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+{"text":"\\section{Introduction}\n\\label{sec.intro}\n\nLet $a$ and $b$ be relatively prime positive integers and let $\\DD_{a,b}$ be the set of $(a,b)$-Dyck paths, lattice paths $\\P$ from $(0,0)$ to $(b,a)$ staying above the line $y=\\frac{a}{b}x$.  These paths are often called {\\em rational Dyck paths} and they generalize the classical and well-studied {\\em Dyck paths}.\n\n\\renewcommand*{\\thefootnote}{\\fnsymbol{footnote}}\n\nWe study a remarkable function $\\zeta$ on rational Dyck paths conjectured to be an automorphism\\footnote{After this article was accepted for publication, we learned that Nathan Williams proved that the zeta map and its sweep map brethren are indeed bijective using other methods. \\cite{Williams}}, which has received considerable attention lately; this ``zeta map'' generalizes the map on standard Dyck paths discovered by Haiman in the study of diagonal harmonics and $q,t$-Catalan numbers~\\cite{haglund2008q}.  Combinatorial definitions of $q,t$-statistics for classical Dyck paths were famously difficult to find, but were nearly simultaneously discovered by Haglund and Haiman.  Interestingly, they discovered two \\emph{different} pairs of statistics: Haiman found $\\ensuremath{\\mathsf{area}}$ and $\\ensuremath{\\mathsf{dinv}}$ shortly after Haglund discovered $\\ensuremath{\\mathsf{bounce}}$ and $\\ensuremath{\\mathsf{area}}$ statistics.  The zeta map was then uncovered, which satisfies $\\ensuremath{\\mathsf{bounce}}(\\zeta(\\P))=\\ensuremath{\\mathsf{area}}(\\P)$ and $\\ensuremath{\\mathsf{area}}(\\zeta(\\P))=\\ensuremath{\\mathsf{dinv}}(\\P)$.\n\nMany details about the zeta map have been gathered and unified in a comprehensive article by Armstrong, Loehr, and Warrington \\cite{ALW-sweep}, including progress on proving its bijectivity in certain cases such as $(a,am\\pm 1)$-Dyck paths \\cite{Loehr,GMII} (which is associated to the Fuss-Catalan numbers).  The zeta map was shown to be a bijection in these special cases by way of a ``bounce path'' by which zeta inverse could be computed.  However, constructing such a bounce path for the general $(a,b)$ case remains elusive.  Armstrong, Loehr, and Warrington showed that there is a much larger family of sweep maps (for which the zeta map is a special case) which extensive computational exploration suggests are also bijective.  A construct of theirs upon which we have relied heavily is the notion of the \\emph{levels} of a lattice path.\n\nRecent progress related to rational Dyck paths has been made in the case when $a\\leq3$ by Gorsky and Mazin and by Kaliszewski and Li \\cite{GMII, KL14}, when $a=4$ by Lee, Li, and Loehr \\cite{LLL14} in connection with the $q,t$-symmetry of the rational Catalan numbers.  A type $C$ analog of the zeta map has been introduced by Sulzgruber and Thiel \\cite{ST14}.  Rational Dyck paths also are intimately entwined in the study of rational parking functions and MacDonald polynomials, with recent work by Gorsky, Mazin, and Vazirani \\cite{GMV} and when $a$ and $b$ are not relatively prime by Bergeron, Garsia, Levin, and Xin~\\cite{BGLX}.\n\nOur goal is to explore the following conjecture:\n\n\\begin{conjecture}[\\cite{ALW-sweep,GMI}]\n\\label{conj.main}\nLet $a$ and $b$ be relatively prime positive integers.  The {\\em zeta map} $\\zeta:\\DD_{a,b}\\rightarrow\\DD_{a,b}$ is a bijection.\n\\end{conjecture}\n\nOur perspective is that there are in fact two maps, the zeta map and the eta map, which jointly contain enough information to recover the original path.  In Section~\\ref{sec:zetaeta}, we provide a straightforward algorithm for recovering $\\P$ from the combined data of $Q=\\zeta(\\P)$ and $R=\\eta(\\P)$.  What we find interesting is that the information contained solely in $\\zeta(\\P)$ does not seem to be enough to reconstruct $\\P$ directly.  Our argument does not give an explicit construction of $\\zeta^{-1}(Q)$, nor do we construct a bounce path.  \n\nThe zeta and eta maps appeared previously in the work of Gorsky and Mazin (See $G_{n,m}$ and~$G_{m,n}$ in \\cite{GMII}) and in the work of Armstrong, Loehr, and Warrington (varying the direction of the sweep map in \\cite{ALW-sweep}). Although, they were never used simultaneously as we do in this paper.  \nThe eta map is based on a natural notion of conjugation on rational Dyck paths explored in Section~\\ref{sec:conjugate} that arises from Anderson's bijection \\cite{And02} between $(a,b)$-Dyck paths and simultaneous $(a,b)$-core partitions, which in turn are related to many more combinatorial interpretations. (See \\cite{AHJ14} for additional background.) One can define the map $\\eta$  by $\\eta(\\P)=\\zeta(\\P^c)$; in most cases $\\zeta(\\P)\\neq \\eta(\\P)$.  Section~\\ref{sec:zeta_map} is devoted to presenting the algorithms for calculating the zeta map and the eta map in multiple fashions.  In particular, we present two new methods involving lasers and interval intersections.\n\nMeanwhile, $\\zeta$ and $\\eta$ combine to induce a new {\\em area-preserving} involution $\\chi$ on the set of Dyck paths defined in Section~\\ref{sec:perp} by \\[\\chi(Q):=\\eta(\\zeta^{-1}(Q))=\\zeta(\\zeta^{-1}(Q)^c).\\]  In Section~\\ref{sec:square}, we give a new proof that in the classical Catalan case, this {\\em conjugate-area map} $\\chi$ is the map that reverses the Dyck path.   Applying our inverse algorithm presents a new construction of the inverse of the zeta map on a Dyck path.  However, we have no explicit description of $\\chi(Q)$ from $Q$ in the general $(a,b)$-case.  Indeed, a concrete construction of $\\chi(Q)$ from $Q$ could be used to construct an explicit inverse for the zeta map.\n\nIn Section~\\ref{sec:inductive_zeta_inverse}, we show that when a rational Dyck path $Q$ visits the lattice point having level equal to 1, $\\zeta^{-1}(Q)$ has a nice decomposition as does its image under the conjugate-area map $\\chi$.  These observations allows us to explicitly find $\\chi$ (and therefore $\\zeta^{-1}$) of any path that has valleys exactly on levels equal to $\\{1,\\hdots,k\\}$ for $k<a$ in Theorem~\\ref{thm:inductive_area}.  We have also constructed $\\chi(Q)$ and $\\zeta^{-1}(Q)$ for paths that bound left-adjusted or up-adjusted partitions in Proposition~\\ref{prop:justified}.\n\nSection~\\ref{sec:9} investigates the poset of rational Dyck paths ordered by when one path is weakly below the other, motivating a new statistic $\\delta(P)$ that appears to be fruitful for recursively computing $\\zeta^{-1}$ from evidence gathered by computer learning algorithms.  Indeed, in the remainder of Section~\\ref{sec:9}, we use $\\delta(P)$ to construct the initial part of a rational bounce path and to give a new algorithm that computes $\\zeta^{-1}$ for $(a,am+1)$-Dyck paths. \n\n\\medskip\nOne of the primary motivations for our research was the study of conjectured statistics for the $q,t$-enumeration of $(a,b)$-Dyck paths. Section~\\ref{sec.notation} sets the stage by introducing key combinatorial concepts and statistics associated to $(a,b)$-Dyck paths.  In Section~\\ref{sec:skew_length} we investigate the \\emph{skew length} statistic $\\sl(\\P)$, originally defined in the context of $(a,b)$-cores in \\cite{AHJ14}.  The original definition of skew length seems to depend on the ordering of $a$ and $b$; we show that skew length is in fact independent of this choice.  The main tools we develop involve a row length filling of the boxes under the $(a,b)$-Dyck path $\\P$ and above the main diagonal, along with the idea of skew inversions and flip skew inversions.   Section~\\ref{sec:conjugate} shows that \\emph{skew length} is preserved under conjugation.  \n \n\\section{Background and Notation}\n\\label{sec.notation}\n\n\\begin{definition}\nAn {\\em $(a,b)$-lattice path} $\\P$ is a lattice path in $\\mathbb{Z}^2$ consisting of north and east steps starting from the origin and ending at the point $(b,a)$.\n\nWe call $\\P$ an {\\em $(a,b)$-Dyck path} if $\\P$ remains (weakly) above the diagonal line connecting the origin to $(b,a)$.  Equivalently, the lattice points $(x,y)$ along $\\P$ satisfy $ax\\leq by$. We draw $(a,b)$-Dyck paths in an $a\\times b$ grid, where the lower left corner is the origin. \n\nWe denote the full collection of $(a,b)$-Dyck paths by $\\DD_{a,b}$, or simply $\\DD$ if there is no confusion about the values of $a$ and $b$.\n\\end{definition}\n\nWe use the English notation for Young diagrams, drawing the largest row at the top.  The \\emph{hook length} of a box $B$ in the Young diagram of a partition is the number of boxes in the \\emph{hook} of boxes directly below or directly to the right of $B$, including the box $B$ itself.  An \\emph{$a$-core partition} (or simply $a$-core) is a partition for which its Young diagram has no boxes with hook length equal to~$a$.  Similarly, a \\emph{simultaneous $(a,b)$-core partition} (or $(a,b)$-core for short) has no hooks equal to $a$ or $b$.\n\nAnderson proved that when $a$ and $b$ are relatively prime there are finitely many $(a,b)$-cores \\cite{And02} by finding a bijection with the set of $(a,b)$-Dyck paths; these are counted by the formula\n\\[\n\\frac{1}{a+b}\\binom{a+b}{a}.\n\\]\nThis formula seems to have been discovered at various times; the earliest reference we know of is~\\cite{DM47} in 1947.  In 1954, Bizley considered the general case of rectangular Dyck paths of which this formula is a special case~\\cite{Biz54}.\n\nBizley's counting method starts from the full set of lattice paths from $(0,0)$ to $(b,a)$, and considers the orbit of the cyclic group $C_{a+b}$ acting by \\emph{cyclic shifts} on paths.  In the case where $a$ and $b$ are relatively prime, there is a unique Dyck path in each such orbit.\n\n\\begin{example} Let $N$ and $E$ represent a north step and an east step, respectively.  Throughout this paper, we will use as our running example the $(5,8)$-Dyck path \\[\\P=NNNENEEENEEEE,\\] shown in Figure~\\ref{fig:example58}.   \n\\end{example}\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[width=0.8\\textwidth]{example58.pdf} \n  \\end{center}\n  \\caption{(Left) A lattice path $\\P$ when $a=5$ and $b=8$.  The hook filling is given by the numbers in the center of the boxes.  The boxes above the path show that the partition bounded by $\\P$ is $(4,1)$. \\newline(Right) The levels of the lattice points along the path.}\n  \\label{fig:example58}\n\\end{figure}\n\n \n\\newpage\\subsection{Dictionary of notation}\\\n\nWe keep track of numerous bits of data associated to an $(a,b)$-Dyck path $\\P$. \n\n\\begin{enumerate}\n\\item {\\bf General constructions:}\n\n\\begin{itemize}\n\\item The \\emph{hook filling} of the boxes in the square lattice is obtained by filling the box with lower-right lattice point $(b,0)$ with the number $-ab$ and increasing by $a$ for every one box west and increasing by $b$ for every one box north. A box is above the main diagonal if and only if the corresponding hook is positive. (See Figure~\\ref{fig:example58}.)  \n\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[width=0.7\\textwidth]{AndersonsBijection.pdf}\n  \\end{center}\n  \\caption{Anderson's bijection gives a correspondence between $(a,b)$-Dyck paths and $(a,b)$-core partitions.  Corresponding to $\\P=NNNENEEENEEEE$ is the $(5,8)$-core $(6,4,3,2,2,1,1,1,1)$.}\n  \\label{fig:AndersonsBijection}\n\\end{figure}\n\n\\item The \\emph{positive hooks} of $\\P$ are the numbers in the hook filling below the path but greater than zero.  (Elsewhere these have been called {\\em beta numbers} or {\\em bead numbers}.)  \n\n\\item We denote by $\\mathfrak{c}(\\P)$ the $(a,b)$-core corresponding to $\\P$ under Anderson's bijection. \nThe hook lengths of the boxes in the first column of $\\mathfrak{c}(\\P)$, its {\\em leading hooks}, are precisely the positive hooks of $\\P$. An example of Anderson's bijection is illustrated in Figure~\\ref{fig:AndersonsBijection}.\n\n\\item The \\emph{row length filling} of $\\P$ are numbers placed in the boxes under $\\P$. They correspond to the number of boxes in the row of $\\mathfrak{c}(\\P)$ with the given hook. This will be developed in Section~\\ref{sec:length_filling}.  (See Figure~\\ref{fig:rowLength}.)\n\\item The \\emph{partition bounded by} $\\P$ is the partition whose Young diagram is the collection of boxes above the path $\\P$.\n\\end{itemize}\n\n\n\\item {\\bf Combinatorial statistics:}\n\\begin{itemize}\n\\item The \\emph{area} of $\\P$, denoted $\\ensuremath{\\mathsf{area}}(\\P)$ is the number of positive hooks of $\\P$. Equivalently, this is the number of rows in $\\mathfrak{c}(\\P)$.\n\n\\item The \\emph{rank} of $\\P$, denoted $\\operatorname{rk}(\\P)$ is the number of rows in the partition bounded by $\\P$.\n\\item The \\emph{skew length} of $\\P$, denoted $\\sl(\\P)$ is a statistic that we discuss in detail in Section~\\ref{sec:skew_length}.\n\\end{itemize}\n\\item {\\bf Sets and sequences of numbers associated to $\\P$:}\n\\begin{itemize}\n\n\\item The \\emph{levels} of $\\P$ are labels associated to the lattice points of $\\P$ defined by Armstrong, Loehr, and Warrington in \\cite{ALW-parking,ALW-sweep}.  Assign level $0$ to $(0,0)$ and label the other lattice points of $\\P$ by adding $b$ after each north step and subtracting $a$ after each east step.  Equivalently, this is the value of the hook filling in the box to the northwest of the lattice point.  Note that the label of the northeast-most lattice point $(b,a)$ is once again $0+a\\cdot b-a \\cdot b=0$.\n\n\\item The path $\\P$ has two reading words obtained by reading the levels in order.  The \\emph{reading word} of $\\P$, denoted $L(\\P)$ (for `levels'), is obtained by reading the levels that occur along the path from southwest to northeast, excluding the final $0$.  (One can imagine assigning to each north and east step of the path the level of the step's initial lattice point.)\n\nThe \\emph{reverse reading word}, denoted $M(\\P)$, is obtained by reading from northeast to southwest, excluding the final $0$. (One can imagine $\\P$ as a path from $(b,a)$ to $(0,0)$ consisting of west and south steps, once again assigning to each step  the level of its initial lattice point.)\n\nReading along $\\P$ in Figure~\\ref{fig:example58} shows that\n\\[L(\\P)=(0,8,16,{24},19,{27},{22},{17},12,{20},{15},{10},{5})\\] and \n\\[M(\\P)=({0},{5},{10},{15},20,{12},{17},{22},27,{19},24,16,8).\\]\nWhen $a$ and $b$ are relatively prime, no value occurs more than once in $L(\\P)$ or $M(\\P)$.