin{equation}\\begin{split}\\label{Eq:DS_SR_41_b_BM}\n\tM &= \\vartheta_{(4,1)}(W)W^{-1}\n\t+ W(\\rho_1H_1+2t_{1,1}\\Lambda_1+4\\Lambda_1^2)W^{-1},\\\\\n\tB_{1,1} &= \\partial_{1,1}(W)W^{-1} + W\\Lambda_1W^{-1}.\n\\end{split}\\end{equation}\n\nLet\n\\[\n\tW = \\exp(\\omega_{-1})\\exp(\\omega_{-2})\\exp(\\omega_{-3})\\exp(\\omega_{-4})\n\t\\exp(\\omega_{<-4}),\n\\]\nwhere\n\\[\\begin{split}\n\t\\omega_{-1} &= -w_2f_2 - w_3f_3,\\\\\n\t\\omega_{-2} &= -w_0f_0 - w_1f_1 - w_4f_4 - w_{2,3}f_{2,3},\\\\\n\t\\omega_{-3} &= -w_{1,2}f_{1,2} - w_{3,4}f_{3,4},\\\\\n\t\\omega_{-4} &= -w_{0,1}f_{0,1} - w_{4,0}f_{4,0} - w_{1,2,3}f_{1,2,3}\n\t- w_{2,3,4}f_{2,3,4},\n\\end{split}\\]\nand $\\omega_{<-4}\\in\\mathfrak{g}_{<-4}(2,2,1,1,2)$.\nThen the system \\eqref{Eq:DS_SR_41_b_BM} gives explicit formulas of $M,B_{1,1}$ as follows:\n\\[\\begin{split}\n\tM &= \\kappa_0\\alpha^{\\vee}_0 + \\kappa_1\\alpha^{\\vee}_1\n\t+ \\kappa_2\\alpha^{\\vee}_2 + \\kappa_3\\alpha^{\\vee}_3\n\t+ \\kappa_4\\alpha^{\\vee}_4 + \\varphi_0e_0 + \\varphi_1e_1\\\\\n\t&\\quad + \\varphi_2e_2 + \\varphi_3e_3 + \\varphi_4e_4\n\t+ \\varphi_{1,2}e_{1,2} + \\varphi_{2,3}e_{2,3} + \\varphi_{3,4}e_{3,4}\n\t+ 4\\Lambda_1^2,\\\\\n\tB_{1,1} &= u_4K + u_0\\alpha^{\\vee}_0\n\t+ \\frac{\\varphi_0-2t_{1,1}}{4}\\alpha^{\\vee}_1 + u_2\\alpha^{\\vee}_2\n\t+ u_3\\alpha^{\\vee}_3 + \\frac{\\varphi_{1,2}}{4}e_2\n\t+ \\frac{\\varphi_{3,4}}{4}e_3 + \\Lambda_1,\n\\end{split}\\]\nwhere\n\\[\\begin{split}\n\t&\\varphi_0 = 4w_1 - 4w_4 + 2t_{1,1},\\quad\n\t\\varphi_1 = -4w_0 + 2w_2w_3 - 4w_{2,3} + 2t_{1,1},\\\\\n\t&\\varphi_2 = -2(2w_1-w_4-t_{1,1})w_3 - 4w_{3,4},\\quad\n\t\\varphi_3 = 2(w_1-2w_4-t_{1,1})w_2 + 4w_{1,2},\\\\\n\t&\\varphi_{1,2} = 4w_3,\\quad \\varphi_{2,3} = -4w_1 + 4w_4 + 2t_{1,1},\\quad\n\t\\varphi_{3,4} = -4w_2,\n\\end{split}\\]\nand\n\\[\\begin{split}\n\t64t_{1,1}u_0 &= (\\varphi_0-4t_{1,1})(4\\varphi_1+\\varphi_{1,2}\\varphi_{3,4})\n\t+ 4\\varphi_2\\varphi_{3,4}\\\\\n\t&\\quad + 16t_{1,1}^2 + 16(\\kappa_0-\\kappa_1+\\kappa_2-\\kappa_4-2\\rho_1),\\\\\n\t64t_{1,1}u_2 &= \\varphi_0(4\\varphi_1+\\varphi_{1,2}\\varphi_{3,4})\n\t+ 4(\\varphi_2-t_{1,1}\\varphi_{1,2})\\varphi_{3,4}\\\\\n\t&\\quad - 16t_{1,1}^2 + 16(\\kappa_0-\\kappa_1+\\kappa_2-\\kappa_4-2\\rho_1),\\\\\n\t64t_{1,1}u_3 &= \\varphi_0(4\\varphi_1+\\varphi_{1,2}\\varphi_{3,4})\n\t+ 4\\varphi_2\\varphi_{3,4}\\\\\n\t&\\quad - 16t_{1,1}^2 + 16(\\kappa_0-\\kappa_1+\\kappa_2-\\kappa_4-2\\rho_1).\n\\end{split}\\]\nNote that $\\kappa_0,\\ldots,\\kappa_4$ are constants.\nWe also remark that\n\\[\\begin{split}\n\t&(\\varphi_0-4t_{1,1})\\varphi_{1,2}\\varphi_{3,4} + 4\\varphi_3\\varphi_{1,2}\n\t+ 4\\varphi_2\\varphi_{3,4} = 16(-\\kappa_2+\\kappa_3+4\\rho_1),\\\\\n\t&4\\varphi_1 + 4\\varphi_4 + \\varphi_{1,2}\\varphi_{3,4} = 16t_{1,1},\\quad\n\t\\varphi_0 + \\varphi_{2,3} = 4t_{1,1}.\n\\end{split}\\]\nHence the system \\eqref{Eq:DS_SR_41_b} can be described as a system of ordinary differential equations in terms of the variables $\\varphi_0,\\varphi_1,\\varphi_2,\\varphi_{1,2},\\varphi_{3,4}$.\n\nLet\n\\[\\begin{split}\n\t&q_1 = \\frac{\\varphi_0}{4t_{1,1}},\\quad p_1 = \\frac{t_{1,1}\\varphi_1}{8},\\\\\n\t&q_2 = \\frac{\\varphi_0}{4t_{1,1}}\n\t+ \\frac{\\varphi_2}{t_{1,1}\\varphi_{1,2}},\\quad\n\tp_2 = \\frac{t_{1,1}\\varphi_{1,2}\\varphi_{3,4}}{32},\\quad\n\tt = -\\frac{t_{1,1}^2}{2}.\n\\end{split}\\]\nWe also set\n\\[\\begin{split}\n\t&\\alpha_1 = \\frac{1}{8}(2-2\\kappa_0+\\kappa_1+\\kappa_4),\\quad\n\t\\alpha_2 = \\frac{1}{8}(2+\\kappa_0-2\\kappa_1+\\kappa_2),\\\\\n\t&\\alpha_3 = \\frac{1}{8}(1+\\kappa_1-2\\kappa_2+\\kappa_3),\\quad\n\t\\alpha_4 = \\frac{1}{8}(\\kappa_2-\\kappa_3-4\\rho_1),\\\\\n\t&\\alpha_5 = \\frac{1}{8}(1-\\kappa_3+\\kappa_4+4\\rho_1).\n\\end{split}\\]\nThen we have\n\n\\begin{thm}\nThe system \\eqref{Eq:DS_SR_41_b} with \\eqref{Eq:DS_SR_41_b_BM} gives the Painlev\\'{e} system $\\mathcal{H}^{A_5^{(1)}}$.\nFurthermore, $\\varphi_{1,2}$ satisfies the completely integrable Pfaffian equation\n\\[\n\tt\\frac{d}{dt}\\log\\varphi_{1,2} = -q_1p_1 - q_2p_2 + tq_2 - \\frac{3}{4}t\n\t- \\frac{1+2\\alpha_1+2\\alpha_3+2\\alpha_5}{4}.\n\\]\n\\end{thm}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe Edelweiss experiment searches for the WIMP candidates of the Dark Matter. The set-up is located in LSM in the French Alps which provides a shielding factor of $\\sim$4800~m.w.e. The detection principle is based on measuring energy of the recoil nucleus originating from the WIMP elastic scattering. Bolometers of pure natural Ge are used both as the detectors and the target material. These detectors cooled down to about 20~mK allow to measure simultaneously heat and ionization signals. Due to the quenching of the ionization signal present for nuclear recoils one achieves a very high discrimination of $\\beta$ and $\\gamma$ background from the recoil candidates \\cite{defay08}. However, neutrons coming from the natural radioactivity or induced by the remaining muons can still mimic the nuclear recoil signal of WIMP events and thus can not be discriminated in the same way as $\\beta$'s and $\\gamma$'s. Therefore, this type of background requires special careful investigation. The knowledge of it also becomes highly important in view of large 1-tonne scale experiments like EURECA \\cite{kraus08}.\n\n\\section{Neutron background}\nFor kinematic reasons, not every neutron can mimic the WIMP nuclear recoil event but only those who have an energy of 0.5\\--10~MeV when they reach the Ge bolometers. Such neutrons appearing due to natural radioactivity in the surrounding (e.g. U\/Th contamination) can be avoided by using a passive hydrogen-rich moderator (50~cm polyethylene shield in case of Edelweiss) and in addition by radiopurity selection of materials to be used. Monitoring of the ambient neutron flux in proximity of the Edelweiss experimental set-up is performed with the help of $^3$He gas detectors. This measurement yields a flux of about $2\\cdot10^{-6}$~n\/cm$^2$\/day \\cite{yakushev08} which is in good agreement with the previously measured value \\cite{fiorucci07}. Another part of the neutron background is caused by muon interactions in the rock and in fact everywhere in the set-up (especially in high-Z materials such as the gamma shield based on lead). High energy neutrons (well above 10~MeV) created in such deep inelastic scattering (DIS) processes further lead to the production of secondary neutrons with energies below 10~MeV. The effect of this $\\mu-$induced neutron component is commonly reduced by tagging the original muons. In Edelweiss experiment the plastic scintillator modules covering the full bolometer set-up act as the muon veto \\cite{chantelauze07}. Full simulations of the Edelweiss set-up including the muon veto were performed in GEANT4 in order to estimate the influence of muons for the Dark Matter search. These simulations involve muon generation to reproduce the muon flux specific for LSM and allow to get complete event topology \\cite{horn07}. It was shown then that muons which miss the veto can still induce some neutrons reaching the bolometers and giving rise to WIMP-like events not vetoed by the muon system. To verify these simulations one has to normalize them to the experimental data, i.e. one needs explicit $\\mu-$induced neutron measurements. One way to achieve this is to check the rate of events which are in coincidence between muon veto and the bolometers. This rate currently measured in Edelweiss is about 0.03~events\/kg\/day and it is reasonably well reproduced in the simulations. However, the rareness of these coincidence events makes it hard to get enough statistics to draw a reliable conclusion. A dedicated detector based on liquid scintillator was thus designed and installed in 2008 in LSM.\n\n\\section{A detector for muon-induced neutrons}\n\\label{sec:nc-detector}\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\t\\includegraphics[width=0.72\\textwidth]{figs\/nc-detector-fin.pdf}\n\t\\caption{General scheme of the neutron counter (side view): 1-tonne of Gd-loaded liquid scintillator viewed by 16 PMTs of 8-inch and 6 PMTs of 2-inch diameter. A layer of lead bricks below the liquid scintillator volume acts as an effective target for muons and high energy neutrons.}\n\t\\label{fig:nc-detector}\t\n\\end{figure}\n\nThe measuring principle of $\\mu$-induced neutrons is based on registering thermalized neutrons in coincidence with the incoming muon or by detecting a multiple neutron event (secondary neutrons produced in a $\\mu$-induced particle showers). To efficiently observe neutrons, a liquid scintillator of 1~m$^3$ volume (50x100x200~cm$^3$) loaded with Gadolinium (St.~Gobain Bicron BC525) is used as a core of the detector. The neutron capture process on Gd results in several gammas with 8~MeV sum energy. The scintillator volume is viewed from each of the two module ends by 8 photomultiplier tubes (PMT) of 8-inch diameter (Fig.~\\ref{fig:nc-detector}). These PMTs are optimized to register the light produced after the neutron capture. In addition, the system is equipped with 6 smaller PMTs (2-inch diameter) to register muons crossing the neutron detector (these muons create much more light and thus the 8-inch PMTs will saturate). The scintillator and PMTs are placed in one plexiglass container divided into three parts: the central one for the scintillator itself and two side ones filled with paraffin in which the PMTs are immersed. This plexiglass chamber is then placed in an aluminum vessel as secondary safety container. Finally, the system is surrounded by iron plates to reflect a fraction of neutrons back to the scintillator. In order to enhance the neutron production (up to a factor of 10 comparing to rock) a 10~cm thick layer of lead bricks is put underneath the detector. On top of the counter, a plastic scintillator module (same type as the muon veto of Edelweiss) is installed. The complete system is positioned right near the western wall of the Edelweiss muon veto. Based on the currently measured muon flux in the lab, the expected count rate of muon-induced neutrons is about few counts per day. \n\nThe GEANT4 simulations mentioned above were extended to optimize the neutron set-up before going for construction. Additionally, a smaller prototype (25x25x250~cm$^3$) was built beforehand in Karlsruhe in order