\n\n\\item The set of levels of $\\P$ is partitioned into the set of \\emph{north levels} $\\N(\\P)$ and \\emph{east levels} $\\E(\\P)$, where when reading from southwest to northeast, levels of lattice points starting north steps of $\\P$ are in $\\N(\\P)$ and levels of lattice points starting east steps of $\\P$ are in $\\E(\\P)$. We order these levels in decreasing order.  In our running example, the north levels of $\\P$ are \n\\[\\N(\\P)=\\{19,16,12,8,0\\},\\] \nand the east levels of $\\P$ are \n\\[\\E(\\P)=\\{27,24,22,20,17,15,10,5\\}.\\]\n\\end{itemize}\n\n\\item {\\bf Permutations associated to $\\P$:}\nThroughout the paper we use square brackets to write permutations in one-line notation, and round parentheses for permutations in cycle notation.\\smallskip\n\n\\begin{itemize}\n\n\\item The \\emph{reading permutation} of $\\P$ is a permutation $\\sigma$ in $S_{a+b}$ that encodes the relative order of the levels recorded in $L(\\P)$. The \\emph{reverse reading permutation} of $\\P$, denoted~$\\tau(\\P)$, encodes the relative order of the values in~$M(\\P)$. In our running example, the one-line notation for $\\sigma(\\P)$ and $\\tau(\\P)$ are \n\\[\\sigma(\\P)=[1,3,7,{12},9,{13},{11},{8},5,{10},{6},{4},{2}]\\] \nand \n\\[\\tau(\\P)=[{1},{2},{4},{6},10,{5},{8},{11},13,{9},12,7,3].\\] \n\n\\item Let $\\gamma(\\P)$ be the permutation in $S_{a+b}$ that when written in cycle notation starting with $1$ has the same order of entries as $\\sigma(\\P)$ written in one-line notation.  In our running example $\\P$ we have\n\\begin{align*} \n\\gamma(\\P)&= (1,3,7,{12},9,{13},{11},{8},5,{10},{6},{4},{2})\\\\ \n &=  [3,1,7,2,10,4,12,5,13,6,8,9,11].\n\\end{align*}\n\\end{itemize}\n\\end{enumerate}\n\n\n\\renewcommand*{\\thefootnote}{\\fnsymbol{footnote}}\n\\setcounter{footnote}{1}\n\n\\begin{remark}\\label{rem:sigma}\nThe path $\\P$ can be recovered knowing only $\\sigma(\\P)$ (or $\\tau(\\P)$ or $\\gamma(\\P)$).  \nThe east steps of~$\\P$ correspond exactly to the right (cyclic) descents\\footnote{A descent of a permutation occurs when $\\sigma(i)>\\sigma(i+1)$.  A cyclic descent is defined in the same way, but considering the indices modulo $a+b$, allowing a descent in the last position of $\\sigma$.} of $\\sigma$; whereas, the north steps of $\\P$ correspond to the right (cyclic) ascents of $\\sigma$. In our running example, the right (cyclic) descents of $\\sigma(\\P)$ occur in positions $4$, $6$, $7$, $8$, $10$, $11$, $12$, and $13$, which are exactly the positions of the east steps in $\\P$.\n\\end{remark}\n\n\\section{Skew length}\\label{sec:skew_length}\n\nIn~\\cite{AHJ14} the \\emph{skew length} statistic is proposed as a $q$-statistic for $(a,b)$-Dyck paths and a related construction is investigated in \\cite[Section 4]{ALW-parking}. In this section, we present the original definition of skew length on cores and two equivalent interpretations on $(a,b)$-Dyck paths using length fillings and skew inversions.  We show that these interpretations are indeed equivalent to the original definition and, as a consequence, we prove that skew length is independent of the ordering of $a$ and $b$.  Further interpretations of skew length are presented in terms of the zeta map in Section~\\ref{sec:zeta_map}.\n\n\\subsection{Skew length on cores and polynomial motivation}\\\n\nWe begin with an observation on ordinary core partitions before discussing simultaneous core partitions.\n\n\\begin{definition}[{\\cite[Definition 2.7]{AHJ14}}]\nLet $\\kappa$ be an $a$-core partition.  Consider the hook lengths of the boxes in the first column of $\\kappa$.  Find the largest hook length of each residue modulo $a$.  The \\emph{$a$-rows} of $\\kappa$ are the rows of $\\kappa$ corresponding to these hook lengths. The \\emph{$a$-boundary} of $\\kappa$ consists of all boxes in its Young diagram with hook length less than $a$.  \\end{definition}\n\n\\begin{proposition}\n\\label{prop:arows}\nLet $\\kappa$ be an $a$-core partition.  The number of boxes in the $a$-rows of $\\kappa$ equals the number of boxes in the $a$-boundary of $\\kappa$.\n\\end{proposition}\n\n\\begin{proof}\nLet $\\operatorname{len}(h)$ be the number of boxes in the row of $\\kappa$ with leading hook $h$.\n\nWe first observe that if $h>a$ is a leading hook of $\\kappa$, then $h-a$ is also a leading hook of $\\kappa$. For this, decompose $h$ into two hooks of lengths $h-a$ and $a$ as illustrated in Figure~\\ref{fig:skewLength_proof}, such that the boxes in the row with leading hook $h$ that are intersected by the hook $a$ are exactly the boxes in the $a$-boundary in that row. This guarantees that the right-end box of the hook $h-a$ is in~$\\kappa$, and therefore that $h-a$ is also a leading hook.\n\nNow, the number of $a$-boundary boxes in the row of $\\kappa$ corresponding to $h$ is $\\operatorname{len}(h)-\\operatorname{len}(h-a)$. Summing over all rows gives the number of $a$-boundary boxes; telescoping over residues modulo $a$ gives the number of boxes in the $a$-rows of $\\kappa$.\n\\end{proof}\n\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=0.5]{skewLength_proof.pdf}\n  \\end{center}\n  \\caption{The number of $a$-boundary boxes in the row of $\\kappa$ corresponding to a leading hook $h$ is $\\operatorname{len}(h)-\\operatorname{len}(h-a)$.}\n  \\label{fig:skewLength_proof}\n\\end{figure}\n\n\\begin{corollary}\nThe number of boxes in the $a$-rows of $\\kappa$ equals the number of boxes in the $a$-rows of $\\kappa^c$\n\\end{corollary}\n\n\\begin{remark}\nFor readers familiar with the abacus diagram interpretation, hook lengths correspond to beads on the abacus; the $a$-rows correspond to the largest bead on each runner of the $a$-abacus.  Proposition~\\ref{prop:arows} gives a way to count the number of boxes in the $a$-boundary of an $a$-core by adding the number of gaps that appear on the abacus before each of these largest beads.\n\\end{remark}\n\n\\begin{definition}[{\\cite[Definition 2.7]{AHJ14}}]\n\\label{def:skewlen}\nLet $\\kappa$ be an $(a,b)$-core partition. The \\emph{skew length} of~$\\kappa$, denoted $\\sl(\\kappa)$, is the number of boxes simultaneously located in the $a$-rows and the $b$-boundary of~$\\kappa$.\n\\end{definition}\n\n\n\\begin{example}\nThe core partition shown in Figure~\\ref{fig:skewLength_cores} is the $(5,8)$-core $\\kappa=\\mathfrak{c}(\\P)$ corresponding to the path $\\P$ in our running example from Figures~\\ref{fig:example58} and \\ref{fig:AndersonsBijection}. \n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=0.6]{skewLength_cores.pdf}\n  \\end{center}\n  \\caption{(Left) The $8$-boundary boxes of our favorite $(5,8)$-core $\\kappa$ are shaded; those in the $5$-rows of $\\kappa$ are darker. \n  (Right) The $5$-boundary boxes of $\\kappa$ are shaded; those in the $8$-rows of $\\kappa$ are darker.\n  Surprisingly, the number of darkly shaded boxes on the left $4+3+2+1=10$ is equal to the number of darkly shaded boxes on the right $3+2+2+1+1+1=10$. (See Corollary~\\ref{cor.absl}.)}\n  \\label{fig:skewLength_cores}\n\\end{figure}\n\nOn the left, the $5$-rows of $\\kappa$ are the rows with leading hook lengths 14, 11, 7, and 3.   The darkly shaded boxes are those boxes in the $5$-rows with hook length less than 8.  The skew length is equal to $4+3+2+1=10$.  \n\nOn the right, we compute of the skew length of $\\kappa$ when considered as an $(8,5)$-core. The $8$-rows of $\\kappa$ are the rows with leading hook lengths 14, 11, 9, 7, 4, and 2. The shaded boxes are those boxes in the $8$-rows with hook length less than 5. The skew length is equal to $3+2+2+1+1+1=10$.\n\nWe will see in Corollary~\\ref{cor.absl} that it is not a coincidence that these two numbers are the same.  \n\nThe number of boxes in the $8$-boundary (shaded boxes, left) equals the number of boxes in the $8$-rows (marked rows, right) and the number of boxes in the $5$-boundary (shaded boxes, right) equals the number of boxes in the $5$-rows (marked rows, left), as proved in general in Proposition~\\ref{prop:arows}.\n\\end{example}\n\n\n\nThe skew length statistic was found by Armstrong; he conjectures it as a key statistic involved in the $q$- and $q,t$-enumeration of $(a,b)$-cores (or $(a,b)$-Dyck paths). Recall that the rank $\\operatorname{rk}(\\kappa)$ of an $(a,b)$-core $\\kappa$ is the number of rows in its corresponding Young diagram. \n\n\\begin{conjecture}\\cite[Conjecture 2.8]{AHJ14}\n\\label{conj.qcount}\nLet $a$ and $b$ relatively prime positive integers. \nThe expression\n\\[\nf_{a,b}(q) = \\frac{1}{[a+b]_q}\\begin{bmatrix}a+b\\\\ a\\end{bmatrix}_q\n\\]\nis equal to the polynomial \\[g_{a,b}(q) = \\sum_\\kappa q^{\\sl(\\kappa)+\\operatorname{rk}(\\kappa)},\\] where the sum is over all $(a,b)$-cores $\\kappa$.\n\\end{conjecture}\n\nHaiman \\cite[Propositions 2.5.2 and 2.5.3]{Haiman93} proved that $f_{a,b}(q)$ is a polynomial if and only if $a$ and $b$ are relatively prime. \n\\cite[Theorem 1.10]{BEG03} provides a proof that $f_{a,b}(q)$ has non-negative coefficients involving representation theory of rational Cherednik algebras, see also~\\cite[Section 1.12]{gordon_catalan_2012}. A proof of Conjecture~\\ref{conj.qcount} would provide a combinatorial interpretation for the coefficients of $f_{a,b}(q)$.\n\n\\begin{proposition}\\cite{Haiman93,BEG03}\n\\label{prop.qdef}\n The expression\n\\[\nf_{a,b}(q) = \\frac{1}{[a+b]_q}\\begin{bmatrix}a+b\\\\ a\\end{bmatrix}_q\n\\]\nis a polynomial if and only if $\\gcd(a,b)=1$.  Furthermore, when $a$ and $b$ are relatively prime, the resulting polynomial has integer coefficients.\n\\end{proposition}\n\nDefine the \\emph{co-skew length} of an $(a,b)$-core $\\kappa$ as \n\\[\n\\sl'(\\kappa):=\\frac{(a-1)(b-1)}{2}-\\sl(\\kappa).\n\\]\nArmstrong conjectures that rank and co-skew length give a $q,t$-enumeration of the $(a,b)$-cores, subject to the following symmetry:\n\\begin{conjecture}\\cite[Conjecture 2.9]{AHJ14}\n\\label{conj.qtsym}\nThe following $q,t$-polynomials are equal:\n\\[\n\\sum q^{\\operatorname{rk}(\\kappa)}t^{\\sl'(\\kappa)} = \\sum q^{\\sl'(\\kappa)}t^{\\operatorname{rk}(\\kappa)}\n\\]\nwhere the sum is over all $(a,b)$-cores $\\kappa$.\n\\end{conjecture}\nThese $q,t$-polynomials are called the \\emph{rational $q,t$-Catalan numbers}.\n\n\\subsection{Skew length on Dyck paths via the row length filling}\\label{sec:length_filling}\\\n\nWe now provide a new method to calculate the skew length of an $(a,b)$-Dyck path $\\P$ which uses a {\\em row length filling} of the boxes below~$\\P$. Our method recovers with the skew length statistic discovered by Armstrong for $(a,b)$-cores. As a consequence, we show that skew length of an $(a,b)$-core is independent of the ordering of $a$ and $b$.\n\nWe provide two equivalent definitions of the {\\em row length filling}.  \n\n\\begin{definition}\\label{def:rlf}\nLet $\\P$ be an $(a,b)$-Dyck path.  The \\emph{row length filling} of $\\P$ is an assignment of numbers to each box below the path~$\\P$.  \n\nFor a box $B$ with positive hook filling $h$, define the row length of $B$ to be the length of the row in $\\mathfrak{c}(\\P)$ with leading hook $h$.  Alternatively, define the row length of $B$ to be $h-p_h$, where $p_h$ is the number of positive entries in the hook  filling strictly less than~$h$.  \n\nFor a box $B$ with non-positive hook filling $h$, define the row length of $B$ to be zero.\n\nFor any hook $h$ in the hook filling of $\\P$, we denote by~$\\operatorname{len}(h)$ the corresponding value of the row length filling of $\\P$.\n\\end{definition}\n\nFigure~\\ref{fig:rowLength} shows in red in the upper left corner the row length of the boxes corresponding to the positive hooks of $\\P$.\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=0.6]{rowLength.pdf}\n  \\end{center}\n  \\caption{The row length filling of boxes below the path $\\P$ is given in red in the upper left corner.  The values correspond to the length of the rows of $\\mathfrak{c}(\\P)$ in Figure~\\ref{fig:AndersonsBijection}.}\n  \\label{fig:rowLength}\n\\end{figure}\n\n\\begin{lemma}\nThe two definitions of row length filling in Definition~\\ref{def:rlf} are equivalent.\n\\end{lemma}\n\n\\begin{proof}\nWhen ordered in increasing order, the entries in the hook filling of $\\P$ correspond to the hook lengths of the boxes in the first column of $\\mathfrak{c}(\\P)$ from shortest to longest. Suppose the first box of the $i$th shortest row has hook length $h$. Then the length of the $i$th shortest row is $h-(i-1)$, which is exactly the corresponding entry in the row length filling.\n\\end{proof}\n\n\\begin{remark}\nFor readers familiar with the abacus diagram interpretation, the row length filling associates to each bead on the abacus the number of gaps that appear before it on the abacus.\n\\end{remark}\n\nThe row length filling is very useful for reading off common core statistics \nfrom the Dyck path.  For example, we can immediately see that:\n\\begin{corollary}\nThe sum of the entries of the row length filling of $\\P$ is equal to the number of boxes of the core $\\mathfrak{c}(\\P)$.\n\\end{corollary}\nFurthermore, because the $a$-rows of $\\mathfrak{c}(\\P)$ correspond to the westmost boxes under $\\P$ and the $b$-rows of $\\mathfrak{c}(\\P)$ correspond to the northmost boxes under $\\P$, the number of boxes in $\\mathfrak{c}(\\P)$ with hook length less than $a$ or less than $b$ can be determined from the row length filling as a direct consequence of Proposition~\\ref{prop:arows}.\n\\begin{corollary}\n\\label{cor.boundaryBoxes}\nThe number of boxes in the $a$-boundary of an $(a,b)$-core $\\mathfrak{c}(\\P)$ is equal to the sum of the row length fillings of the westmost boxes under $\\P$.  Likewise, the number of boxes in the $b$-boundary of $\\mathfrak{c}(\\P)$ is equal to the sum of the row length fillings of the northmost boxes under $\\P$.\n\\end{corollary}\n\nIn the same vein, the skew length of $\\P$ can also be easily computed, as follows: \n\n\\begin{theorem}\n\\label{thm.skewLengthCompute}\nThe skew length of an $(a,b)$-core $\\mathfrak{c}(\\P)$ may be computed from the row length filling of~$\\P$ by adding all lengths at peaks of $\\P$ and subtracting all lengths at valleys of $\\P$.\n\\end{theorem}\n\n\\begin{proof}\nBy the argument in the proof of Proposition~\\ref{prop:arows}, we see that when $h$ is a positive hook of an $(a,b)$-Dyck path $\\P$ (so that $h-a$ is the hook of the box directly east of the box with hook $h$ and $h-b$ is the hook of the box directly south of the box with hook $h$), then\n\\begin{enumerate}[(i)]\n\\item The number of $a$-boundary boxes in the row of $\\mathfrak{c}(\\P)$ corresponding to $h$ is $\\operatorname{len}(h)-\\operatorname{len}(h-a)$.\n\\item The number of $b$-boundary boxes in the row of $\\mathfrak{c}(\\P)$ corresponding to $h$ is $\\operatorname{len}(h)-\\operatorname{len}(h-b)$.