to test mechanical properties, handling of liquids and gas as well as to study light collection, PMTs and overall performance. This prototype also allowed to develop a LED system to monitor over time the light properties of the scintillator and stability of PMTs. There are in total 8 LEDs ($\\lambda=$425~nm) placed at different positions. These LEDs are operated via VME-based PC commands and regularly fired one by one. The data from the groups of opposite PMTs are then analyzed. \n\nThe neutron counter is also equipped with safety sensors because of the pseudocumene based scintillator. This includes vapor sensors to check the internal and surrounding atmosphere, two leak sensors in the aluminum vessel, one temperature meter immersed in the paraffin volume and one outside of the counter. Signals of the vapor sensors are incorporated into LSM safety system which takes care of an alarm activation in the lab. One can as well monitor these sensors using the LabVIEW$^{\\tt{TM}}$-based program (\\underline{Ka}rlsruhe \\underline{C}ontrol of \\underline{S}afety or KA-CS) installed on Linux computer (SuSE~10.3) (Fig.~\\ref{fig:nc-ka-cs}). This software notifies users by email in case of an alarm due to a failure or passing of specified thresholds.\n\n\nThe neutron detector described was successfully installed in LSM in September 2008 and as for the time of writing, it is under intensive commissioning.\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=0.62\\textwidth]{figs\/nc-ka-cs.pdf}\n\t\\caption{Principle scheme of KA-CS system for safety monitoring: sensors are continuously read via NI-6221 DAQ card by the LabVIEW program installed on Linux computer.}\n\t\\label{fig:nc-ka-cs}\n\\end{figure}\n\n\\section{Conclusion}\nImproved sensitivity of Dark Matter search experiments requires much better knowledge of the background conditions. Activity of the Edelweiss collaboration concerning the neutron background studies was presented, in particular the new detector for the $\\mu$-induced neutrons was described. \n\n\\section{Acknowledgements}\nThis work is in part supported by the German Research Foundation (DFG) through the Trans\\-regional Collaborative Research Center SFB-TR27 as well as by the EU contract RII3-CT-2004-506222.\n\n\n\\section{...}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nLet $G$ be a finite group which is a transitive subgroup of a certain symmetry group $S_{d+1}$. A number field $K$ of degree $d+1$ is called a $G$-field if its Galois closure $\\widehat{K}$ over $\\mathbb{Q}$ is a $G$-Galois extension. For a $G$-field $K$, we attach the Artin L-function\n$$\nL(s,\\rho,K)=\\frac{\\zeta_K(s)}{\\zeta(s)}=\\sum_{n=1}^\\infty a_{\\rho}(n)n^{-s},\n$$\nwhere $\\rho$ is $d$-dimensional representation of $G$. Note that $-1\\leq a_{\\rho}(p)\\leq d$.\nIf $G=S_{d+1}$, $\\rho$ is the $d$-dimensional standard representation of $S_{d+1}$.\nLet $L(X)^{r_2}$ be the set of $G$-fields $K$ with $ |d_K| x^a$.\n\nBy Theorem 25.8 and Theorem 30.2 (the method of moments) in \\cite{B}, it is enough to consider $h(x)=x^r$.\nConsider\n\\begin{equation}\\label{central-limit}\n\\sum_{\\rho\\in L(X)} \\left(\\frac {\\sum_{p\\leq x} a_\\rho(p)}{\\sqrt{\\pi(x)}}\\right)^r.\n\\end{equation}\n\nBy multinomial formula,\n$$\\left(\\sum_{p\\leq x} a_{\\rho}(p)\\right)^r=\\sum_{u=1}^r {\\sum}^{(1)}_{(r_1,...,r_u)} \\frac {r!}{r_1!\\cdots r_u!} \\frac 1{u!} {\\sum}^{(2)}_{(p_1,...,p_u)} a_{\\rho}(p_1)^{r_1}\\cdots a_{\\rho}(p_u)^{r_u},\n$$\nwhere $\\sum_{(r_1,...,r_u)}^{(1)}$ means the sum over the $u$-tuples $(r_1,...,r_u)$ of positive integers such that $r_1+\\cdots+r_u=r$, and $\\sum_{(p_1,...,p_u)}^{(2)}$ means the sum over the $u$-tuples $(p_1,...,p_u)$ of distinct primes such that $p_i\\leq x$ for each $i$.\nThen\n$$(\\ref{central-limit})=\\pi(x)^{-\\frac r2} \\sum_{u=1}^r \\frac 1{u!} {\\sum}^{(1)}_{(r_1,...,r_u)} \\frac {r!}{r_1!\\cdots r_u!} {\\sum}^{(2)}_{(p_1,...,p_u)} \\left(\\sum_{\\rho\\in L(X)} a_{\\rho}(p_1)^{r_1}\\cdots a_{\\rho}(p_u)^{r_u}\\right).\n$$\n\nNow we claim that except when $r$ is even, $u=\\frac r2$, and $r_1=\\cdots=r_u=2$, it gives rise to the error term.\n\nNow suppose $r_i\\geq 2$ for all $i$, and $r_j>2$ for some $j$. Then since $r_1+\\cdots+r_u=r$, $u\\leq\\frac {r-1}2.$ Hence by the trivial estimate,\nsuch term is majorized by\n$$\\pi(x)^{-\\frac r2} \\sum_{u=1}^r \\frac 1{u!} {\\sum}_{(r_1,...,r_u)}^{(1)}\\frac {r!}{r_1!\\cdots r_u!} d^{r_1+\\cdots+r_u} |L(X)| \\pi(x)^u\n\\ll_r \\pi(x)^{-\\frac 12} |L(X)| \\sum_{u=1}^r \\frac 1{u!} d^r \\ll_{r,d} X \\pi(x)^{-\\frac 12}.\n$$\nThis gives rise to the error term.\n\nSuppose $r_i\\leq 2$ for all $i$.\nSuppose $r_i=1$ for some $i$. We may assume that $r_1=1$.\n\nLet $N$ be the number of conjugacy classes of $G$, and partition the sum $\\sum_{\\rho\\in L(X)}$ into $(N+w)^{u}$ sums, namely, given\n$(\\mathcal S_1,...,\\mathcal S_{u})$, where $\\mathcal S_i$ is either $\\mathcal S_{p_i,C}$ or $\\mathcal S_{p_i,r_j}$,\nwe consider the set of $\\rho\\in L(X)$ with the local conditions $\\mathcal S_i$ for each $i$. Note that in each such partition, $a_{\\rho}(p_1)^{r_1}\\cdots a_{\\rho}(p_u)^{r_u}$ remains a constant.\n\nSuppose $p_1$ is unramified, and fix the splitting types of $p_2,\\cdots,p_u$, and let $\\text{Frob}_{p_1}$ runs through the conjugacy classes of $G$. Then by (\\ref{estimate1}), the sum of such $N$ partitions is\n$$\\sum_C \\left(\\frac{|C|a_\\rho(p_1)}{|G|(1+f(p_1))} A(\\mathcal S_2,...,\\mathcal S_{u})X + O((p_1\\cdots p_u)^\\gamma X^\\delta) \\right),\n$$\nfor a constant $A(\\mathcal S_2,...,\\mathcal S_u)$.\nLet $\\chi_\\rho$ be the character of $\\rho$. Then $a_{\\rho}(p)=\\chi_{\\rho}(g)$, where $g=\\text{Frob}_p$. By orthogonality of characters,\n$\\sum_C |C| a_{\\rho}(p_1)=\\sum_{g\\in G} \\chi_\\rho(g)=0$. Hence the above sum is\n$O((p_1\\cdots p_u)^\\gamma X^\\delta)$, and it is majorized by $\\pi(x)^{-\\frac r2+u}x^{\\gamma u} X^{\\delta}.$\n\nHence we can assume that $r_i\\leq 2$ for each $i$, and $p_j$ is ramified when $r_j=1$. Suppose $r_1+\\cdots+r_{v}+r_{v+1}+\\cdots+r_u=r$, $r_1=\\cdots=r_v=1$ and $r_{v+1}=\\cdots=r_u=2$. Then $u-v\\leq \\frac {r-1}2$, and $p_1,...,p_v$ are ramified.\nThe partition of fixed splitting types of $p_{v+1},...,p_u$ is majorized by\n$$\n\\prod_{i=1}^v \\frac {f(p_i)}{1+f(p_i)} B(\\mathcal S_{v+1},...,\\mathcal S_u)X + O((p_1\\cdots p_u)^\\gamma X^\\delta),\n$$\nfor some constant $B(\\mathcal S_{v+1},...,\\mathcal S_u)$.\nSince $\\frac {f(p)}{1+f(p)}\\ll \\frac 1p$, it contributes to\n$$\\pi(x)^{u-v-\\frac r2}(\\log\\log x)^v X+ \\pi(x)^{-\\frac r2+u}x^{\\gamma u} X^{\\delta}\\ll X (\\log\\log x)^v \\pi(x)^{-\\frac 12}+\\pi(x)^{-\\frac r2+u}x^{\\gamma u} X^{\\delta}.\n$$\n\nNow let $r$ be even, $u=\\frac r2$, and $r_1=\\cdots=r_u=2$. If one of $p_1,p_2, \\cdots , p_u$ is ramified, their contribution is\nmajorized by $X \\pi(x)^{-1}\\log\\log x.$\nNow we assume that all primes are unramified. Then the corresponding term is\n\\begin{equation}\\label{main}\n\\pi(x)^{-\\frac r2} \\frac 1{u!} \\frac {r!}{2^u} \\sum_{(p_1,...,p_u)}^{(2)} \\left(\\sum_{L(s,\\rho)\\in L(X)} a_{\\rho}(p_1)^2\\cdots a_{\\rho}(p_u)^2\\right).\n\\end{equation}\n\nLet $N$ be the number of conjugacy classes of $G$, and partition the sum $\\sum_{\\rho\\in L(X)}$ into $N^u$ sums\nwhere $(C_1,...,C_u)$ is the set of $\\rho\\in L(X)$ such that $\\text{Frob}_{p_i}\\in C_i$ for each $i$.\nThen,\n\\begin{eqnarray*}\n&& \\sum_{\\rho\\in L(X)} a_{\\rho}(p_1)^2\\cdots a_{\\rho}(p_u)^2=\\sum_{(C_1,...,C_u)} \\chi_{\\rho}(p_1)^2\\cdots \\chi_{\\rho}(p_u)^2\n\\left(\\sum_{\\rho\\in L(X)\\atop \\text{Frob}_{p_i}\\in C_i} 1\\right) \\\\\n&& =\\sum_{(C_1,...,C_u)} \\chi_{\\rho}(p_1)^2\\cdots \\chi_{\\rho}(p_u)^2 \\left(\\prod_{i=1}^u \\frac {|C_i|}{|G|(1+f(p_i))} |L(X)|+O((p_1\\cdots p_u)^\\gamma X^{\\delta})\\right).\n\\end{eqnarray*}\nNow\n$$\n\\sum_{(C_1,...,C_u)} \\chi_{\\rho}(p_1)^2\\cdots \\chi_{\\rho}(p_u)^2 \\prod_{i=1}^u \\frac {|C_i|}{|G|(1+f(p_i))}=\\prod_{i=1}^u \\left(\\sum_{C_i} \\frac {\\chi_{\\rho}(p_i)^2|C_i|}{|G|(1+f(p_i))}\\right).\n$$\nHere $\\chi_{\\rho}(p)^2=\\chi_{\\rho^2}(p)=\\chi_{Sym^2\\rho}(p)+\\chi_{\\wedge^2\\rho}(p)$. We observed in \\cite{CK} that since $\\rho$ is an irreducible real self-dual representation,\n$Sym^2\\rho$ contains the trivial representation and\n$\\wedge^2\\rho$ does not contain the trivial representation (\\cite{JL}, page 274). Hence $\\chi_{\\rho}(p)^2=1+\\sum_{j=1}^l \\eta_j(p)$, where $\\eta_j$'s are non-trivial irreducible characters of $G$. By the orthogonality of characters, for each $j$, $\\sum_C |C|\\eta_j(p)=\\sum_{g\\in G} \\eta_i(g)=0$. Hence $\\sum_{C} \\chi_{\\rho}(p)^2|C|=|G|.$\nTherefore,\n\\begin{eqnarray*}\n&& {\\sum}_{(p_1,...,p_u)}^{(2)} \\left(\\sum_{\\rho\\in L(X)} a_{\\rho}(p_1)^2\\cdots a_{\\rho}(p_u)^2\\right) \\\\\n&=& \\pi(x)^u |L(X)| + O(\\pi(x)^{u-1} |L(X)|\\log\\log x) + O(\\pi(x)^u x^{\\gamma u} X^{\\delta}).\n\\end{eqnarray*}\n\nNote\n$$\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty} t^r e^{-\\frac{t^2}{2}}dt = \\begin{cases} \\frac{r!}{(r\/2)! 2^{r\/2}}, & \\text{if $r$ is even,}\\\\\n0, & \\text{if $r$ is odd}\\end{cases}.\n$$\nHence we have proved\n\\begin{theorem}\\label{Artin}\nSuppose $\\frac {\\log X}{\\log x}\\longrightarrow \\infty$ as $x\\to\\infty$. Then\n$$\n\\frac 1{|L(X)|} \\sum_{\\rho\\in L(X)} \\left(\\frac {\\sum_{p\\leq x} a_{\\rho}(p)}{\\sqrt{\\pi(x)}}\\right)^r\n=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty} t^r e^{-\\frac{t^2}{2}}dt + O\\left(\\frac {(\\log\\log x)^r}{\\pi(x)^{\\frac 12}}\\right).\n$$\n\\end{theorem}\n\nThis proves (\\ref{main-id}).\n\n\\section{Central Limit Theorem for Hecke eigenforms; Level aspect}\n\nIn this section, in analogy to (\\ref{N}),\nwe consider central limit theorem for modular form $L$-functions with the trivial central character with respect to congruence subgroups as the level goes to infinity. We follow \\cite{N} closely. For $k\\geq 2$, let $S_k(N)$ be the set of normalized Hecke eigen cusp forms of weight $k$ with respect to $\\Gamma_0(N)$ with the trivial central character.\nLet $f(z)=\\sum_{n=1}^\\infty a_f(n)n^{\\frac {k-1}2} e^{2\\pi inz}$; $a_f(mn)=a_f(m)a_f(n)$, if $(m,n)=1$; $a_f(1)=1$; $a_f(p^j)=a_f(p)a_f(p^{j-1})-a_f(p^{j-2})$.