\n\\end{enumerate}\nBy restricting to the $a$-rows or $b$-rows, we see that the skew length of $\\mathfrak{c}(\\P)$ is given by:\n\\begin{equation}\\label{eqn.lenSum_a}\n\\sum \\operatorname{len}(h)-\\operatorname{len}(h-b),\n\\end{equation}\nwhere the sum is over all westmost boxes under $\\P$, or alternatively the skew length of $\\mathfrak{c}(\\P)$ is given by:\n\\begin{equation}\\label{eqn.lenSum_b}\n\\sum \\operatorname{len}(h)-\\operatorname{len}(h-a),\n\\end{equation}\nwhere the sum is over all northmost boxes under $\\P$.\nWhen one westmost box under $\\P$ is directly north of another, Formula~\\eqref{eqn.lenSum_a} telescopes.  After cancelling terms, we are left with the lengths at peaks of $\\P$ minus the lengths at valleys of $\\P$.  An equivalent argument can be made from Formula~\\eqref{eqn.lenSum_b}.\n\\end{proof}\n\n\\begin{example}\nIn Figure~\\ref{fig:rowLength}, we see that the sum of the row length fillings is 21, which is the number of boxes of $\\mathfrak{c}(\\P)$.  Adding the row lengths of the westmost boxes under $\\P$ gives $2+6+4+1+0=13$ boxes in the $5$-boundary of $\\mathfrak{c}(\\P)$, while adding the row lengths of the northmost boxes under $\\P$ gives $4+6+3+1+2+1+0+0=17$ boxes in the $8$-boundary of $\\mathfrak{c}(\\P)$, as expected from Figure~\\ref{fig:skewLength_cores}. Our path $\\P$ has three peaks with row lengths 2, 6, and 4 and two valleys with row lengths 2 and 0. \nThe skew length of our path is then\n\\[\\sl(\\P)=(2+6+4)-(2+0)=10.\\]  \n\\end{example}\n\nWhen computing skew length directly from the core, it is not obvious that the number of boxes in $a$-rows and the $b$-boundary should be equal to the number of boxes in $b$-rows and the $a$-boundary (see Figure~\\ref{fig:skewLength_cores}).  But the method of computing the skew length given by Theorem~\\ref{thm.skewLengthCompute} is independent of the ordering of $a$ and $b$: Switching $a$ and $b$ flips the rectangle to a $b\\times a$ rectangle in which peaks are still peaks, valleys are still valleys, and the hook filling and row length filling are otherwise unaffected.\n\n\\begin{corollary}\n\\label{cor.absl}\nThe skew length of an $(a,b)$-core $\\kappa$ is independent of the ordering of $a$ and $b$.\n\\end{corollary}\n\n\n\n\\subsection{Skew length via skew inversions}\\\n\nThis section presents another interpretation of the skew length of an $(a,b)$-Dyck path $\\P$ in terms of the number of its {\\em skew inversions} or the number of its {\\em flip skew inversions}. \n\nRecall that the north levels of $\\P$ are the levels $\\N(\\P)=\\{n_1,\\dots,n_a\\}$ of the initial lattice points of the north steps in the path, and that the east levels of $\\P$ are the levels $\\E(\\P)=\\{e_1,\\dots,e_b\\}$ of the initial lattice points of the east steps.\n\n\\begin{definition}\nA \\emph{skew inversion} of $\\P$ is a pair of indices $(i,j)$ such that $n_i>e_j$.  \nA \\emph{flip skew inversion} of $\\P$ is a pair of indices $(i,j)$ with $n_i+b<e_j-a$. \n\\end{definition}\n\n\\begin{theorem}\\label{thm:skewLength-skewInversions} Let $\\P$ be an $(a,b)$-Dyck path. The skew length of $\\P$ equals the number of skew inversions of $\\P$, which is equal to the number of flip skew inversions of $\\P$.\n\\end{theorem}\n\nThe key to the proof of Theorem~\\ref{thm:skewLength-skewInversions} is recognizing the relationship between westmost boxes under $\\P$ and north levels in $\\N(\\P)$ and the relationship between northmost boxes under $\\P$ and east levels in $\\E(\\P)$. \n\n\\begin{remark}\n\\label{rem:hook-north-east}\nFigure~\\ref{fig:hook-north-east} shows that when $h$ is the hook filling of a westmost box under $\\P$, then the associated north level $n_h$ (corresponding to the lattice point at its southwest corner) is $h+a$.  When $h$ is the hook filling of a northmost box under $\\P$, then the associated east level $e_h$ (corresponding to the lattice point at its northwest corner) is $h+a+b$. \n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=0.5]{hook-north-east.pdf}\n  \\end{center}\n  \\caption{When $h$ is the hook filling of a westmost box under $\\P$, the associated north level is $n_h=h+a$.  When $h$ is the hook filling of a northmost box under $\\P$, the associated east level is $e_h=h+a+b$.}\n  \\label{fig:hook-north-east}\n\\end{figure}\n\\end{remark}\n\n\\begin{lemma}\\label{lem:skew_inversions_length_dif}\nLet $h$ be the hook filling of a westmost box under an $(a,b)$-Dyck path $\\P$. The length difference $\\operatorname{len}(h)-\\operatorname{len}(h-b)$ is equal to the number of skew inversions involving the associated north level $n_h$, which equals the number of $b$-boundary boxes in the $a$-row corresponding to $h$.\n\\end{lemma}\n\n\\begin{proof}\nRecall that $\\operatorname{len}(h)=h-p_h$, where $p_h$ is the number of positive hooks in the hook filling of $\\P$ less than $h$.  Then:\n\\begin{eqnarray*}\n\\operatorname{len}(h)-\\operatorname{len}(h-b)&=&h-p_h - (h-b) + p_{h-b}\\\\\n  &=&b-(p_h-p_{h-b})\\\\\n  &=&b-\\#\\{g\\mid h-b\\leq g< h\\}.\n\\end{eqnarray*}\n\nEach box with hook filling $g$ satisfying the inequalities $h-b\\leq g< h$ is in a distinct column of the diagram of $\\P$. If two were in the same column, then the difference of their hooks would be a multiple of $b$, so that both could not satisfy the inequality.  As a result, we may add a multiple of $b$ to each $g$ satisfying the inequalities to obtain a unique northmost box  under $\\P$ with hook filling $\\bar{g}$ satisfying $h-b\\leq \\bar{g}$. Conversely, for every northmost box under $\\P$ with hook filling $\\bar{g}$ satisfying this inequality there is a unique box with hook filling $g$ in the same column satisfying $h-b\\leq g< h$. Therefore,\n\n\\begin{equation*}\n\\begin{aligned}\n\\operatorname{len}(h)-\\operatorname{len}(h-b)&=b-\\#\\{\\bar{g}\\mid h-b\\leq \\bar{g}\\}\\\\\n&=\\#\\{\\bar{g}\\mid h-b> \\bar{g}\\}.\n\\end{aligned}\n\\end{equation*}\n\nBy Remark~\\ref{rem:hook-north-east}, this is equivalent to $\\operatorname{len}(h)-\\operatorname{len}(h-b)=\\#\\{e_j\\mid n_h> e_j\\}$, as desired.  The last clause of the statement of the lemma is given in the proof of Theorem~\\ref{thm.skewLengthCompute}.\n\\end{proof}\n\nSimilar arguments prove the following.\n\\begin{lemma}\\label{lem:skew_inversions_length_dif_b}\nLet $h$ be the hook filling of a northmost box under an $(a,b)$-Dyck path $\\P$. The length difference $\\operatorname{len}(h)-\\operatorname{len}(h-a)$ is equal to the number of flip skew inversions involving the associated east level $e_h$, which equals the number of $a$-boundary boxes in the $b$-row corresponding to $h$.\n\\end{lemma}\n\nTheorem~\\ref{thm:skewLength-skewInversions} now follows directly from Definition~\\ref{def:skewlen} by summing over all westmost boxes in Lemma~\\ref{lem:skew_inversions_length_dif} and all northmost boxes in Lemma~\\ref{lem:skew_inversions_length_dif_b}.\n\n\\begin{example}\\label{ex:skew_inversions}\nIn our running example, the north levels are~$\\N=\\{19,16,12,8,0\\}$ and the east levels are~$\\E=\\{27,24,22,20,17,15,10,5\\}$. There are 10 skew inversions because there are 4 east levels less than $n_1=19$, 3 east levels less than~$n_2=16$, 2 east levels less than~$n_3=12$, 1 east level less than~$n_4=8$, and 0 east levels less than~$n_5=0$. The total number of skew inversions is then $4+3+2+1+0=10$. \nThese numbers correspond to the number of $b$-boundary boxes in the $a$-rows of the core $\\mathfrak{c}(\\P)$ in Figure~\\ref{fig:skewLength_cores}.\n\nTo calculate the flip skew inversions, consider the sets $\\N+b=\\{27,24,20,16,8\\}$ and $\\E-a=\\{22,19,17,15,12,10,5,0\\}$.  There are 10 flip skew inversions because there are 3 elements of the form~$n_i+b$ less than~$e_1-a=22$, there are 2 less than~$e_2-a=19$, 2 less than~$e_3-a=17$, 1 less than~$e_4-a=15$, 1 less than~$e_5-a=12$, 1 less than~$e_6-a=10$, 0 less than~$e_7-a=5$, and 0 less than~$e_8-a=0$. The total number of flip skew inversions is then $3+2+2+1+1+1+0+0=10$. \nThese numbers correspond to the number of $a$-boundary boxes in the $b$-rows of the of the core $\\mathfrak{c}(\\P)$.\n\\end{example}\n\n\\begin{remark} Skew inversions in an $(a,b)$-Dyck path arise from pairs of north levels and east levels where $n_i>e_j$.  Note that $n_i+b$ is the level of the terminal lattice point of the corresponding north step (instead of initial lattice point), while $e_j-a$ is the level of the terminal lattice point of the corresponding east step.  So flip skew inversions are best understood by a reverse reading of $\\P$ as a sequence of west and south steps, counting the pairs where the south level is less than the west level.  Alternatively, flip skew inversions of $\\P$ correspond to skew inversions of $\\P$ when $\\P$ is reflected (flipped) to be a $(b,a)$-Dyck path. \n\\end{remark}\n\n\n\\section{The conjugate map}\n\\label{sec:conjugate}\n\nFor any partition $\\kappa$, its conjugate partition $\\kappa^c$ is obtained by reflecting along its main diagonal. (See Figure~\\ref{fig:conjugate_cores}.)  Since hook lengths are preserved under this reflection, when $\\kappa$ is an $(a,b)$-core, so is $\\kappa^c$.  When $a$ and $b$ are relatively prime, there is a natural conjugate map on $(a,b)$-Dyck paths~$\\P$.  Apply cyclic shifts to the path $\\P$ until we encounter a path strictly \\emph{below} the diagonal, the conjugate path $\\P^c$ is the result of rotating this path~$180^\\circ$. (See Figure~\\ref{fig:conjugate_paths}.) The first main result of this section (Theorem~\\ref{thm:conjugation}) shows that these conjugations are equivalent under Anderson's bijection, and the second (Theorem~\\ref{thm.slconj}) shows that conjugation preserves skew length. \nThese two results were simultaneously found in independent work by Xin in~\\cite{xin_rank_2015}.  Lemmas \\ref{lem:conjugate_hooks} and \\ref{lem:conjugate_positive_hooks} mirror the notion of conjugation of the semimodule of leading hooks presented by Gorsky and Mazin \\cite{GMII}. \n\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=0.5]{conjugate_cores.pdf}\n  \\end{center}\n  \\caption{The conjugate map on $(a,b)$-cores.}\n  \\label{fig:conjugate_cores}\n\\end{figure}\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=0.5]{conjugate_paths.pdf}\n  \\end{center}\n  \\caption{The conjugate map on $(a,b)$-Dyck paths.}\n  \\label{fig:conjugate_paths}\n\\end{figure}\n\n\\begin{theorem}\\label{thm:conjugation}\nConjugation on $(a,b)$-cores coincides with conjugation on $(a,b)$-Dyck paths via Anderson's bijection:\n\\[\n\\mathfrak{c}(\\P)^c = \\mathfrak{c}(\\P^c).\n\\]\n\\end{theorem}\n\nThis follows directly by showing the equivalence between the leading hooks of $\\mathfrak{c}(\\P)^c$ and the positive hooks of $\\P^c$.  A result of Olsson gives the leading hooks of $\\mathfrak{c}(\\P)^c$; we include a proof for completeness.\n\n\\begin{lemma}\\cite[Lemma 2.2]{Olsson93}\\label{lem:conjugate_hooks}\n Let $\\kappa$ be any partition with leading hooks given by the set $H$, with $m=\\max(H)$. The conjugate partition $\\kappa^c$ has leading hooks given by $\\{m-n : n\\in \\{0,1,\\dots ,m\\}\\setminus H\\}$. \n\\end{lemma}\n\n\\begin{proof}\nLet $\\kappa$ be any partition with leading hooks (hooks in the first column) given by the set $H$, with $m=\\max(H)$. The leading hooks of its conjugate partition are the hooks in the top row of~$\\kappa$. This partition has one column for each number $n$ in the set $\\{0,1,\\dots ,m\\}\\setminus H$. The hook of the upper box in the column corresponding to $n$ is equal to $m-n$ as illustrated in Figure~\\ref{fig:conjugate_proof}.  \n\\end{proof}\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[width=0.9\\textwidth]{conjugate_proof.pdf}\n  \\end{center}\n  \\caption{Illustration of the proof of Lemmas~\\ref{lem:conjugate_hooks} and \\ref{lem:conjugate_positive_hooks}}\n  \\label{fig:conjugate_proof}\n\\end{figure}\n\n\\begin{lemma}\\label{lem:conjugate_positive_hooks}\n Let $\\P$ be an $(a,b)$-Dyck path with positive hooks given by $H$, with $m=\\max(H)$. The conjugate path $\\P^c$ has positive hooks given by $\\{m-n : n\\in \\{0,1,\\dots ,m\\}\\setminus H\\}$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\P$ be an $(a,b)$-Dyck path with positive hook set given by $H$ and where $m=\\max(H)$. Fill all the boxes on the left of the path with the hooks that are less than~$m$. Hooks appearing in the same row are equivalent mod $a$. Furthermore, the rows contain all the residues $0,1,\\dots, a-1$ modulo $a$ because $a$ and $b$ are relatively prime and, as a consequence, the filled hooks contain all the numbers from 0 to $m$. \n\nDraw a diagonal parallel to the main diagonal passing through the upper left corner of the box  below $\\P$ with the largest hook $m$. Consider the area $A$ below this diagonal directly on the left of $\\P$ as illustrated in Figure~\\ref{fig:conjugate_proof}. The boxes in $A$ are exactly the boxes on the left of the path with hook length~$n$ less than $m$. Applying cyclic shifts to $P$ to obtain a path below the main diagonal transforms the area $A$ to the area between the main diagonal and the shifted path. Since this transformation maps the box with hook length $m$ to the box with hook length 0 (when rotated~$180$ degrees), the hook length $n$ gets transformed to the hook length $m-n$.\n\\end{proof}\n\n\n\n\\begin{example}\nIn both Figure~\\ref{fig:conjugate_cores} and Figure~\\ref{fig:conjugate_paths}, the set of hooks on the left is $H=\\{1,2,3,4,6,7,9,11,14\\}$, with $m=14$. The set $\\{0,1,\\dots , m\\}\\setminus H=\\{0,5,8,10,12,13\\}$, and subtracting these numbers from~$14$ we get that the leading and positive hooks of the conjugate are $\\{14,9,6,4,2,1\\}$ as desired.\n\\end{example}\n\n\\begin{theorem}\n\\label{thm.slconj}\nThe skew length of $\\P$ is equal to the skew length of $\\P^c$.\n\\end{theorem}\n\n\\begin{proof}\nLet $n_i>e_j$ be a skew inversion for the path $\\P$, with largest level $m$. The north and east steps of the conjugate path are in correspondence with the north and east steps in the original path, respectively. The corresponding north and east levels are given by $n_i'=m-n_i-b$ and $e_j'=m-e_j+a$. A simple calculation shows that these satisfy $n_i'+b<e_j'-a$, giving a flip skew inversion for $\\P^c$. Thus, there is a one-to-one correspondence between skew inversions for $\\P$ and flip skew inversions in $\\P^c$ (and a similar correspondence between flip skew inversions for $\\P$ and skew inversions in $\\P^c$). The result follows directly from Theorem~\\ref{thm:skewLength-skewInversions}. \n\\end{proof}\n\n\\begin{remark}\\label{rem:conj_flip}\nAs explained in the proof of Theorem~\\ref{thm.slconj} the number of skew inversions of $\\P^c$ is equal to the number of flip skew inversions of $\\P$.  