\n\nWe show\n\\begin{theorem} \\label{Hecke}\nFor a continuous real function $h$ on $\\Bbb R$, (assume that $\\frac {\\log N}{\\log x}\\longrightarrow \\infty$ as $x\\to\\infty$.)\n\n$$\n\\frac 1{\\#S_k(N)} \\sum_{f\\in S_k(N)} h\\left(\\frac {\\sum_{p\\leq x} a_f(p)}{\\sqrt{\\pi(x)}}\\right)\\longrightarrow \\frac 1{\\sqrt{2\\pi}}\\int_{-\\infty}^\\infty h(t) e^{-\\frac {t^2}2}\\, dt\\quad \\text{as $x\\to\\infty$}.\n$$\n\\end{theorem}\n\nWe have, from \\cite{Se},\n\n\\begin{lemma} Suppose $k\\geq 2$. Let $S_k(N,\\chi)$ be the set of normalized Hecke eigen cusp forms of weight $k$ with respect to $\\Gamma_0(N)$ with a character $\\chi$ (mod $N$). Then\n$$\\sum_{f\\in S_k(N,\\chi)} a_f(n)=\\frac {k-1}{12} \\chi(\\sqrt{n}) n^{-\\frac 12} \\psi(N)+O(n^{c}N^{\\frac 12} d(N)),\n$$\nfor some constant $c$, independent of $n, N$.\n\\end{lemma}\n\nHere $\\psi(N)=N \\prod_{l | N} (1+\\frac 1l)$,\nand $d(N)$ is the number of positive divisors of $N$. Note that $\\psi(N)=|SL_2(\\Bbb Z): \\Gamma_0(N)|$.\nHere $\\chi(x)=0$ if $x$ is not a positive integer prime to $N$. In particular, if $n$ is not a square,\n$\\sum_{f\\in S_k(N,\\chi)} a_f(n)=O(n^{c}N^{\\frac 12} d(N)).$ Taking $n=1$ and $\\chi=1$, we have\n$$\\#S_k(N)=\\frac {k-1}{12} \\psi(N)+O(N^{\\frac 12}d(N)).\n$$\n\nWe need to compute, for a positive integer $r$,\n\\begin{equation}\\label{central-limit-h}\n\\sum_{f\\in S_k(N)} \\left(\\frac {\\sum_{p\\leq x} a_f(p)}{\\sqrt{\\pi(x)}}\\right)^r.\n\\end{equation}\n\nBy multinomial formula,\n$$(\\ref{central-limit-h})=\\pi(x)^{-\\frac r2} \\sum_{u=1}^r \\frac 1{u!} {\\sum}_{(r_1,...,r_u)}^{(1)} \\frac {r!}{r_1!\\cdots r_u!} {\\sum}_{(p_1,...,p_u)}^{(2)} \\left(\\sum_{f\\in S_k(N)} a_f(p_1)^{r_1}\\cdots a_f(p_u)^{r_u}\\right).\n$$\n\nNow we claim that except when $r$ is even, $u=\\frac r2$, and $r_1=\\cdots=r_u=2$, it gives rise to the error term.\n\nBy \\cite{N}, Lemma 2, we can show that\n$a_f(p)^n=\\sum_{j=0}^n h_n(j)a_f(p^j)$, where $h_n(j)=0$ if $n$ is odd and $j$ is even, or if $n$ is even and $j$ is odd.\nFor $u$-tuples $(r_1,...,r_u)$ and $(p_1,...,p_u)$, we define\n\\begin{eqnarray*}\nA(r_1,...,r_u) &=& {\\sum}_{(p_1,...,p_u)}^{(2)} B(r_1,...,r_u; p_1,...,p_u),\\\\\n B(r_1,...,r_u; p_1,...,p_u) &=& \\sum_{f\\in S_k(N)} a_f(p_1)^{r_1}\\cdots a_f(p_u)^{r_u}.\n\\end{eqnarray*}\n\nThen\n$$\nB(r_1,...,r_u; p_1,...,p_u)=\\sum_{0\\leq j_r\\leq r_1,...,0\\leq j_u\\leq r_u} h_{r_1}(j_1)\\cdots h_{r_u}(j_u) \\sum_{f\\in S_k(N)} a_f(p_1^{j_1}\\cdots p_u^{j_u}).\n$$\n\nAs in \\cite{N}, if $r_l$ is odd for some $l$, $A(r_1,...,r_u)\\ll N^{\\frac 12}d(N) \\pi(x)^u x^{cur}.$\n\t\nNow let $r_1=\\cdots=r_u=2$. Then $r$ is even, and $u=\\frac r2$.\n\n$$A(r_1,...,r_u)=\\pi(x)^{\\frac r2} \\#S_k(N)+O(\\pi(x)^{\\frac r2-1}(\\log\\log x)^{\\frac r2}\\#S_k(N)).\n$$\n\nNow suppose that all $r_i$'s are even, and $r_i>2$ for some $i$. Then $u\\leq \\frac r2-1$.\nThen\n$$A(r_1,...,r_u)\\ll \\pi(x)^{-1}\\#S_k(N).\n$$\n\nHence, as in Theorem \\ref{Artin}, we have\n\\begin{prop} Assume that $\\frac {\\log N}{\\log x}\\longrightarrow \\infty$ as $x\\to\\infty$. Then\n$$\n\\frac{1}{\\#S_k(N)} \\sum_{f\\in S_k(N)} \\left(\\frac {\\sum_{p\\leq x} a_f(p)}{\\sqrt{\\pi(x)}}\\right)^r\n=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty} t^r e^{-\\frac{t^2}{2}}dt + O\\left(\\frac {(\\log\\log x)^{\\frac r2}}{\\pi(x)}\\right).\n$$\n\\end{prop}\nThis proves Theorem \\ref{Hecke}\n\n\\section{Analogues of Sato-Tate distribution}\n\nFor a Hecke eigenform $f\\in \\mathcal F_k$, Sato-Tate conjecture says that for a continuous real function $h$ on $[-2,2]$,\n$$\n\\frac 1{\\pi(x)} \\sum_{p\\leq x} h(a_{f}(p))\\longrightarrow \\frac 1{2\\pi}\\int_{-2}^2 h(t) \\sqrt{4-t^2}\\, dt,\\quad \\text{as $x\\to\\infty$}.\n$$\nLet $a_f(p)=2\\cos\\theta_f(p)$ for $\\theta_f(p)\\in [0,\\pi]$. Then\n$\\{\\theta_f(p)\\}$ is uniformly distributed with respect to the measure $\\frac 2{\\pi}\\sin^2\\theta \\, d\\theta$ on $[0,\\pi]$. This is proved in \\cite{BGHT}.\n\nFor a vertical Sato-Tate distribution, one can consider, for a fixed prime $p$,\n\\begin{equation} \\label{CDF}\n\\sum_{f\\in \\mathcal F_k} a_{f}(p)^n.\n\\end{equation}\n\nConrey-Duke-Farmer \\cite{CDF} proved, for a holomorphic form of weight $k$,\n$$\\sum_{f\\in \\mathcal F_k} a_f(p)^n=\\frac k{6\\pi} \\left(1+\\frac 1p\\right)\\int_0^\\pi 2^n \\cos^n\\theta \\frac {\\sin^2\\theta}{(1-\\frac 1{p})^2+\\frac 4p \\sin^2\\theta} \\, d\\theta+O(p^{\\frac n2+\\epsilon}).\n$$\n\nThis implies that $\\{\\theta_f(p), f\\in \\mathcal F_k\\}$ is uniformly distributed with respect to the measure\n$$\\frac 2{\\pi} \\left(1+\\frac 1p\\right)\\frac {\\sin^2\\theta}{(1-\\frac 1p)^2+\\frac 4p \\sin^2 \\theta} \\, d\\theta.\n$$\n\nFor Artin $L$-function analogue of Sato-Tate distribution, we consider, for $r\\geq 1$,\n\n\\begin{equation}\\label{Sato-Tate}\n\\frac 1{\\pi(x)} \\sum_{p\\leq x} a_{\\rho}(p)^r.\n\\end{equation}\n\nIn our case, note that $-1\\leq a_{\\rho}(p)\\leq d$.