Therefore, the skew length of a conjugate path may be thought of as the skew length of the original path when flipped to a $(b,a)$-Dyck path. \n\\end{remark}\n\nConsider the hook lengths of the boxes in the first row of an $(a,b)$-core partition $\\kappa$.  Find the largest hook length of each residue modulo $a$.  The \\emph{$a$-columns} of $\\kappa$ are the columns of $\\kappa$ corresponding to these hook lengths. Theorem~\\ref{thm.slconj} implies the following result, which is illustrated in Figure~\\ref{fig:conjugate_cores_skewlenght}.\n\n\\begin{corollary}\\label{cor:slconj_cores}\nLet $\\kappa$ be an $(a,b)$-core partition. The number of boxes in the $a$-rows and $b$-boundary of $\\kappa$ is equal to the number of boxes in the $a$-columns and $b$-boundary of $\\kappa$.\n\\end{corollary}\n\n\\begin{proof}\nThe number of boxes in the $a$-rows and $b$-boundary of $\\kappa$ is equal to the skew length of $\\kappa$. The number of boxes in the $a$-columns and $b$-boundary of $\\kappa$ is equal to the skew length of $\\kappa^c$. The result then follows from Theorem~\\ref{thm:conjugation} and Theorem~\\ref{thm.slconj} by applying Anderson's bijection.\n\\end{proof}\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=0.5]{conjugate_cores_skewlength}\n  \\end{center}\n  \\caption{(Left) The $8$-boundary boxes of our favorite $(5,8)$-core $\\kappa$ are shaded; those in the $5$-rows of $\\kappa$ are darker. \n  (Right) The $8$-boundary boxes of $\\kappa$ are shaded; those in the $5$-columns of $\\kappa$ are darker.\n  The number of darkly shaded boxes on the left $4+3+2+1=10$ is equal to the number of darkly shaded boxes on the right~$6+3+1=10$. (See Corollary~\\ref{cor:slconj_cores}.)}\n  \\label{fig:conjugate_cores_skewlenght}\n\\end{figure}\n\n\\section{The zeta map (and eta)}\n\\label{sec:zeta_map}\n\nThe zeta map is an intriguing map from $\\DD_{a,b}$ to $\\DD_{a,b}$ which can be defined in a wide variety of ways. See, for example, \\cite{AHJ14,ALW-sweep,ALW-parking,GMII}, with equivalence of many definitions given in~\\cite{ALW-sweep}.  The precise description of zeta depends on making some choices; in our experience, these choices always resolve into one of two distinct maps, which we call zeta and eta.  The eta map  can be interpreted as the zeta map applied to the conjugate of $\\P$, as reproved in Proposition~\\ref{prop:eta_zeta_conjugate} by appealing to skew inversions. The joint dynamics of zeta and eta will be used to present a combinatorial description of the inverse of zeta in Section~\\ref{sec:inverse}.\n\nIn this section we present four combinatorial descriptions for computing the zeta and eta maps, starting with an interpretation involving core partitions implicit in \\cite{AHJ14}, followed by with an equivalent description via the sweep maps considered in~\\cite{ALW-sweep}.  Our main contributions are two new combinatorial descriptions of the zeta map involving interval intersections and a laser filling, along with the study of the eta map in all four contexts.  \n\n\\subsection{Zeta and eta via cores}\\\n\nDrew Armstrong conjectured a combinatorial interpretation for the zeta map by way of core partitions, drawing inspiration from Lapointe and Morse's bounded partitions \\cite{LM}, after learning of Loehr and Warrington's sweep map discussed in the next section.  We present his definition and provide a parallel definition for the eta map.\n\n\\begin{definition}\\label{def:zeta_eta}\nLet $\\P$ be an  $(a,b)$-Dyck path and let $\\mathfrak{c}(\\P)$ be its corresponding $(a,b)$-core. From~$\\P$ define two partitions $\\lambda(\\P)$ and $\\mu(\\P)$ and corresponding lattice paths $\\zeta(\\P)$ and $\\eta(\\P)$:\n\\begin{itemize}\n\\item $\\lambda(\\P)=(\\lambda_1, \\dots , \\lambda_a)$ is the partition that has parts equal to the number of $b$-boundary boxes in the $a$-rows of $\\mathfrak{c}(\\P)$. \n\\item $\\mu(\\P)=(\\mu_1,\\dots,\\mu_b)$ is the partition that has parts equal to the number of  $a$-boundary boxes in the $b$-rows of $\\mathfrak{c}(\\P)$.\n\\item $\\zeta(\\P)$ is the $(a,b)$-Dyck path that bounds the partition $\\lambda(\\P)$. \n\\item $\\eta(\\P)$ is the $(a,b)$-Dyck path that bounds the conjugate of the partition $\\mu(\\P)$.  \n\\end{itemize}\nThe \\emph{zeta map} $\\zeta:\\DD_{a,b}\\rightarrow\\DD_{a,b}$ is defined by $\\zeta:\\P\\mapsto\\zeta(\\P)$.  The \\emph{eta map} $\\eta:\\DD_{a,b}\\rightarrow\\DD_{a,b}$ is defined by $\\eta:\\P\\mapsto\\eta(\\P)$.   \n\\end{definition}\n\nOne can see from the definition of zeta and eta, via the sweep map described below, that $\\zeta(\\P)$ and $\\eta(\\P)$ are indeed paths that stay above the main diagonal. We refer to~\\cite{ALW-sweep} for a proof.\n\nAn alternative method for calculating $\\lambda(\\P)$ and $\\mu(\\P)$ follows from Lemmas~\\ref{lem:skew_inversions_length_dif} and~\\ref{lem:skew_inversions_length_dif_b}.\n\n\\begin{lemma}\\label{lem:lambda_mu_inversions}\nThe entries of the partitions $\\lambda(\\P)$ and $\\mu(\\P)$ satisfy:\n\\begin{enumerate}[(i)]\n\\item $\\lambda_i$ is the number of skew inversions of $\\P$ involving the north level $n_i$.\n\\item $\\mu_j$ is the number of flip skew inversions of $\\P$ involving the east level $e_j$.\n\\end{enumerate}\n\\end{lemma}\n\nIn the $(n,n+1)$ case, the zeta map specializes to the map studied in~\\cite{haglund2008q} for classical Dyck paths, which sends the $\\ensuremath{\\mathsf{dinv}}$ and $\\ensuremath{\\mathsf{area}}$ statistics considered by Haiman to the $\\ensuremath{\\mathsf{area}}$ and $\\ensuremath{\\mathsf{bounce}}$ statistics considered by Haglund. One of the main interests on the zeta map is the fact that it sends skew length to co-area, or equivalently, co-skew length to area. \n\n\\begin{corollary}\nThe skew length of $\\P$ is equal to the co-area of $\\zeta(\\P)$.\n\\end{corollary}\n\n\\begin{proof}\nThe co-area of $\\zeta(\\P)$ is by definition equal to the number of boxes in the partition $\\lambda$. By Lemma~\\ref{lem:lambda_mu_inversions}, this number of boxes counts the number of skew inversions of $\\P$, and thus is equal to the skew length of $\\P$.  \n\\end{proof}\n\n\n\\begin{remark}\\label{rem:dinv}\nThe $\\ensuremath{\\mathsf{dinv}}$ statistic for classical Dyck paths can be generalized to the rational Catalan case as the number of boxes $B$ above the path satisfying \n\\[\n\\frac{\\text{arm}(B)}{\\text{leg}(B)+1} \\leq \\frac{b}{a} < \\frac{\\text{arm}(B)+1}{\\text{leg}(B)},\n\\]\nwhere arm denotes the number of boxes directly on the right of $B$ above the path, and leg denotes the number of boxes directly below $B$ above the path. This intriguing statistic also satisfies~\\mbox{$\\ensuremath{\\mathsf{dinv}}(P)=\\ensuremath{\\mathsf{area}}(\\zeta(P))$}, see~\\cite[Theorem~16]{loehr_continuous_2009} and~\\cite{GMI}. As a consequence the co-skew length and dinv statistics are the same,\n\\begin{equation}\n\\sl'(P) = \\ensuremath{\\mathsf{dinv}}(P).\n\\end{equation}\nNote that the definition of $\\ensuremath{\\mathsf{dinv}}$ is preserved by flipping an $(a,b)$-Dyck path to a $(b,a)$-Dyck path, and therefore skew length is preserved by flipping (as alternatively proved in Corollary~\\ref{cor.absl}).\nBy Remark~\\ref{rem:conj_flip}, the skew length of the conjugate of $\\P$ is equal to the skew length of $\\P$ when flipped to a $(b,a)$-Dyck path. This provides an alternative proof that skew length is preserved under conjugation (Theorem~\\ref{thm.slconj}). \n\\end{remark}\n\nThe work of Gorsky and Mazin~\\cite[Theorem 8]{GMII} and of Armstrong, Loehr, and Warrington~\\cite[Table~1]{ALW-sweep} include the following proposition; we present a new proof involving skew inversions.\n\n\\begin{proposition}[\\cite{GMII,ALW-sweep}]\\label{prop:eta_zeta_conjugate} Let $\\P$ be an $(a,b)$-Dyck path.  Then\n\\[\n\\eta(\\P) = \\zeta(\\P^c).\n\\]\n\\end{proposition}\n\n\\begin{proof}\nThere is a one-to-one correspondence between the skew inversions of $\\P^c$ and the flip skew inversions of $\\P$, as shown in the proof of Theorem~\\ref{thm.slconj}.  Through Lemma~\\ref{lem:lambda_mu_inversions}, one deduces that~$\\lambda(\\P^c)$ is the conjugate of $\\mu(\\P)$. As a consequence, $\\zeta(\\P^c)=\\eta(\\P)$.\n\\end{proof}\n\n\\begin{remark}\nIn~\\cite{GMII}, conjugation is considered in terms of normalized dual semimodules. The zeta and eta maps correspond to the maps $G_{m,n}$ and $G_{n,m}$ in~~\\cite[Section~2.3]{GMII}. \n\\end{remark}\n\nDenote by $\\P^\\mathrm{flip}$ the result of flipping an $(a,b)$-Dyck path $\\P$ to a $(b,a)$-Dyck path. The example corresponding to the path $\\P$ in Figure~\\ref{fig:zetaEta_cores} and the following result are illustrated in Figure~\\ref{fig:zetaEta_flip}. This result can also be essentially found in~\\cite[Table~1]{ALW-sweep}.\n\n\\begin{proposition}[\\cite{ALW-sweep}]\nLet $\\P$ be an $(a,b)$-Dyck path. Then,\n\\begin{align*}\n \\zeta(\\P^\\mathrm{flip}) &= \\eta(\\P)^\\mathrm{flip}, \\\\\n \\eta(\\P^\\mathrm{flip}) &= \\zeta(\\P)^\\mathrm{flip}. \n\\end{align*}\n\\end{proposition}\n\n\\begin{proof}\nThe skew inversions of $\\P^\\mathrm{flip}$ are in correspondence with the flip skew inversions of $\\P$, and therefore $\\lambda(\\P^\\mathrm{flip})=\\mu(P)$. As a consequence, $\\zeta(\\P^\\mathrm{flip}) = \\eta(\\P)^\\mathrm{flip}$. A similar argument shows that~$\\mu(\\P^\\mathrm{flip})=\\lambda(P)$ and $\\eta(\\P^\\mathrm{flip}) = \\zeta(\\P)^\\mathrm{flip}$.\n\\end{proof}\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=0.55]{zetaEta_cores.pdf}\n  \\end{center}\n  \\caption{In our running example, $\\zeta(\\P)$ bounds the partition $\\lambda(\\P)=(4,3,2,1,0)$ and $\\eta(\\P)$ bounds the conjugate of the partition $\\mu(\\P)=(3,2,2,1,1,1,0,0)$. }\n  \\label{fig:zetaEta_cores}\n\\end{figure}\n\n\\begin{example}\n\\label{ex:zetaEta_cores}\nFigure~\\ref{fig:zetaEta_cores} illustrates an example of the zeta map and the eta map applied to our running example path $\\P$. From Example~\\ref{ex:skew_inversions}, the $8$-boundary boxes in the $5$-rows of $\\mathfrak{c}(\\P)$ give $\\lambda(\\P)=(4,3,2,1,0)$ and the $5$-boundary boxes in the $8$-rows of the core $\\mathfrak{c}(\\P)$ give $\\mu(\\P)=(3,2,2,1,1,1,0,0)$.  Then $\\zeta(\\P)$ is the path that bounds $\\lambda(\\P)$ and $\\eta(\\P)$ is the path that bounds the conjugate partition $\\mu(\\P)^c=(6,3,1,0,0)$.  We often combine $\\lambda(\\P)$, $\\zeta(\\P)$, $\\mu(\\P)$, and $\\eta(\\P)$ as on the right hand side of Figure~\\ref{fig:zetaEta_intervals}.\n\nThe core partition $\\mathfrak{c}(\\P^c)$ corresponding to the conjugate path $\\P^c$ is illustrated in the right part of Figure~\\ref{fig:conjugate_cores}. The $a$-rows of this core are the rows with leading hooks 14, 6, and 2. Counting the number of $b$-boundary boxes in these rows shows that $\\lambda(\\P^c)=(6,3,1,0,0)$, which equals $\\mu(\\P)^c$. We see that $\\eta(\\P)=\\zeta(\\P^c)$. \n\\end{example}\n\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=0.55]{zetaEta_flip}\n  \\end{center}\n  \\caption{Zeta and eta applied to the flipped Dyck path of our running example path $P$. }\n  \\label{fig:zetaEta_flip}\n\\end{figure}\n\n\n\\subsection{Zeta and eta via sweep maps}\\label{sec:zetaEta_sweep}\\\n\n\\renewcommand*{\\thefootnote}{\\fnsymbol{footnote}}\nThis section presents the combinatorial description of the zeta map on rational Dyck paths as a \\emph{sweep map} created by Loehr and Warrington in~\\cite{ALW-sweep}. \n\nHeuristically, this map `sweeps' the line of fixed slope $\\frac{a}{b}$  across $\\P$ starting on the main diagonal moving to the northwest, recording north and east steps in the order in which they are met. \nAnalogously, the eta map `sweeps' the line of slope $\\frac{a}{b}$ across $\\P$ starting at the farthest point from the main diagonal moving to the southeast, recording south and west steps in the order in which they are met\\footnote{These definitions exhibit the choice of `east-north' or `west-south' convention in~\\cite{ALW-sweep}.}. This procedure is illustrated for our running example in Figure~\\ref{fig:zetaEta_sweep}.\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=0.55]{zetaEta_sweep.pdf}\n  \\end{center}\n  \\caption{Zeta and eta via sweep maps. \n  The steps of $\\zeta(\\P)$ are labeled by the levels of the lattice points of $\\P$ in order, recording whether they correspond to north or east levels.\n  The steps of $\\eta(\\P)$ are labeled by the levels of the lattice points of $\\P$ in reverse order starting from the upper right corner, recording whether they correspond to south or west levels.}\n  \\label{fig:zetaEta_sweep}\n\\end{figure}\n\nRecall that the reading word $L(\\P)$ is obtained by reading the levels that occur along the path from southwest to northeast, excluding the final $0$, and the reverse reading word $M(\\P)$ is obtained by reading from northeast to southwest, excluding the final $0$. \n\n\\begin{theorem}[{\\cite{ALW-sweep}}]\\label{thm:zeta_sweep}\nThe zeta map can be computed as follows:\n\\begin{enumerate}\n\\item[$(1)$] Place a bar over each of the entries of $L(\\P)$ corresponding to an east step; these occur exactly at the \\emph{right (cyclic) descents} of $\\sigma$.\n\\item[$(2)$] Sort $L(\\P)$ in increasing order, keeping track of the bars on various values.\n\\item[$(3)$] Read the resulting sequence of labels (bars and non-bars) to produce a new northeast lattice path, which we denote $\\zeta(\\P)$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:eta_sweep}\nThe eta map can be computed as follows:\n\\begin{enumerate}\n\\item[$(1')$] Place a bar over each of the entries of $M(\\P)$ corresponding to a {west} step; these occur exactly at the \\emph{right (cyclic) {ascents}} of $\\tau$.\n\\item[$(2')$] Sort $M(\\P)$ in increasing order, keeping track of the bars on various values.\n\\item[$(3')$] Read the resulting sequence of labels (bars and non-bars) to produce a new southwest lattice path from $(b,a)$ to $(0,0)$, which we denote $\\eta(\\P)$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{example}\nIn our running example in Figure~\\ref{fig:zetaEta_sweep}, we mark the reading word \n\\[\nL(\\P)=(0,8,16,\\bar{24},19,\\bar{27},\\bar{22},\\bar{17},12,\\bar{20},\\bar{15},\\bar{10},\\bar{5}),\n\\] \nwhich sorts to $(0,\\bar{5},8,\\bar{10},12,\\bar{15},16,\\bar{17},19,\\bar{20},\\bar{22},\\bar{24},\\bar{27})$. Thus $\\zeta(\\P)$ is the path \n\\[NENENENENEEEE.\\]\nWe mark the reverse reading word\n\\[M(\\P)=(\\bar{0},\\bar{5},\\bar{10},\\bar{15},20,\\bar{12},\\bar{17},\\bar{22},27,\\bar{19},24,16,8),\\] \nwhich sorts to  \n$(\\bar{0},\\bar{5},8,\\bar{10},\\bar{12},\\bar{15},16,\\bar{17},\\bar{19},20,\\bar{22},24,27)$.  Thus $\\eta(\\P)$ is the path \n\\[WWSWWWSWWSWSS \\textup{, which is equivalent to } NNENEENEEENEE.\\]\n\\end{example}\n\n\\begin{remark}\nNote that both computations in Theorem~\\ref{thm:zeta_sweep} and Theorem~\\ref{thm:eta_sweep} can be performed just as easily on the standardization $\\sigma(\\P)$ of $L(\\P)$, since only the relative values of the labels matter.\n\\end{remark}\n\n\\begin{proof}[Proof of Theorems~\\ref{thm:zeta_sweep} and~\\ref{thm:eta_sweep}]\nConsider the path $\\zeta(\\P)$ described in Theorem~\\ref{thm:zeta_sweep}. The number of boxes on the left of the north step corresponding to a north level $n_i$ of $\\P$ is equal to the number of east levels smaller that $n_i$. This number is equal to the number of skew inversions involving~$n_i$, which coincides with $\\lambda_i$ by Lemma~\\ref{lem:lambda_mu_inversions}~$(i)$. Therefore the described algorithm to compute $\\zeta(\\P)$ coincides with the definition of zeta in Definition~\\ref{def:zeta_eta}.\n\nConsider the (rotation of the) path $\\eta(\\P)$ described in Theorem~\\ref{thm:eta_sweep}. The number of boxes below a given west step of $\\P$ is equal to the number of south levels smaller than the corresponding west level. This number is equal to the number of flip skew inversions involving the corresponding level~$e_j$, which coincides with $\\mu_j$ by Lemma~\\ref{lem:lambda_mu_inversions}~$(ii)$. Therefore, the described algorithm to compute $\\eta(\\P)$ coincides with the definition of eta in Definition~\\ref{def:zeta_eta}.\n\\end{proof}\n\n\\subsection{Zeta and eta via the laser filling}\\\n\nThis section presents a new interpretation of zeta and eta that are read from a \\emph{laser filling} in the boxes below the path $\\P$ and above the main diagonal. Our main result in this section describes the partitions $\\lambda$ and $\\mu$ in terms of the laser filling. \nThis result will be used in Section~\\ref{sec:square} to give a new combinatorial description of the inverse of the zeta map in the square case without the use of bounce paths.  \n\nFigure~\\ref{fig:zetaEta_laser} illustrates the following definition.\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=0.55]{zetaEta_laser.pdf}\n  \\end{center}\n  \\caption{Laser filling of a path $\\P$. The laser pointed from the lower right corner of the box with filling 2 crosses two vertical walls of the path, while all other lasers cross only one. \n  The entries of the partition $\\lambda(\\P)=(4,3,2,1,0)$ are the sums of the laser fillings on the rows. The entries of the partition $\\mu(\\P)=(3,2,2,1,1,1,0,0)$ are the sums of the laser fillings on the columns. \n  }\n  \\label{fig:zetaEta_laser}\n\\end{figure}\n\n\\begin{definition}\nLet $\\P$ be an $(a,b)$-Dyck path and let $B$ be a box below $\\P$ and above the line $y=\\frac{a}{b}x$.  Draw the line of slope $\\frac{a}{b}$ through the southeast corner of $B$ (a bi-directional laser).  The {\\em laser filling} of $B$ is equal to the number of vertical walls of $P$ crossed by the laser. Equivalently, it is equal to the number of horizontal walls of $P$ crossed by the laser. \n\\end{definition}\n\n\\begin{remark}\nLasers also appear in Armstrong, Rhoades, and Williams's \\cite{ARW13}.  Their \nlasers stop at the first wall they meet; by contrast, our\nlasers traverse (and count!)\\ the walls of the path~$\\P$.\n\\end{remark}\n\n\\begin{theorem}\\label{thm:laser}\nThe partitions $\\lambda$ and $\\mu$ associated to $\\P$ can be computed as follows:\n\\begin{enumerate}[(i)]\n\\item The parts of $\\lambda(\\P)$ are the sums of the laser fillings in the rows.\n\\item The parts of $\\mu(\\P)$ are the sums of the laser fillings in the columns.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nThe entries of $\\lambda$ count the skew inversions involving the north levels of each of the vertical steps in the path. For a given vertical step, this number is equal to the number of horizontal steps in the path that are strictly below the laser through its starting point. Each of these horizontal steps is crossed by exactly one of the lasers through the lower right corners of the boxes below the path that are in the same row of the vertical step in consideration. Statement {\\em(i)} follows and Statement {\\em (ii)} is proved similarly. \n\\end{proof}\n\n\\begin{corollary}\nThe skew length of $\\P$ is equal to the sum of the laser fillings of $\\P$.\n\\end{corollary}\n\n\\begin{proof}\nThe skew length of $\\P$ is equal to the area of $\\lambda(\\P)$. By the previous theorem, this area is equal to the sum of all laser fillings of $\\P$.\n\\end{proof}\n\n\n\\subsection{Zeta and eta via interval intersections}\\label{sec:zetaEta_intervals}\\\n\nThis section presents a second new combinatorial interpretation of zeta and eta in terms of {\\em interval intersections}.  Each step of the path $\\P$ has an associated closed interval whose endpoints are the levels of its starting and ending points.  When the intervals are ordered in increasing order, zeta and eta can be directly determined.\n\n\\begin{definition}\n\\label{def:intervals}\nLet $\\P$ be an $(a,b)$-Dyck path.  Let $\\N(\\P)$ be the north levels of $\\P$ and $\\E(\\P)$ be the east levels of $\\P$.  Define the {\\em north intervals} of $\\P$ to be the set $\\calI_N=\\{[n_i,n_i+b] \\text{ for } n_i\\in\\N(\\P)\\}$ and the {\\em east intervals} of $\\P$ to be the set $\\calI_E=\\{[e_j-a,e_j] \\text{ for } e_j\\in\\E(\\P)\\}$.\n\\end{definition}\n\n\\begin{theorem}\\label{cor:zetaEta_intervals}\nCreate an $a\\times b$ grid.  Label the rows of the grid by the north intervals of $\\P$ increasing from bottom to top, and the columns of the grid by the east intervals of $\\P$ increasing from left to right.  Fill in the boxes in this grid when the corresponding row and column intervals do not intersect.  The boundary path of the shaded boxes above the main diagonal is $\\zeta(\\P)$ and the boundary path of the shaded boxes below the main diagonal is $\\eta(\\P)$, rotated 180 degrees.\n\\end{theorem}\n\n\\begin{proof}\nThis theorem is a straightforward consequence of Lemma~\\ref{lem:lambda_mu_inversions}.\n\\end{proof}\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=0.55]{zetaEta_intervals.pdf}\n  \\end{center}\n  \\caption{Zeta and eta via interval intersections. The intervals on the left correspond to the ordered level intervals of the vertical steps in the path. The intervals on the top correspond to the level intervals of the horizontal steps. The shaded boxes of $\\lambda$ and $\\mu$ are the boxes whose corresponding row and column intervals do not intersect.}\n  \\label{fig:zetaEta_intervals}\n\\end{figure}\n\n\\begin{example}\n  For our running example path $\\P$, the north intervals are $[0,8]$, $[8,16]$, $[12,20]$, $[16,24]$, and $[19,27]$, which can be read directly from the north steps of $\\P$, or calculated from the north levels as in Definition~\\ref{def:intervals}.  Similarly, the east intervals of $\\P$ are $[0,5]$, $[5,10]$, $[10,15]$, $[12,17]$, $[15,20]$, $[17,22]$, $[19,24]$, and $[22,27]$.  Labeling the rows of a $5\\times 8$ grid with the north levels and the columns with the east levels gives the right side of Figure~\\ref{fig:zetaEta_intervals}.  The shaded boxes are the those where the corresponding row interval does not intersect the corresponding column interval, from which $\\lambda(\\P)$, $\\mu(\\P)$, $\\zeta(\\P)$, and $\\eta(\\P)$ can be read posthaste.\n\\end{example}\n\n\n\\section{Pairing the zeta map with the eta map}\\label{sec:inverse}\n\nBy considering the zeta map together with the eta map, we gain two new ideas: a new approach for proving that the zeta map is a bijection and (if $\\zeta$ is a bijection) a new area-preserving involution on the set of $(a,b)$-Dyck paths.  For clarity and consistency, we have decided to use the letter $P$ to denote a path that is in the domain of $\\zeta$ and use the letter $Q$ to denote a path that is in the image of $\\zeta$.\n\n\\subsection{Inverse of the zeta map knowing eta}\\label{sec:zetaeta}\\\n\n For the image $\\calZ$ under the pair of maps \\[(\\zeta,\\eta):\\DD\\rightarrow\\DD\\times\\DD,\\] we define a map $\\iota:\\calZ\\rightarrow\\DD$ such that $(\\zeta,\\eta)\\circ\\iota$ is the identity map.  \nFurther, we conjecture that for every $(a,b)$-Dyck path $Q$ that appears as the image of $\\zeta$, there exists a unique $(a,b)$-Dyck path $R$ such that $(Q,R)\\in \\calZ$.  This would imply that in~$\\calZ$ every element of $\\DD$ appears exactly once as the initial entry in the pair, from which it would follow that the zeta map is a bijection.\n\n\\begin{definition} \nA pair of $(a,b)$-Dyck paths $(Q,R)$ is an {\\em admissible pair} if $(Q,R)=\\big(\\zeta(\\P),\\eta(\\P)\\big)$ for some $(a,b)$-Dyck path $\\P$.  The {\\em set of admissible pairs} $\\calZ\\subset\\DD\\times\\DD$ is the image under the pair of maps $(\\zeta,\\eta):\\DD\\rightarrow\\DD\\times\\DD$.  \n\\end{definition}\n\nWe now describe a simple combinatorial description of the inverse map $\\iota$ that recovers $\\P$ from the pair~$(Q,R)$ or, equivalently, from the pair of partitions ($\\lambda$, $\\mu$) they bound.  \n\n\\begin{definition} \\label{def:iota}\nLet $(Q,R)$ be an admissible pair.  Define $\\iota(Q,R)$ as follows.\n\\begin{enumerate}\n\\item[(1)] Draw the path $Q$ above the diagonal and rotate the path $R$ 180 degrees so that it embeds below the diagonal in the same diagram.  Label the steps of each path from~$1$ to~$a+b$ starting at the bottom-left corner and ending at the top-right corner in the order in which they appear in the path. \n\\item[(2)] Create the permutation $\\gamma:[a+b]\\to [a+b]$ as follows. If $l$ is a label of a horizontal step in $Q$, define $\\gamma(l)$ to be the label of the horizontal step in $R$ that is in the same column of~$l$.  If $l$ is a label of a vertical step in $Q$, define $\\gamma(l)$ to be the label of the vertical step in $R$ that is in the same row of $l$.\n\\item[(3)] For admissible pairs $(Q,R)$, $\\gamma$ is a cycle permutation.  Interpret $\\gamma$ in cycle notation as $(\\sigma_1,\\sigma_2,\\hdots,\\sigma_{a+b})$, fixing $\\sigma_1=1$.\nDefine $\\P=\\iota(Q,R)$ to be the path whose east steps correspond to the cyclic descents of $\\sigma$.\\footnote{A descent occurs when $\\sigma_i>\\sigma_{i+1}$.  A cyclic descent is defined in the same way, but considering the indices modulo~$a+b$, allowing a descent in the last position of $\\sigma$.}\n\\end{enumerate}\n\\end{definition}\n\n\n\\begin{theorem}\n\\label{thm:inverse}\n$\\gamma$ is a cycle permutation and the map $\\iota$ is the inverse map for the pair $(\\zeta,\\eta)$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose $(Q,R)$ is an admissible pair, so that there exists a $\\P\\in\\DD$ such that $(Q,R)=(\\zeta(\\P),\\eta(\\P))$.  Label the steps of $Q$ and $R$ with the levels of $\\P$ as determined by the sweep map algorithm given in Theorems~\\ref{thm:zeta_sweep} and \\ref{thm:eta_sweep} (as illustrated in Figure~\\ref{fig:zetaEta_sweep}).  The definition of the permutation $\\gamma$ using these labels instead of on $[a+b]$ induces a permutation on the set of levels of the lattice points of $\\P$. We will prove that this permutation is the cycle permutation given by the reading word $L(\\P)$ of $\\P$. \n\nBecause of the relationship between the forward reading word $L(\\P)$ and the reverse reading word $M(\\P)$, the labels of the vertical steps of $R$ are exactly the labels of the vertical steps of $Q$ plus $b$, while the labels of the horizontal steps of $R$ are exactly the labels of the horizontal steps of $Q$ minus $a$. This implies that the permutation $\\gamma$ maps the level of a lattice point in $\\P$ to the level of the next lattice point along $\\P$, forming a permutation on the set of labels that is a cycle ordered by the reading word~$L(\\P)$. \n\nSince the level labels appear in order as we walk along $Q$, only the relative order of the labels matters; returning all labels to the numbers from~$1$ up to $a+b$ recovers $\\gamma(\\P)$, which when interpreted as a permutation in one line notation is the reading permutation $\\sigma(\\P)$.  By Remark~\\ref{rem:sigma}, we recover $\\P$ directly from $\\sigma(\\P)$ and the result follows. \n\\end{proof}\n\nTaken with Theorem~\\ref{thm:inverse}, the following conjecture would imply that $\\zeta$ is a bijection.\n\n\\begin{conjecture}\nSuppose that $Q\\in\\DD_{a,b}$.  There exists at most one $R\\in\\DD_{a,b}$ such that $(Q,R)\\in\\calZ$.\n\\end{conjecture}\n\n\\begin{example}\nFigure~\\ref{fig:zetaEta_inverse} illustrates  the procedure outlined in Definition~\\ref{def:iota} for the pair~$(Q,R)=(\\zeta(\\P),\\eta(\\P))$ from our running example $\\P$. After labeling the paths $Q=\\zeta(\\P)$ and $R=\\eta(\\P)$ from $1$ to $13$, we see that $\\gamma(1)=3$, $\\gamma(2)=1$, $\\gamma(3)=7$, etc.  Writing $\\gamma$ in cycle notation gives \\[\\gamma=(1,3,7,{\\bf 12},9,{\\bf 13},{\\bf 11},{\\bf8},5,{\\bf10},{\\bf6},{\\bf4},{\\bf2}).\\]  If we instead interpret this sequence of numbers as the one line notation of a permutation $\\sigma$, the cyclic descents of $\\sigma$ are bolded and correspond to the east steps of $\\iota(Q,R)$. We see that $\\iota(Q,R)=\\P$.\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=0.5]{zetaEta_inverse}\n  \\end{center}\n  \\caption{Calculating $\\P=\\iota(Q,R)$ using the method in Definition~\\ref{def:iota}.}\n  \\label{fig:zetaEta_inverse}\n\\end{figure}\n\\end{example}\n\n\\begin{remark}\nThe essence of the proof of Theorem~\\ref{thm:inverse} is that the $\\zeta$ and $\\eta$ maps track the positions of the right cyclic descents of $L(\\P)$ and  $M(\\P)$.  Using these two sets of data, and the precise relationship between $L(\\P)$ and $M(\\P)$, we are able to solve for the levels of $\\P$.  Interestingly, $\\zeta(\\P)$ does not obviously contain enough information to reconstruct $\\P$. We cannot construct a unique permutation solely from its collection of right descents, and need additional information to recover $\\P$.  In the standard Catalan case, this additional information is essentially implied by the particular structure of the $n\\times (n+1)$ rectangle; for the general case, we obtain the extra information necessary from $\\eta(\\P)$.\n\\end{remark}\n\n\\begin{remark}\nWhen pairing arbitrary $Q$ and $R$ paths a number of things can go wrong.  First, Theorem~\\ref{thm.slconj} implies that in order to come from an actual path, we must have $\\ensuremath{\\mathsf{area}}(Q)=\\ensuremath{\\mathsf{area}}(R)$.  Second, we know that $\\gamma$ must have a single cycle; it is simple to construct examples where this does not occur.  