\nBy effective Chebotarev density theorem (cf. \\cite{Se1}, page 132), for $\\log x\\gg |G|(\\log \\left|d_{\\widehat{K}}\\right| )^2$,\n\\begin{equation*}\\label{chebo}\n\\sum_{p\\leq x \\atop \\text{Frob}_p\\in C} 1=\\frac {|C|}{|G|} \\pi(x)+O\\left(\\pi(x^{\\beta})\\right)+O\\left(x e^{-c |G|^{-\\frac 12} (\\log x)^{\\frac 12}}\\right),\n\\end{equation*}\nwhere $\\beta$ is an exceptional zero of $\\zeta_{\\widehat{K}}(s)$ such that $1-\\beta\\leq \\frac 14\\log d_{\\widehat{K}}$, if it exists. Hence\n\n$$\\sum_{p\\leq x} a_{\\rho}(p)^r=\\sum_C a_{\\rho}(p)^r \\left(\\sum_{p\\leq x\\atop \\text{Frob}_p\\in C} 1\\right)\n=\\sum_C a_{\\rho}(p)^r \\frac {|C|}{|G|} \\pi(x)+ O(\\pi(x^{\\beta})+x e^{-c |G|^{-\\frac 12} (\\log x)^{\\frac 12}}).\n$$\n\nNow $\\sum_C |C| a_{\\rho}(p)^r=\\sum_{g\\in G} \\chi_{\\rho}(g)^r$ and $\\chi_{\\rho}(g)^r=\\chi_{\\rho^r}(g)$.\nNote that\n$$\\frac 1{|G|}\\sum_{g\\in G} \\chi_{\\rho^r}(g)=n_r,\n$$\nwhich is the multiplicity of the trivial representation in $\\rho^r$. Hence\n$$\\sum_{p\\leq x} a_{\\rho}(p)^r=n_r \\pi(x)+O(\\pi(x^{\\beta})+x e^{-c |G|^{-\\frac 12} (\\log x)^{\\frac 12}}).\n$$\nTherefore,\n$$\\frac 1{\\pi(x)} \\sum_{p\\leq x} a_{\\rho}(p)^r\\longrightarrow n_r, \\quad \\text{as $x\\to\\infty$}.\n$$\n\nFor vertical Sato-Tate distribution, for a fixed prime $p$, consider\n$$\\frac 1{|L(X)|} \\sum_{\\rho\\in L(X)} a_{\\rho}(p)^r.\n$$\nThen by (\\ref{estimate1}),\n\\begin{eqnarray*}\n&& \\sum_{\\rho\\in L(X)} a_{\\rho}(p)^r=\\sum_C a_{\\rho}(p)^r \\left(\\sum_{\\rho\\in L(X)\\atop \\text{Frob}_p\\in C} 1\\right)+ a_{\\rho}(p)^r\n\\left(\\sum_{\\rho\\in L(X)\\atop \\text{$p$ is ramified}} 1 \\right) \\\\\n&&\n=\\frac {|L(X)|}{|G|(1+f(p))} \\sum_C |C| a_{\\rho}(p)^r +O(p^\\gamma X^\\delta)+O\\left(\\frac Xp\\right)\n=\\frac {|L(X)| n_r}{1+f(p)}+O(p^\\gamma X^\\delta)+O\\left(\\frac Xp\\right).\n\\end{eqnarray*}\nSo if $X>p^{\\frac {1+\\gamma}{1-\\delta}}$,\n$$\\frac 1{|L(X)|} \\sum_{\\rho\\in L(X)} a_{\\rho}(p)^r=\\frac {n_r}{1+f(p)}+O(p^{-1}).\n$$\n\n\\section{Counting $S_5$ quintic fields with local conditions}\\label{S_5}\n\nShankar and Tsimerman \\cite{ST} recently counted $S_5$ quintic fields with a power saving error terms. For $i=0,1,2$, let $N_5^{(i)}(X)$ be the number of $S_5$ quintic fields of signature $(5-2i,i)$ with $|d_K| < X$. Then they showed\n\\begin{eqnarray*}\nN_5^{(i)}(X)= D_i X+ O_{\\epsilon}\\left( X^{\\frac{399}{400}+\\epsilon }\\right),\n\\end{eqnarray*}\nwhere $D_i=d_i \\prod_p (1+ p^{-2} -p^{-4} -p^{-5})$ and $d_0,d_1,d_2$ are $\\frac{1}{240}, \\frac{1}{24}$ and $\\frac{1}{16}$, respectively.\n\nWe can count quintic fields with finitely many local conditions. Let $C$ be a conjugacy class of $S_5$ and $f(p)=p^{-1}+2p^{-2}+2p^{-3}+p^{-4}$. Let $\\mathcal S=\\{ LC_p \\}$ be a finite set of local conditions. Define\n $|\\mathcal S_{p,C}|=\\frac {|C|}{|G|(1+f(p))}$, $|\\mathcal S_{p,r_i} |=\\frac {c_i(p)}{(1+f(p))}$, and $|\\mathcal S|=\\prod_p |LC_p|$, where $c_i(p)$'s are given explicitly at the end of this section.\n\n\\begin{theorem} Let $N_5^{(i)}(X,\\mathcal S)$ be the number of $S_5$ quintic fields of signature $(5-2i,i)$ with $|d_K| < X$, and with the local condition $\\mathcal S$. Then\n\\begin{eqnarray*}\nN_5^{(i)}(X,\\mathcal S)= |\\mathcal S| D_i X+ O_{\\epsilon}\\left(\\left(\\prod_{p\\in \\mathcal S} p \\right)^{2-\\epsilon} X^{\\frac{199}{200}+\\epsilon }\\right).\n\\end{eqnarray*}\n\\end{theorem}\n\nWe follow the notations in \\cite{ST}. Let $V_{\\mathbb{Z}}$ be the space of $4$-tuples of $5 \\times 5$ alternating matrices with integer coefficients. The group $G_\\mathbb{Z} = GL_4(\\mathbb{Z}) \\times SL_5(\\mathbb{Z})$ acts on $V_\\mathbb{Z}$ via\n$$\n(g_4,g_5) \\cdot ( A, B, C, D)^t = g_4 ( g_5 A g_5^t, g_5 B g_5^t,g_5 C g_5^t,g_5 D g_5^t)^t.\n$$\nHere $g_4\\cdot (A,B,C,D)^t$ means $(a_1 (A,B,C,D)^t, a_2 (A,B,C,D)^t, a_3 (A,B,C,D)^t, a_4 (A,B,C,D)^t)$, where $a_i$ is the $i$th row of $g_4$.\n\nThere is a canonical bijection between the set of $G_\\mathbb{Z}$-equivalence classes of elements $(A,B,C,D) \\in V_{\\mathbb{Z}}$, and the set of isomorphism classes of pairs of $(R,R')$, where $R$ is a quintic ring and $R'$ is a sextic resolvent ring of $R$. (See \\cite{B08}.)\nLet $\\mathcal V$ be an element of $V_{\\mathbb{Z}}$. Over the residue field $\\mathbb{F}_p$, the element $\\mathcal V$ determines a quintic\n$\\mathbb{F}_p$-algebra $R(\\mathcal V)\/(p)$. Let us define the splitting symbol $(\\mathcal V,p)$ by\n$$\n(\\mathcal V,p)=(f_1^{e_1}f_2^{e_2} \\cdots ),\n$$\nwhenever $R(\\mathcal V)\/(p) \\cong \\mathbb{F}_{p^{f_1}}[t_1]\/(t_1^{e_1}) \\oplus \\mathbb{F}_{p^{f_2}}[t_2]\/(t_2^{e_2}) \\oplus \\cdots$. Then there are 17 possible splitting types for $(\\mathcal V,p)$; $(11111),$ $(1112),$ $(122),$ $(113),$ $(23),$ $(14),$ $(5),$ $(1^2111),$ $(1^212),$ $(1^23),$ $(1^21^21),$ $(2^21),$ $(1^311),$ $(1^32),$ $(1^31^2),$ $(1^41),$ and $(1^5)$. Let $\\sigma$ be one of 17 splitting types. Then define $T_p(\\sigma)$ to be the set of $\\mathcal V\\in V_{\\mathbb{Z}}$ such that $(\\mathcal V,p)=\\sigma$ and $U_p(\\sigma)$ to be the set of elements in $T_p(\\sigma)$ corresponding to quintic rings that are maximal at $p$. The set $U_p(\\sigma)$ is defined by congruence conditions on coefficients of $\\mathcal V$ modulo $p^2$. Let $\\mu(U_p(\\sigma))$ be the $p$-adic density of $\\mathcal S$ in $V_{\\mathbb{Z}_p}$. They are computed in Lemma 4 in \\cite{B08}. Let ${U}_p$ denote the union of the 17 $U_p(\\sigma)$. Then Lemma 20 of \\cite{B08} implies that\n\\begin{eqnarray*}\n\\mu({U_p})=(p-1)^8p^{12}(p+1)^4(p^2+1)^2(p^2+p+1)^2(p^4+p^3+p^2+p+1)(p^4+p^3+2p^2+2p+1)\/p^{40}.\n\\end{eqnarray*}\n\nNote that\n$$d_i \\zeta(2)^2\\zeta(3)^2\\zeta(4)^2\\zeta(5) \\prod_p \\mu({U_p}) = d_i \\prod_p (1+p^{-2}-p^{-4}-p^{-5}),\n$$\nwhich is the coefficient of the main term in counting quintic fields. Here we need to interpret $\\mu({U_p})$ in the following way:\n$U_p$ can be considered as a subset of $(\\mathbb{Z}\/q^2 \\mathbb{Z})^{40}$, or the union of $k$ translates of $p^2 V_{\\mathbb{Z}}$, where $k$ is the size of the set. Here $k$ is $\\mu(U_p)q^{80}$.\nLet ${W}_p$ be the complement of ${U_p}$ in $V_{\\mathbb{Z}}$, then $\\mu({W}_p)=1- \\mu({U_p})$. Then $W_p$ is the union of $\\mu({W}_p)q^{80}$ translates of $q^2V_{\\mathbb{Z}}$.\n\nFor $q$ square-free, let $W_q \\subset V_{\\mathbb{Z}}$ be the set of elements corresponding to quintic rings that are not maximal at each prime dividing $q$. Then $W_q$ is the union of $\\prod_{p \\mid q} \\mu(W_p) \\cdot q^{80}$ translates of $q^2 V_{\\mathbb{Z}}$ by the Chinese Remainder Theorem.\n\nLet $V_{\\mathbb{Z}}^{ndeg}$ denote the set of elements in $V_{\\mathbb{Z}}$ that correspond to orders in $S_5$-fields, and let $V_{\\mathbb{Z}}^{deg}$ be the complement of $V_{\\mathbb{Z}}^{ndeg}$.\nA point in $V_{\\mathbb{Z}}$ corresponds to a maximal order in an $S_5$ quintic fields precisely if it is in $ \\cap_p U_p \\cap V_{\\mathbb{Z}}^{ndeg}$. For a $G_\\mathbb{Z}$-invariant subset $S$ of $V_{\\mathbb{Z}}$, let $N(S,X)$ denote the number of irreducible $G_\\mathbb{Z}$-orbits in $S^{ndeg}:=S \\cap V_{\\mathbb{Z}}^{ndeg}$ having discriminant bounded by $X$. For a set $S$ which is not $G_\\mathbb{Z}$-invariant, we can define $N^*(S,X)$ which also counts the orbits of degenerate points in $S$.\n\nNow we choose a finite set of primes $\\{p_1,p_2, \\cdots, p_n \\}$. And choose a splitting type $\\sigma_{p_k}$ for each $p_k$, $k=1,2,\\cdots, n$. Define $U'_p$ to be $U_p$ if $p \\neq p_k$, $k=1,2,\\cdots, n$. If $p=p_k$ for some $k$, then $U'_p=U_p(\\sigma_{p_k})$. Let $W'_p$ be the complement of $U'_p$. Then $W'_q$ is the union of $\\prod_{p \\mid q} \\mu(W'_p) \\cdot q^{80}$ translates of $q^2 V_{\\mathbb{Z}}$.\n\nLet $N_5^{(i)}(X, \\{ \\sigma_{p_k} \\}_{k=1,2,\\cdots,n})$ be the number of $S_5$ quintic fields of signature $(5-2i,i)$ with $|d_K| Q} O_\\epsilon \\left( (p_1 p_2 \\cdots p_n)^{2-\\epsilon} \\frac{X}{q^{2-\\epsilon} } \\right)\\\\\n&&=\\sum_{q \\leq Q} \\left( \\mu(W'_q) \\cdot c_i X - \\mu(q)N_{12}^*(W_q \\cap V_{\\mathbb{Z}}^{deg,(i)},X) \\right)\n+ O_\\epsilon \\left( (p_1 p_2 \\cdots p_n)^{2-\\epsilon} X\/Q^{1-\\epsilon} + X^{\\frac{39}{40}}Q^{3+\\epsilon} \\right)\\\\\n&&= \\sum_q c_i \\mu(W'_q)X + (p_1 p_2 \\cdots p_n)^{2-\\epsilon} O_\\epsilon \\left( X\/Q^{1-\\epsilon}+X^{\\frac{39}{40}}Q^{3+\\epsilon} + X^{\\frac{199}{200}Q^{1+\\epsilon}}\\right)\\\\\n&&= c_i \\prod_p (1- \\mu(W'_q)) X + (p_1 p_2 \\cdots p_n)^{2-\\epsilon} O_\\epsilon \\left( X\/Q^{1-\\epsilon}+X^{\\frac{39}{40}}Q^{3+\\epsilon} + X^{\\frac{199}{200}Q^{1+\\epsilon}}\\right)\\\\\n&&= c_i \\prod_p (\\mu(U'_q)) X + (p_1 p_2 \\cdots p_n)^{2-\\epsilon}O_\\epsilon \\left( X\/Q^{1-\\epsilon}+X^{\\frac{39}{40}}Q^{3+\\epsilon} + X^{\\frac{199}{200}Q^{1+\\epsilon}}\\right).\n\\end{eqnarray*}\n\nPutting $Q=X^{\\frac{1}{400}}$, we have\n$$(p_1 p_2 \\cdots p_n)^{2-\\epsilon}O_\\epsilon \\left( X\/Q^{1-\\epsilon}+X^{\\frac{39}{40}}Q^{3+\\epsilon} + X^{\\frac{199}{200}Q^{1+\\epsilon}}\\right) \\ll_\\epsilon (p_1 p_2 \\cdots p_n)^{2-\\epsilon} X^{\\frac{399}{400}+\\epsilon}.\n$$\n Note that\n \\begin{eqnarray*}\n c_i \\prod_p (\\mu(U'_q)) &=& \\prod_{k=1}^n \\frac{\\mu(U_p(\\sigma_{p_k}))}{\\mu(U_{p_k})} c_i \\prod_p \\mu(U_p)= \\prod_{k=1}^n \\frac{\\mu(U_p(\\sigma_{p_k}))}{\\mu(U_{p_k})} d_i \\prod_p \\left( 1+ p^{-2} -p^{-4} -p^{-5} \\right).\n\\end{eqnarray*}\n\nFrom Lemma 20 in \\cite{B08}, we can see that, for $f(p)=p^{-1}+2p^{-2}+2p^{-3}+p^{-4}$,\n$$\n\\frac{\\mu(U_p(\\sigma))}{\\mu(U_p)} = \\frac{1\/120}{1+f(p)},\\:\\frac{1\/12}{1+f(p)},\\:\\frac{1\/8}{1+f(p)},\\:\\frac{1\/6}{1+f(p)},\\: \\frac{1\/6}{1+f(p)},\\:\\frac{1\/4}{1+f(p)},\\:\\mbox{and }\\frac{1\/5}{1+f(p)}\n$$\nfor $\\sigma=(11111),(1112),(122),(113),(23),(14),(5),$ respectively, and\n\\begin{eqnarray*}\n\\frac{\\mu(U_p(\\sigma))}{\\mu(U_p)}& =& \\frac{1\/6 \\cdot 1\/p}{1+f(p)},\\:\\frac{1\/2 \\cdot 1\/p}{1+f(p)},\\:\\frac{1\/3 \\cdot 1\/p}{1+f(p)},\\:\\frac{1\/2 \\cdot 1\/p^2}{1+f(p)},\\: \\frac{1\/2 \\cdot 1\/p^2}{1+f(p)},\\:\\frac{1\/2 \\cdot 1\/p^2}{1+f(p)},\\frac{1\/2 \\cdot 1\/p^2}{1+f(p)},\\\\\n & & \\frac{1\/p^3}{1+f(p)},\\:\\frac{1\/p^3}{1+f(p)},\\:\\mbox{ and } \\frac{1\/p^4}{1+f(p)}\n\\end{eqnarray*}\nfor $\\sigma=(1^2111),(1^212),(1^23),(1^21^21),(2^2 1),(1^3 11),(1^3 2),(1^3 1^2), (1^4 1), (1^5),$ respectively. Hence we have proved the theorem. \n\n\\section{Counting $S_4$ quartic fields with local conditions}\\label{S_4}\n\nIn \\cite{BBP}, Belabas, Bhargava and Pomerance obtained a power saving error term for counting $S_4$ quartic fields. For $i=0,1$,\nlet $N_4^{(i)}(X)$ be the number of isomorphism