It is also possible to find pairs $(Q,R)$ where $\\gamma$ has a single cycle, but the labels $l_i$ obtained from the reverse bijection are in the wrong relative order.  In other words, we may have $\\zeta(\\iota(Q,R))\\neq Q$. \n\\end{remark}\n\nWe propose the problem of characterizing all possible permutations $\\gamma(P)$. As a straightforward consequence of the description of this permutation in terms of the pair $Q$ and $R$, we conclude Proposition 6.8 without proof.\n\n\\begin{proposition}\n\\label{prop.exc}\nThe positions of the exceedences of $\\gamma(\\P)$ give the collection of north steps in~$\\zeta(\\P)$, and the values of the exceedences of $\\gamma(\\P)$ are the north steps in~$\\eta(\\P)$ when rotated $180^\\circ$.\n\\end{proposition}\n\n\\subsection{An area-preserving involution on rational Dyck paths}\\\n\\label{sec:perp}\n\nIf $\\zeta$ is invertible, we can use $\\eta$ to define a new area-preserving involution on the set of $(a,b)$-Dyck paths, induced by the conjugate map under~$\\zeta$ which we call the conjugate-area map.  This involution sends the path $\\zeta(\\P)$ to the path $\\eta(\\P)=\\zeta(\\P^c)$ and is predictable for certain families of $(a,b)$-Dyck paths. \n\n\\begin{definition}\nThe \\emph{conjugate-area map} \napplied to an $(a,b)$-Dyck path $Q$ is the path \\[\\chi(Q):=\\zeta \\circ c \\circ \\zeta^{-1}(Q).\\] \nIf $\\lambda$ is the partition bounded by $Q$, we define $\\chi(\\lambda)$ to be the partition bounded by $\\chi(Q)$.\n\\end{definition}\n\n\\begin{figure}[h]\n\\begin{tikzpicture}\n  \\matrix (m) [matrix of math nodes, row sep=4em, column sep=6em]\n    { \\P & \\zeta(\\P)   \\\\\n     \\P^c & \\eta(\\P)  \\\\ };\n  { [start chain] \\chainin (m-1-1);\n  \\chainin (m-1-2) [join={node[above,labeled] {\\text{zeta}}}]; \n  \\chainin (m-2-2) [join={node[right,labeled] {\\text{conj-area}}}]; }\n  { [start chain] \\chainin (m-1-1);\n  \\chainin (m-2-1) [join={node[left,labeled] {\\text{conjugate}}}];\n  \\chainin (m-2-2) [join={node[below,labeled] {\\text{zeta}}}]; }\n  { [start chain] \\chainin (m-1-1);\n  \\chainin (m-2-2) [join={node[right,labeled] {\\raisebox{.1in}{\\text{\\tiny eta}}}}]; }\n\\end{tikzpicture}\n\\caption{Diagrammatic description of the conjugate-area involution.}\n\\label{fig:alpha}\n\\end{figure}\n\n\\begin{remark}\nFor partitions $\\lambda$ and $\\mu$ bounded by $\\zeta(\\P)$ and $\\eta(\\P)$ we have\n$\\chi(\\lambda) = \\mu^c$.\n\\end{remark}\n\n\\begin{proposition}\nIf the zeta map is a bijection then the conjugate-area map is an area-preserving involution on the set of $(a,b)$-Dyck paths.\n\\end{proposition}\n\\begin{proof}\nSince conjugation is an involution, we see that applying the operator $\\zeta \\circ c \\circ \\zeta^{-1}$ twice is equal to the identity, and therefore $\\chi(\\chi(Q)) = Q$. Furthermore, conjugation preserves skew length~(Theorem~\\ref{thm.slconj}), which is mapped to co-area via the zeta map. Thus, $\\chi$ must be an area-preserving involution.\n\\end{proof}\n\n\nOne possible approach to prove that $\\zeta$ is a bijection would be to directly construct the involution~$\\chi$.  In Section~\\ref{sec:square} we show that in the square case $\\chi$ is exactly the map that reverses the path~$\\P$; equivalently one finds $\\chi(\\lambda)$ by simple conjugation.  In the rational case, conjugation must fail in general because conjugates of partitions may not sit above the main diagonal.  Although, Proposition~\\ref{prop:justified} exhibits our empirical observation that for `small' partitions $\\lambda$, $\\chi(\\lambda)$ is often the conjugate.\n\nWe have found that $\\chi$ is predictable in other families of examples as well; in Section~\\ref{sec:inductive_zeta_inverse} we present an inductive combinatorial description of the inverse of the zeta map and of the area-preserving involution for a nice family of examples.\n\n\\begin{example}[Left-justified and up-justified partitions]\nConsider two families of partitions whose Young diagrams fit above the main diagonal in the $a\\times b$ grid.  Let $n\\in \\mathbb{N}$ be a number no bigger than the number of boxes above the main diagonal in the $a\\times b$ grid.  Define the {\\em left-justified partition}~$\\lambda^n$ to be the unique partition whose Young diagram has $n$ boxes as far to the left as possible and the {\\em up-justified} partition $\\nu^n$ to be the unique partition whose Young diagram has $n$ boxes as far up as possible.  Figure~\\ref{fig:leftDown_partitions} shows $\\lambda^8=(3,2,2,1)$ embedded above the diagonal and $\\nu^8=(6,2)$ rotated 180 degrees and embedded below the diagonal.  We use the notation $\\nu^n$ because it is the conjugate of what one might expect if we called it $\\mu^n$, as pointed out by an astute referee.\n\n\\begin{proposition}\n\\label{prop:justified}\nThe left-justified and up-justified partitions are related by the conjugate-area map:\n\\[\n\\chi(\\lambda^n) = \\nu^n. \n\\]\nMoreover, $\\zeta^{-1}(\\lambda^n)$ is the path with area $n$ containing the first $n$ positive hooks in the grid.\n\\end{proposition}\n\\begin{proof}\nLet $\\P^n$ be the path containing the first $n$ positive hooks in the grid. This path consists of all the boxes below a line parallel to the main diagonal sitting in the highest level of the path, and therefore all the labels in the laser filling are equal to 1. Adding the labels in the rows and the columns we obtain the partitions $\\lambda^n$ and $(\\nu^n)^c$.\n\\end{proof}\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=0.5]{leftDown_partitions}\n  \\end{center}\n  \\caption{The left-justified partition $\\lambda^8$, up-justified partition $\\nu^8$, and corresponding path~$\\P^8$.}\n  \\label{fig:leftDown_partitions}\n\\end{figure}\n\n\nFigure~\\ref{fig:leftDown_partitions} illustrates and example of left-justified and up-justified partitions $\\lambda^n$ and $\\nu^n$ together with their corresponding path $\\P^n$ for $n=8$. The reader is invited to verify that $\\zeta(\\P^8)$ and $\\eta(\\P^8)$ are given by the paths bounding $\\lambda^8$ and $\\nu^8$ using any of the methods described in Section~\\ref{sec:zeta_map}, as well as to verify that the inverse map $\\iota$ presented in Section~\\ref{sec:inverse} gives $\\P^8$ when applied to the paths bounding $\\lambda^8$ and $\\nu^8$.\n\\end{example}\n\n\\section{The square case}\\label{sec:square}\n\nIn this section, we consider $(n,n+1)$-Dyck paths, lattice paths in an $n\\times (n+1)$ grid staying above the main diagonal.  They are in bijection with classical Dyck paths in an $n\\times n$ grid by simply forgetting the last east step of the path.\nHaglund and Haiman~\\cite{haglund2008q} discovered a beautiful description of the inverse of the zeta map in this case using a bounce path that completely characterizes the area sequence below the path. We present a new combinatorial description of the inverse of the zeta map in this case in terms of an area-preserving involution. This approach opens a new direction in proving that the zeta map is a bijection in the general $(a,b)$ case. \n\n\\subsection{The conjugate-area involution, conjugate partitions and reverse paths.}\\\n\nLet $Q$ be an $(n,n+1)$-Dyck path. The area-preserving involution $\\chi$ conjugates the partition~$\\lambda$ bounded by the path $Q$.  This was proved in \\cite[Theorem 9]{GMII}; we provide a new proof using our laser interpretation of zeta and eta.  For simplicity, denote by ${Q}^r$ the path whose bounded partition is $\\lambda^c$. We refer to $P^r$ as the \\emph{reverse path} of $Q$. Forgetting the last east step of the path, the reverse operation acts by reversing the path in the $n\\times n$ grid. An example of the conjugate-area involution, conjugate partition and reverse path is illustrated in Figure~\\ref{fig:square_perp}. \n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=0.5]{square_perp}\n  \\end{center}\n  \\caption{The conjugate-area involution in the $(n,n+1)$ case. }\n  \\label{fig:square_perp}\n\\end{figure}\n\n\\begin{theorem}[\\cite{GMII}]\\label{thm:square_perp}\nFor a Dyck path $Q$ and the partition $\\lambda$ it bounds, we have $\\chi(Q) = Q^r$ and~$\\chi(\\lambda) = \\lambda^c$. \n\\end{theorem}\n\n\\begin{proof}\nWe need to show that the partitions $\\lambda$ and $\\mu$ bounded by the images $\\zeta(\\P)$ and $\\eta(\\P)$ of any $(n,n+1)$-Dyck path $\\P$ satisfy \n\\[\n\\chi(\\lambda) = \\mu^c = \\lambda^c. \n\\]\nEquivalently, we need to show that $\\lambda=\\mu$. \nThe entries of the partitions $\\lambda$ and $\\mu$ are the sums of the labels in the laser filling of $\\P$ over the rows and columns respectively (Theorem~\\ref{thm:laser}). We will show that the values of the sums over the rows are in correspondence with the values of the sums over the columns, and therefore $\\lambda=\\mu$. This correspondence is illustrated for an example in Figure~\\ref{fig:square_perp_proof}.\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=0.5]{square_perp_proof}\n  \\end{center}\n  \\caption{Argument in the proof of Theorem~\\ref{thm:square_perp}. The values of the sums over the rows are in correspondence with the values of the sums over the columns, and therefore $\\lambda=\\mu$.}\n  \\label{fig:square_perp_proof}\n\\end{figure}\n\nFor every row, draw a line of slope 1 in the northeast direction pointing from the starting point of the north step in that row. This line hits the path for the first time in the ending point of an east step of the path. The labels of the laser filling in the boxes in the column corresponding to this east step are exactly the same as the labels of the laser filling in the row in consideration.  (This is because the lasers are lines with slope $\\frac{n}{n+1}$, which implies that for any two boxes on the same diagonal of slope 1 that are not interrupted in line of sight by the path $\\P$, they will have the same laser filling.)  Thus, their corresponding sums are equal.  Doing this for all the rows gives the desired correspondence between the entries of the partition $\\lambda$ and the entries of the partition $\\mu$.  \n\\end{proof}\n\n\\subsection{The inverse of the zeta map}\\\n\nBecause Theorem~\\ref{thm:square_perp} provides the explicit formula for $\\chi$, the method to find inverse of the zeta map in the $(n,n+1)$ case follows as a direct consequence of Theorem~\\ref{thm:inverse}. The description of the map $\\iota$ is presented in Definition~\\ref{def:iota}.\n\n\\begin{theorem}\\label{thm_square_inverse}\nLet $Q$ be an $(n,n+1)$-Dyck path. Then, $\\zeta^{-1}(Q)=\\iota(Q,Q^r)$.\n\\end{theorem}\n\nAn example of this result is illustrated in Figure~\\ref{fig:square_zeta_inverse1}. The laser filling of the path $\\zeta^{-1}(Q)$ in this example is shown in Figure~\\ref{fig:square_perp_proof}. One can verify that the sum of the labels of the laser filling on the rows and columns gives rise to the partitions $\\lambda$ and $\\mu$ bounded by $Q$ and $\\chi(Q)$ (Theorem~\\ref{thm:laser}).\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=0.5]{square_zeta_inverse1}\n  \\end{center}\n  \\caption{The inverse of $\\zeta$ by way of conjugate partitions.}\n  \\label{fig:square_zeta_inverse1}\n\\end{figure}\n\n\nAn alternative way to obtain the cycle permutation $\\gamma$ directly from $Q$ is as follows. Shade the boxes in the $n\\times (n+1)$ rectangle that are crossed by the main diagonal as illustrated in Figure~\\ref{fig:square_zeta_inverse2}. Move east from a vertical step labeled $i$ until the center of the first shaded box you see, and then move up until hitting an horizontal step of the path. The image $\\gamma(i)$ is equal to the label of this horizontal step plus 1. In the example of Figure~\\ref{fig:square_zeta_inverse2}, the path starting at the vertical step labeled 7 hits the horizontal step 12, therefore $\\gamma(7)=12+1=13$. \n\nIn order to determine $\\gamma(i)$ of a label of an horizontal step, we move down until the center the last shaded box we see, and then move left until hitting a vertical step of the path. As before, $\\gamma(i)$ is equal to the label of this vertical step plus 1. In the example, $\\gamma(15)=10+1=11$.  The image of the label of the first horizontal step of the path is by definition equal to 1. Interpret $\\gamma$ in cycle notation as $(\\sigma_1,\\sigma_2,\\dots,\\sigma_{2n+1})$ where we fix $\\sigma_1=1$. \nAs a direct consequence of Theorem~\\ref{thm_square_inverse} we get:\n\n\\begin{theorem}\\label{thm_square_inverse2}\nLet $Q$ be an $(n,n+1)$-Dyck path. The inverse $\\zeta^{-1}(Q)$ is the path whose east steps correspond to the cyclic descents of the permutation~$\\gamma$ when interpreted in one line notation.\n\\end{theorem}  \n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=0.5]{square_zeta_inverse2}\n  \\end{center}\n  \\caption{Alternative description of the cycle permutation $\\gamma$.}\n  \\label{fig:square_zeta_inverse2}\n\\end{figure}\n\n\n\\section{Zeta inverse and area-preserving involution for a nice family of examples}\\label{sec:inductive_zeta_inverse}\n\nIn this section we present an inductive combinatorial description of the inverse of the zeta map and of the conjugate-area involution~$\\chi$ for a nice family of $(a,b)$-Dyck paths. This family consists of the Dyck paths that contain the lattice point with level $1$. Such Dyck paths are obtained by concatenating two Dyck paths in the $a'\\times b'$ and $a'' \\times b''$ rectangles illustrated in Figure~\\ref{fig:base_induction}. The sides of these two rectangles are the unique positive integers $0<a',a''<a$ and $0<b',b''<b$ such that \n\\begin{align*} \na' b - b' a &=1, \\\\ \nb'' a - a'' b &=1. \n\\end{align*}\nAs a consequence, $a'$ and $b'$ are relatively prime as well as $a''$ and $b''$, allowing us to apply induction.\n\n\\begin{figure}[htbp]\n\\includegraphics[width=0.6\\textwidth]{base_induction}\n\\caption{Base induction for zeta inverse and the area-preserving involution.}\n\\label{fig:base_induction}\n\\end{figure}\n\n\\subsection{Zeta inverse}\nLet $P$ be an $(a,b)$-Dyck path congaing the lattice point at level $1$, and let $P'$ and $P''$ be the two Dyck paths in the  $a'\\times b'$ and $a'' \\times b''$ rectangles whose concatenation is equal to~$P$.\nDefine the \\emph{star product} of $P' \\star P''$ as the path obtained by cutting~$P'$ at its highest level and infixing $P''$. This special product is illustrated in Figure~\\ref{fig:star_product}. Note that the highest level of $P'$ can be equivalently obtained by sweeping the main diagonal of either the $a\\times b$ rectangle or the $a'\\times b'$ rectangle.\n\n\\begin{figure}[htbp]\n\\includegraphics[width=0.7\\textwidth]{star_product}\n\\caption{Star product of rational Dyck paths.}\n\\label{fig:star_product}\n\\end{figure}\n\n\\begin{theorem}\\label{thm:inductive_inverse}\nIf $Q$ is an $(a,b)$-Dyck path containing the lattice point at level 1, zeta inverse of $Q$ is equal to the star product of the zeta inverses of $Q'$ and $Q''$:\n\\[\\zeta^{-1}(Q) = \\zeta^{-1}(Q') \\star \\zeta^{-1}(Q''). \\]\n\\end{theorem}\n\n\\begin{proof}\nWe will show that $\\zeta(P'\\star P'')$ is the concatenation of $\\zeta(P')$ and $\\zeta(P'')$, the theorem then follows by applying zeta to both sides of the equation. Since the path $P'$ is cut at its highest level, sweeping the main diagonal of the $a\\times b$ rectangle crosses the levels of $P'\\star P''$ corresponding to the path $P'$ first, followed by all the levels corresponding to the path $P''$. Therefore, $\\zeta(P'\\star P'')$ is the concatenation of $\\zeta(P')$ and $\\zeta(P'')$.\n\\end{proof}\n\n\n\\subsection{Area-preserving involution} The conjugate-area map of $Q$ can be obtained by induction in this case as well.\n\n\\begin{lemma}\\label{lem:areadif_level}\nLet $l$ be the level of a lattice point $p$ in the $a\\times b$ grid. If $U_l$ is the rectangle composed by the boxes northwest of $p$ and $\\tilde U_l$ is the rectangle composed by the boxes southeast of $p$, then \n\\[\n\\ensuremath{\\mathsf{area}}(\\tilde U_l)-\\ensuremath{\\mathsf{area}}(U_l) = l.\n\\]  \n\\end{lemma}\n\n\\begin{proof}\nIf $p=(p_1,p_2)$, then $l = p_2b-p_1a$. Furthermore,\n\\[\n\\ensuremath{\\mathsf{area}}(\\tilde U_l) - \\ensuremath{\\mathsf{area}}(U_l)= (b-p_1)p_2 - p_1(a-p_2) = p_2b-p_1a =l.\n\\]\n\\end{proof}\n\n\\begin{theorem}\\label{thm:inductive_area}\nLet $Q$ is an $(a,b)$-Dyck path containing the lattice point at level 1. The bounded partition of $\\chi(Q)$ is the partition whose restriction to the $a'\\times b'$ and $a''\\times b''$ rectangles gives the bounded partitions of $\\chi(Q')$ and $\\chi(Q'')$, and which contains all boxes below the main diagonal outside the two rectangles.\n\\end{theorem}\n\n\\begin{proof}\nWe first observe that $\\chi(Q)$ defined this way has the same area of $Q$. This is equivalent to show that the bounded partitions of $\\chi(Q)$ and $Q$ have the same area, when restricted to the complement of the $a'\\times b'$ and $a''\\times b''$ rectangles. These restrictions are exactly the rectangle $\\tilde U_1$ after removing the box on its upper left corner, and the rectangle $U_1$. The claim then follows by Lemma~\\ref{lem:areadif_level}. \n\n Now, let $\\gamma'$ and $\\gamma''$ be the cycle permutations arising from the pairs $(Q',\\chi(Q'))$ and $(Q'',\\chi(Q''))$, and $\\gamma$ be the cycle permutation of $(Q,\\chi(Q))$.\n The cycle permutation $\\gamma$ can be obtained by cutting~$\\gamma'$ exactly before its highest value and putting $\\gamma''$ in between, with all its values increased by $a'+b'$. In the example in Figure~\\ref{fig:inductive_area} we get\n\\begin{align*}\n \\gamma' &= ({\\color{blue} 1, 3\\ |\\ 5, 4, 2}) \\\\\n \\gamma'' &= ({\\color{red} 1, 3, 7, 5, 8, 6, 4, 2}) \\\\\n \\gamma &= ( {\\color{blue}1, 3,} \\ {\\color{red}6, 8, 12, 10, 13, 11, 9, 7} \\ {\\color{blue} 5, 4, 2})\n\\end{align*}\n\nThe cyclic descents of $\\gamma$ correspond exactly to the cyclic descents of $\\gamma'$ and $\\gamma''$. Moreover, the descent at the highest value of $\\gamma'$ corresponds to the east step at the highest level of $\\zeta^{-1}(Q')$. Thus, replacing cyclic descents in $\\gamma$ by east steps and ascents by north steps gives rise to the start product~$\\zeta^{-1}(Q')\\star \\zeta^{-1} (Q'')$, which is equal to $\\zeta^{-1}(Q)$ by Theorem~\\ref{thm:inductive_inverse}. \n\\end{proof}\n\n\\begin{figure}[htbp]\n\\includegraphics[width=0.3\\textwidth]{inductive_area}\n\\caption{The inductive conjugate-area map for paths containing level 1.}\n\\label{fig:inductive_area}\n\\end{figure}\n\n\\subsection{$k$th valley Dyck paths}\nOne interesting family of $(a,b)$-Dyck paths is the family of \\emph{$k$th valley Dyck paths}, paths $Q_k$ with valleys at levels $0,1,2,\\dots , k$ for some $k < a$. The area-conjugate map for these paths behaves very nice and can be described in terms of the rectangles $U_l$ and $\\tilde U_l$ in Lemma~\\ref{lem:areadif_level}.\n\nFor $0<l<a$, consider the collections of boxes $V_l$ and $\\hat V_l$ defined by\n\\[\n V_l = U_l \\smallsetminus \\bigcup_{i=1}^{l-1} U_i, \\hspace{1cm}\n \\hat V_l = \\hat U_l \\smallsetminus \\bigcup_{i=1}^{l-1} \\hat U_i,\n\\]\nwhere $\\hat U_i$ is composed of the boxes of  $\\tilde U_i$ that are below the main diagonal. Equivalently, $\\hat U_i$ is the result of removing the box in the upper left corner of  $\\tilde U_i$. An example is illustrated in Figure~\\ref{fig:kthDyckpaths}.\n\n\\begin{lemma}\nFor $0<l<a$, the area of $V_l$ is equal to the area of $\\hat V_l$.\n\\end{lemma}\n\n\\begin{proof}\nSince $V_1=U_1$ and $\\hat V_1=\\hat U_1$, which is $\\tilde U_1$ after removing one box, Lemma~\\ref{lem:areadif_level} implies that~$V_1$ and $\\hat V_1$ have the same area. The level 2 becomes level 1 in the smaller $a''\\times b''$ rectangle, and~$V_2,\\hat V_2$ are given by $U_1,\\hat U_1$ in this smaller rectangle. Again, Lemma~\\ref{lem:areadif_level} implies that $V_2$ and $\\hat V_2$ have the same area. Continuing the same argument in the smaller rectangles that appear in the process finishes the proof.\n\\end{proof}\n\n\\begin{figure}[htbp]\n\\includegraphics[width=0.45\\textwidth]{kthDyckpaths}\n\\caption{Example of the conjugate-area map for $k$th valley Dyck paths for $k=3$. The area of $V_i$ is equal to the area $\\hat V_i$.}\n\\label{fig:kthDyckpaths}\n\\end{figure}\n\nNote that the bounded partition of $Q_k$ is the (disjoint) union of $V_1,\\dots,V_k$. \n\n\\begin{proposition}\nThe bounded partition of $\\chi(Q_k)$ is the (disjoint) union of $\\hat V_1,\\dots,\\hat V_k$.\n\\end{proposition}\n\n\\begin{proof}\nNote that the restriction of $Q_k$ to the $a'\\times b'$ and $a''\\times b''$ rectangles gives two smaller $k$th valley Dyck paths $Q'_{k'}$ and $Q''_{k''}$. The result then follows directly from Theorem~\\ref{thm:inductive_area} by induction on~$k$.  \n\\end{proof}\n\nFigure~\\ref{fig:kthDyckpaths_inverse} illustrates an example of the inverse of the zeta map for $k$th valley Dyck paths obtained by applying Theorem~\\ref{thm:inverse}.\n\n\\begin{figure}[htbp]\n\\includegraphics[width=0.9\\textwidth]{kthDyckpaths_inverse}\n\\caption{Example of the inverse of the zeta map for $k$th valley Dyck paths for $k=3$.}\n\\label{fig:kthDyckpaths_inverse}\n\\end{figure}\n\n\n\n\n\n\n\n\\section{The delta statistic and initial bounce paths}\\label{sec:9}\n\nWe have tried a number of approaches for showing that the zeta map is a bijection. Our last approach uses a delta statistic which can be estimated by means of an initial bounce path.\n\n\\begin{definition}\nDefine the \\emph{delta statistic} $\\delta(\\P)$ to be the number of levels $l_i<a+b$ along $\\P$. \n\\end{definition}\n\nGeometrically, $\\delta(\\P)$ counts the number of lattice points in $P$ that belong to the diagonal path (closest to the diagonal). Surprisingly, this is sufficient to construct the inverse of the zeta map.\n\n\\begin{theorem}\\label{cor:delta_inverse}\nIf $\\delta(\\P)$ is determined uniquely by $\\zeta(\\P)$ for all $\\P$, then the map $\\zeta$ is invertible.\nIn such case, the inverse path $\\P$ is determined from $\\gamma(\\P)$ as in Proposition~\\ref{rem:zetainverse_delta}. \n\\end{theorem}\n\n\n\\subsection{Box math and the inclusion poset of rational Dyck paths}\\label{sec:82}\\\nOur approach for proving Theorem~\\ref{cor:delta_inverse} \nrelies on careful analysis of the poset structure on the set of rational Dyck paths under the usual inclusion relation: we say $\\P<Q$ if the path $\\P$ is weakly below the path $Q$, or, equivalently, if the set of positive hooks of $\\P$ is contained in the set of positive hooks of $Q$.  This poset is graded by the area statistic, with covering relation given by adding a single box. \n\n\\begin{definition}\nThe \\emph{maximal level} $m$ of a path $\\P$ is the largest level appearing in the reading word of $L(\\P)$.  Likewise, the \\emph{maximal box} is the box under the peak of $\\P$ labeled by the maximal level~$m$.  For any path with area greater than $0$, we define the \\emph{predecessor} of $\\P$ as the path obtained by removing the box under the peak of $\\P$ farthest from the diagonal. This replaces the maximal level $m$ with~$m-a-b$ in $L(P)$.  \n\\end{definition}\n\n\\begin{lemma}\nSuppose that $\\P$ is an $(a,b)$-Dyck path with predecessor $\\P'$. We have~$\\sl(\\P')<\\sl(\\P)$.\n\\end{lemma}\n\\begin{proof}\nSince~$\\P'$ is obtained by removing the maximal box of~$\\P$, the laser filling of $\\P'$ is equal to the laser filling of $\\P$ when removing the laser filling of its maximal box. Since skew length is equal to the sum of the entries in the laser filling, the result follows.\n\\end{proof}\n\n\n\nSince every path has a unique maximal hook, we induce a spanning tree $T$ in the Hasse diagram of $\\DD$ with the property that if $\\P'<\\P$ in $T$, then $\\sl(\\P')<\\sl(\\P)$.\nWe can also precisely describe the combinatorial effect of removing the maximal box from $\\P$ on~$\\zeta(\\P)$.\n\n\\begin{proposition}\nAll of the following operations are equivalent ways to remove the maximal box:\n\\begin{enumerate}\n\\item Remove the box whose associated hook length is greatest (furthest box from the diagonal).\n\\item Remove the longest row from $\\mathfrak{c}(\\P)$.\n\\item In the reading word $(l_1, l_2, \\ldots, l_{a+b})$ of $\\P$, reduce the maximal level $m$ by $(a+b)$, leaving all other levels unchanged.\n\\item In the standardization $\\sigma(\\P)$, let $\\alpha$ be the number of levels of~$\\P$ greater than $m-a-b$ excluding $m$.  Replace the entry $(a+b)$ in $\\sigma$ with $a+b-\\alpha$, and increase all entries greater than or equal to $a+b-\\alpha$ by one.  Equivalently, multiply $\\sigma(\\P)$ on the left by the cycle permutation $\\rho_{a+b-\\alpha,a+b}$ with cycle notation $(a+b-\\alpha, a+b-\\alpha+1, \\ldots, a+b)$.\n\\item Conjugate the permutation $\\gamma(\\P)$ by the cycle $\\rho_{a+b-\\alpha,a+b}$ to obtain:\n\\[\\rho_{a+b-\\alpha,a+b} \\gamma(\\P)  \\rho_{a+b-\\alpha,a+b}^{-1}.\\]\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nThe second operation follows directly from the definition of $\\mathfrak{c}(\\P)$.  The third method is clear from the effect on the labels $l_i$ of applying the first method.  The fourth item follows from third, and the fifth item follows from the effect of conjugation on the cycle notation of a permutation.\n\\end{proof}\n\nWe can thus try to understand the structure of the tree $T$ by understanding certain conjugations of the permutation $\\gamma(\\P)$.\nWe can observe that adding the box at the label $l_1=0$ is equivalent to removing the maximal box from $\\P^c$.  This reduces all labels $l_i$ by $a+b$ except for the label $l_1=0$, which remains the same.  As a result, the relative value of the label $0$ is increased from $1$ to \nthe number $\\delta(P)$ of labels $l_i<a+b$, \nwhile all other labels $\\leq \\delta(P)$ are reduced by one.  \n\n\\begin{proposition}\\label{prop:predecessor1}\nLet $\\P'$ be the conjugate of the path obtained by removing the maximal box from~$\\P^c$. The permutation~$\\gamma(\\P')$ is the conjugate \n\\[\n\\gamma(\\P') = \\rho_{1,\\delta(\\P)}^{-1} \\gamma(\\P) \\rho_{1,\\delta(\\P)}.\n\\]\n\\end{proposition}\n\nThe action of removing the maximal box from~$\\P^c$ on~$Q=\\zeta(\\P)$ is also completely determined by~$\\delta(\\P)$. For simplicity, we call the path $Q'=\\zeta(P')$ the \\emph{$\\zeta$-predecessor} of $Q$. \n\\begin{proposition}\\label{prop:predecessor2}\nThe $\\zeta$-predecessor of $Q=\\zeta(P)$ is completely determined by $\\delta=\\delta(P)$ as follows:\n\\begin{enumerate}[1.]\n\\item All the steps in $Q$ after the first $\\delta$ steps remain unchanged.\n\\item The first $\\delta$ steps are rotated as follows:\n\\begin{enumerate}[(a)]\n\\item the first east step that appears is changed to a north step,\n\\item the first north step is changed to an east step,\n\\item the first $\\delta$ steps are rotated once (rotating the first step to the end of the first delta steps). \n\\end{enumerate}\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{example}\nAn example of this procedure is illustrated in Figure~\\ref{fig:delta_Q_predecessor} for the path~$Q=\\zeta(\\P)$ associated to our running example path $\\P$. The number of levels of~$\\P$ smaller than $a+b$ is~$\\delta(P)=5$. The cycle permutation $\\gamma'$ is obtained from $\\gamma$ by rotating the labels $1,\\dots,5$. \n\\end{example}\n\n\\begin{figure}[htb]\n  \\includegraphics[width=0.8\\textwidth]{delta_Q_predecessor} \n  \\caption{Determining the $\\zeta$-predecessor of a path $Q=\\zeta(\\P)$ with $\\delta(\\P)$.}\n  \\label{fig:delta_Q_predecessor}\n\\end{figure}\n\n\\begin{proof}\nAdding a box at the level $\\ell_1=0$ of $P$ increases this level by $a+b$ and all other levels remain unchanged. The level $\\ell_1=0$ is transformed from a north level in $P$ to an east level in $P'$, while the east level $\\ell_i=a$ is transformed to a north level. Observing that the levels $\\ell_1$ and $\\ell_i$ correspond to the first north and first east steps in $Q$ respectively, and that the relative order of the levels less than $a+b$ is rotated once, one concludes the result. \n\\end{proof}\n\n\nThe two formulations in Proposition~\\ref{prop:predecessor1} and Proposition~\\ref{prop:predecessor2} have the advantage that we do not actually need to know the value of the maximal label $m$, and reduces the problem of showing that $\\zeta$ is a bijection to computing a single statistic.  