classes of $S_4$-quartic fields of signature $(4-2i,i)$ with $|d_K| 7$ Gyr) and are born near the galactic center, but separate as a function of age (see Figures \\ref{fig:contour_3panel} and \\ref{fig:enlink_clusters}). \n \n \\item Using a simple second order polynomial regression, we quantify the relationship between observable abundance labels and birth property outputs (Section \\ref{sec:results_regression}). We model age as a function of (\\mbox{$\\rm [Fe\/H]$}, \\mbox{$\\rm [X\/Fe]$}), and can infer a star's age to a precision of $\\pm 0.52$ Gyr for the 2-dimensional abundance simulation and $\\pm 0.06$ Gyr for the 15-dimensional abundance simulation. We also model $R_\\text{birth}$\\ as a function of (\\mbox{$\\rm [Fe\/H]$}, age), and infer it to a precision of $\\pm 1.24$ kpc and $\\pm 1.17$ kpc for the 2- and 15-dimensional abundance simulations respectively. \n \n \\item The ability to reconstruct stellar groups born in different times and places from their abundances is determined by the formation history of the galaxy. When formation conditions lead to age and $R_\\text{birth}$\\ trends in the abundance plane with small dispersion, we find that there is a simple connection between clustered abundances and discrete birth times and places. Under clumpy star formation however, the simple relationship vanishes (Section \\ref{sec:victor}). \n \n \\item Our comparison of three simulations implies that the low dispersion of age across the \\mbox{$\\rm [\\alpha\/Fe]$}-\\mbox{$\\rm [Fe\/H]$}\\ plane of the Milky Way indicates that the Milky Way's star formation history is sufficiently quiet and that clustering in abundance will correspond to birth associations in time and location (Figure \\ref{fig:ageDisp}).\n \n\\end{itemize}\n\nWe seek to examine how abundance structure links to birth properties. We find that there is a simple relationship between age and chemical abundances, which agrees with previous work \\citep[e.g.][]{Ness2019}. $R_\\text{birth}$\\ cannot be tested as we can do for age --- we never have direct access to this quantity in observations. From our regression however, we see age and ([Fe\/H], \\mbox{$\\rm [X\/Fe]$}) link us to $R_\\text{birth}$\\ in the simulations. Indeed this analytical formalism has been adopted in models of radial migration \\citep[e.g.][]{Frankel2018,2019minchev}. We examine the $R_\\text{birth}$--age distribution further using the idea of abundance clustering, which we seek to see if it links to underlying physical processes. \n\nThis work highlights how we might use clustering of high dimensional abundance measurements in large surveys to infer groups of different birth place and time, and the impact of measurement uncertainty in working with the observational data. \n\n\\section{Acknowledgements} \nMelissa K Ness acknowledges support from the Sloan Foundation Fellowship. \nTobias Buck acknowledges support from the European Research Council under ERC-CoG grant CRAGSMAN-646955. This research made use of {\\sc{pynbody}} \\citet{pynbody}.\nWe gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (www.lrz.de).\nThis research was carried out on the High Performance Computing resources at New York University Abu Dhabi. We greatly appreciate the contributions of all these computing allocations.\nK.V.J. is supported by NSF grant AST-1715582.\nB.S. is supported by NSF grant DMS-2015376.\nVPD and LBS are supported by STFC Consolidated grant \\#ST\/R000786\/1\n\n\n\\section{Appendix - additional figures}\nHere we include abundance--age plots colored by $R_\\text{birth}$\\ for both the 2d and 15d simulations. These plots are similar to Figures \\ref{fig:Rbirthxfe_2d} and \\ref{fig:Rbirthxfe_hd}, however the coloring and x-axis are switched. \n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=\\textwidth]{buck_AgeXfe.pdf}\n\\caption{(\\textbf{Left}) The \\mbox{$\\rm [Fe\/H]$}--age plane colored by $R_\\text{birth}$\\ for the 2-dimensional simulation. The black lines and grey area mark off the solar metallicity stars, which we consider to be $\\pm$0.05 dex in \\mbox{$\\rm [Fe\/H]$}. (\\textbf{Right}) the running mean of [O\/Fe] of the solar metallicity stars across age colored by $R_\\text{birth}$\\ selected from within the horizontal lines at left. For a given bin of metallicity, stars clearly have a polynomial trend in \\mbox{$\\rm [X\/Fe]$}--age.}\n\\label{fig:Agexfe}\n\\end{figure*}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.92\\textwidth]{buckHD_AgeXfe.pdf}\n\\caption{(\\textbf{Top left}) The \\mbox{$\\rm [Fe\/H]$}--age plane colored by $R_\\text{birth}$\\ for the 15-dimensional simulation. The black lines and grey area mark off the solar metallicity stars, which we consider to be $\\pm$0.05 dex in \\mbox{$\\rm [Fe\/H]$}. All other plots show the running mean of \\mbox{$\\rm [X\/Fe]$}\\ of the solar metallicity stars accross age colored by $R_\\text{birth}$. Similar to the 2-dimensional simulation shown in Figure \\ref{fig:Agexfe}, solar metallicity stars of different ages separate into different polynomial curves.}\n\\label{fig:Agexfe_hd}\n\\end{figure*}\n\n\\pagebreak\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}