To wit, if~$\\delta(\\P)$ can be directly computed from~$Q=\\zeta(\\P)$, then we can  obtain the $\\zeta$-predecessor of $Q$, and repeat until we arrive at the diagonal path~$\\P_0$. This would completely determine $\\P$ from $\\zeta(\\P)$. \n\n\\begin{proposition}\\label{rem:zetainverse_delta}\nLet $Q=\\zeta(P)$ and $Q=Q_1,\\dots,Q_l$ be the list of $\\zeta$-predecessors of $Q$ with $Q_l$ being the final path containing all boxes above the main diagonal. If $\\delta_i$ is the $\\delta$-statistic corresponding to~$Q_i$, then the permutation $\\gamma(P)$ is determined by \n\\[\n\\gamma(P) = \\rho\\  \\gamma_0\\ \\rho^{-1},\n\\]\nwhere $\\rho= \\rho_{1,\\delta_1}\\dots \\rho_{1,\\delta_{l-1}}$, the permutation $\\rho_{1,i}$ has cycle notation $(1, \\dots , i)$, and $\\gamma_0=\\gamma(\\P_0)$. \nThe east steps of the path $\\P$ are encoded by the cyclic descents of $\\gamma$ when considered in one-line notation, as described in Section~\\ref{sec:zetaeta}. \n\\end{proposition}\n\n\\begin{remark}\nThe $\\delta$ statistic has also been considered shortly after this paper by Xin in~\\cite{xin_efficient_2015}. This statistic, and another related statistic called ``key\", is used by the author to present a search algorithm for inverting the zeta map. This algorithm shows an alternative proof that the zeta map is a bijection in the cases~$(a,ak\\pm 1)$, by giving a recursive construction of~$\\zeta^{-1}(Q)$. However, the general case remains open. The results of Xin in~\\cite{xin_efficient_2015} are very similar to those presented in this section, and also use the operations of removing the maximal box in $\\P$ and $\\P^c$. Corollary~\\ref{cor:delta_inverse} should be compared with~\\cite[Corollary~19]{xin_efficient_2015}. We also present estimates for the~$\\delta$ statistic and a precise formula in the Fuss-Catalan case $(a,ak+1)$ in Proposition~\\ref{prop:funnydelta} and Corollary~\\ref{cor:delta_Fuss}, which should be compared with~\\cite[Theorem~16]{xin_efficient_2015}. The estimates we present in Proposition~\\ref{prop:funnydelta} determine the number of children of the nodes in the search tree in the ``ReciPhi algorithm\" in~\\cite[Section~5]{xin_efficient_2015}.\nOur algorithm for describing the inverse of zeta in the Fuss-Catalan case $(a,ak+1)$ \nshould be compared with the ReciPhi algorithm for the Fuss-Catalan case in~\\cite[Section~5]{xin_efficient_2015}.\n\\end{remark}\n\n\n\\subsection{Initial part of a rational bounce path}\nThe zeta map has been shown to be a bijection in the special cases $(a,am\\pm 1)$ by way of a ``bounce path'' by which zeta inverse could be computed~\\cite{Loehr,GMII}. However, constructing such a bounce path for the general~$(a,b)$ case remains elusive. In this section, we construct the initial part of a rational bounce path and show its relation to the $\\delta$ statistic. In particular, we explicitly compute $\\delta$ in the Fuss-Catalan case~$(a,ak+1)$.\n\n\\begin{definition}\nLet $Q$ be an $(a,ak+r)$-Dyck path with $0\\leq k$ and $0<r<a$. The rational \\emph{initial bounce path} of $Q$ consists of a sequence of alternating $k+1$ \\emph{vertical moves} and~$k$ \\emph{horizontal moves}. We begin at $(0,0)$ with a vertical move followed by a horizontal move, and continue until eventually finish with the $(k+1)$th vertical move.  \nLet $v_1,\\dots,v_{k+1}$ denote the lengths of the successive vertical moves and $h_1,\\dots, h_k$ denote the lengths of the successive horizontal moves. These lengths are determined as follows.\n\nWe start from $(0,0)$ and move north $v_1$ steps until reaching an east step of $Q$. Next, move~$h_1=v_1$ steps east. Next, move north $v_2$ steps from the current position until reaching an east step of $Q$.  Next, move $h_2=v_1+v_2$ steps east. In general, we move north $v_i$ steps from the current position until reaching an east step of the path, and then move east $e_i=v_1+\\dots +v_i$ steps. This is done until obtaining the last vertical move $v_{k+1}$. \n\\end{definition}\n\n\\begin{remark}\nThe definition of the initial bounce path is exactly the same as an initial part of the bounce path in the Fuss-Catalan case~\\cite{Loehr,GMII}. The description of this initial part remains the same for the general $(a,b)$-case but we still do not know how to extend it to a complete bounce path in general. \n\\end{remark}\n\nIt turns out that the initial bounce path is closely related to the $\\delta$ statistic. Denote by $|v|=v_1+\\dots +v_{k+1}$ and by $|h|=h_1+\\dots +h_k$. In all the results of this section we always assume $0\\leq k$ and $0<r<a$. \n\n\\begin{proposition}\\label{prop:funnydelta}\nLet $Q=\\zeta(P)$ be an $(a,ak+r)$-Dyck path and $\\widetilde \\delta(\\P) \\leq \\delta(\\P)$ be the number of levels in~$\\P$ that are less than or equal to $a(k+1)$. The two following equations hold:\n\\begin{equation}\\label{eq:bounce1}\n\\widetilde \\delta(\\P) = |v|+|h|+1,\n\\end{equation}\n\\begin{equation}\\label{eq:bounce2}\n|v|+|h|+1 \\leq \\delta(\\P) \\leq |v|+|h|+r.\n\\end{equation}\n\\end{proposition}\n\nThis proposition will follow from the following lemma. Note that every such a path $\\P$ contains the east levels $a,2a,\\dots,(k+1)a$ at the end of the path. Moreover,\n\n\\begin{lemma}\nThe east steps of $Q=\\zeta(P)$ that are reached by the vertical moves $v_1,\\dots,v_{k+1}$ of its initial bounce path correspond to the east levels $a,2a,\\dots,(k+1)a$ of $P$.  \n\\end{lemma}\n\n\\begin{proof}\nDenote by $A_0=\\{0,1,\\dots , a-1\\}$ the set of natural numbers between $0$ and $a-1$, and let~$A_i=A_0+ia$ be the translation of $A_0$ by $ia$. Note that the number of boxes above the main diagonal that are directly on the right of a north step with north level in $A_i$ is exactly equal to $i$. So, these sets can be used to encode the ``area-vector\" of a Dyck path.\n\nAs in the known Fuss-Catalan bounce path description, we will show that the vertical steps of~$Q$ that are directly on the left of the $v_i$ vertical move contribute area $i-1$ in $\\P$. More precisely, we will show:\n\n\\begin{enumerate}[1.]\n\\item The vertical steps of~$Q$ that are directly on the left of the $v_i$ vertical move correspond to north levels in $\\P$ that belong to $A_{i-1}$.\n\\item The horizontal steps of~$Q$ that are directly above of the $h_i$ horizontal move correspond to east levels in $\\P$ that belong to $A_i$.\n\\end{enumerate}\n\nDenote by $N_i$ the set of north levels in $\\P$ that belong to $A_i$, for $i=0,\\dots ,k$. Similarly, denote by $E_i$ the set of east levels in $\\P$ that belong to $A_i$, for $i=1,\\dots, k$. We first show that $E_i$ can be obtained as the disjoint union \n\\begin{equation}\\label{eq:ENcorrespondence}\nE_i = \\bigcup_{j=0}^{i-1} \\left(N_j + (i-j)a\\right). \n\\end{equation}\n\nFor this, note that the first north level in $\\P$ that appears after an east level $e\\in E_i$ should be a north level $n\\in N_j$ for some $j \\in \\{0,\\dots i-1\\}$, and that $e=n+(i-j)a$. Reciprocally, every north level $n\\in N_j$ for some $j \\in \\{0,\\dots i-1\\}$ forces the east levels $e_{j+1},e_{j+2}\\dots, e_{k}$ to appear as east levels in $\\P$, where $e_{j+l}=n+l a$ (indeed, $e_{j+l}\\in A_{j+l}$ which means that $e_{k-1}<ak<b=ak+r$. Then, all the lattice points one step below $e_{j+1},e_{j+2}\\dots, e_{k-1}$ are below the main diagonal). Thus, the north level $n\\in N_j$ contributes with exactly one east level $e=n+(i-j)a$ in $E_i$. \n\nItems 1 and 2 above are now equivalent to prove that $v_i=|N_{i-1}|$ and $h_i=|E_i|$.  Equation~\\ref{eq:ENcorrespondence} implies that \n$|E_i|=|N_0|+\\dots +|N_{i-1}|.$\nSince $h_i=v_1+\\dots+v_i$, it suffices to prove $v_i=|N_{i-1}|$. \nNote that $v_1$ is clearly the number of elements in $N_0$, since the smallest east level of $\\P$ (which corresponds to the first east step in $Q$) is equal to $a$. Moving $h_1=v_1=|E_1|$ steps horizontally covers all the east steps of $Q$ corresponding to the east levels in $\\P$ that belong to $A_1$. Moving up $v_2$ units from the current position hits the path at the east step corresponding to the first east level of $\\P$ that belongs to $A_2$. This east level is exactly equal to $2a$, and all the north steps on the left of $v_2$ in $Q$ correspond to the north levels in $\\P$ that belong to $A_1$, that is $v_2=|N_1|$. In general, $\\P$ contains all east levels $a,2a,\\dots,(k+1)a$. Therefore, the initial value of $E_i$ is equal to $ia$ and is smaller than all values in $N_i$. As a consequence the vertical move $v_i$ of the bounce path hits the path $Q$ precisely at the east step corresponding to the level $ia$ of $\\P$ as desired, and $v_i=|N_{i-1}|$. This finishes the proof of the proposition and the proof of items 1 and 2.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:funnydelta}]\nThe east step of $Q$ that is reached by the vertical move~$v_{k+1}$ corresponds to the east level $(k+1)a$ in $\\P$. So, $\\widetilde \\delta(\\P)$ is equal to the position of this east step in $Q$, which is equal to $|v|+|h|+1$. Since $a+b=(k+1)a+r$ and $a+b$ never appears as a level in $\\P$, then~$\\delta\\leq \\widetilde \\delta + r-1 = |v|+|h|+r$.\n\\end{proof}\n\nReplacing $r=1$ in Equation~\\eqref{eq:bounce2}, we obtain.\n\\begin{corollary}\\label{cor:delta_Fuss}\nIn the Fuss-Catalan case $(a,ak+1)$, the statistic $\\delta(\\P)$ is determined by the initial bounce path of $Q=\\zeta(\\P)$ by\n\\begin{equation}\n\\delta(\\P) = |v|+|h|+1.\n\\end{equation}\n\\end{corollary}\n\nAs a consequence of Corollary~\\ref{cor:delta_inverse} and Corollary~\\ref{cor:delta_Fuss} we obtain an alternative proof that the zeta map is a bijection in the Fuss-Catalan case~$(a,ak+1)$, as previously shown by Loehr in~\\cite{Loehr}. \n\n\\begin{corollary}\\label{cor:zetabijection_Fuss}\nThe zeta map is a bijection in the Fuss-Catalan case $(a,ak+1)$.\n\\end{corollary}\n\n\\section*{Acknowledgements}\n\\label{sec.ack}\n\nSome of the results in this work are partially the result of working sessions at the Algebraic Combinatorics Seminar at the Fields Institute with the active participation of Farid Aliniaeifard, Nantel Bergeron, Eric Ens, Sirous Homayouni, Sahar Jamali, Shu Xiao Li, Trueman MacHenry, and Mike Zabrocki. \n\nWe thank two anonymous referees for their comments and for pointing us to some references of previously known results.\n\nWe especially thank Rishi Nath for discussions involving core partitions that helped simplify arguments in Section~\\ref{sec:skew_length}, Drew Armstrong for sharing his knowledge of the history of this subject, and Greg Warrington for pointing us the references about the $\\ensuremath{\\mathsf{dinv}}$ statistic in Remark~\\ref{rem:dinv}.  We are also grateful to York University for hosting a visit of the third author.\n\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nUltracold diatomic molecules, namely with translational motion cooled down to temperatures well below one millikelvin, and internal degrees of freedom reduced to a single quantum level \\cite{bahns1996}, are nowadays well recognized as promising systems for quantum simulation, quantum computation, ultracold chemistry and precision measurements. This is particularly true for those species which possess additional internal properties like a permanent electric dipole moment (PEDM) in their own frame, and\/or a magnetic dipole moment, as they can be manipulated by external electric and magnetic fields \\cite{Review_cold_molecule_1,Review_cold_molecule_2,Quantum_comp_RbSr,polar_paramagnetic_molecules_field,Micheli_lattice_spin}.\n\nThe first translationally-ultracold molecular species ever produced were homonuclear, namely Cs$_2$ \\cite{Cs2_Comparat}, and Rb$_2$ molecules \\cite {gabbanini2000}. The formation process relied on photoassociation (PA) of ultracold atomic pairs followed by radiative emission (RE) down to the electronic ground state \\cite{thorsheim1987}. Shortly after, several groups were able to create heteronuclear diatomic species with the same approach \\cite{jones2006}. A first breakthrough came with the direct observation of ultracold molecules formation (UMF) in the lowest rovibrational level ($v=0, J=0$) of their ground state \\cite{deiglmayr2008a}, with some ability of control of their internal state \\cite{viteau2008,wakim2012,manai2012}. The fully-controlled creation of ultracold dipolar molecules was demonstrated at about the same time on the KRb polar species \\cite{KRb,KRb_2,borsalino2014}, but using the alternative approach of magnetoassociation of an atom pair into a weakly-bound molecule, followed by a stimulated Raman adiabatic passage (STIRAP) to transfer population into the lowest energy level of the KRb electronic ground state. \n\nGround-state species exhibiting an additional magnetic moment are for instance diatomic molecules composed of an alkali-metal atom and an alkaline-earth atom (or an Ytterbium atom), in which the magnetic dipole moment comes from the existence of an unpaired electron. Surprisingly, the spectroscopy of such diatomic molecules is still poorly known. The recent interest for such ultracold species triggered several investigations at relatively low resolution \\cite{krois2013,krois2014,pototschnig2015,gerschmann2017,schwanke2017,ciamei2018}. However these species are still challenging to create in the ultracold domain. After an initial prediction \\cite{zuchowski2010a}, magnetic Feshbach resonances have been observed for $^{87}$Rb$^{88}$Sr and $^{87}$Rb$^{87}$Sr molecules \\cite{barbe2017}, but they are not yet used for the formation of weakly-bound ground-state $^{87}$Rb$^{88}$Sr .\n\nAll-optical methods are attractive to create ultracold molecules as they do not rely on peculiarities of the molecular structure like the presence of Feshbach resonances in the ground state at moderate magnetic fields. Here we model the PA+RE sequence mentioned above for the $^{87}$Rb$^{84}$Sr bosonic species. As expected we find that it is not selective enough to populate a single quantum level in the molecular ground state. Therefore we consider the motional levels of a tight trap \\cite{jaksch2002} to implement a STIRAP transfer, as previously demonstrated with Sr$_2$ molecules \\cite{STIRAP_production_Sr2_1,STIRAP_production_Sr2_2}. Our calculations are based on the RbSr electronic structure data previously obtained in our group \\cite{RbSr_FCI_GS,Zuchowski_RbSr}, which are recalled in Section \\ref{sec:structure}. We compute in Section \\ref{sec:PA} PA and UMF rates, when the PA laser is tuned to the red of either the ($5\\,^2S_{1\/2} \\rightarrow 5\\,^2P_{1\/2,3\/2}$) resonant transitions in $^{87}$Rb, or the ($5\\,^1S_{0} \\rightarrow 5\\,^3P_{0,1,2}$) intercombination transitions in $^{84}$Sr. The transition $^1$S$_0$ $\\rightarrow$ $^3$P$_1$ is indeed employed for the cooling of Sr atoms in ongoing experiments for quantum degenerate mixtures of strontium and rubidium atoms \\cite{Mixture_BEC_Sr_Rb}. Relying on the obtained knowledge about transition dipole moment, we propose promising candidate levels to implement STIRAP process in a tight optical trap to create weakly-bound ultracold $^{87}$Rb$^{84}$Sr ground state molecules (Section \\ref{sec:STIRAPtrap}), and their transfer into the lowest vibrational level of their ground state using a second STIRAP sequence (Section \\ref{sec:STIRAPgroundstate}). For convenience purpose in the calculations, atomic units of distance (1~a.u.$=a_0=0.052917721067$ $10^{-10}$ ~m), energy (1~a.u.=1 hartree=219474.6313702~cm$^{-1}$) and electric dipole moment (1~a.u.=2.54175~D) will be used throughout the paper, except otherwise stated.\n\n\\section{Electronic structure of the RbSr molecule}\n\\label{sec:structure}\n\nIn this work, we are interested in the states correlated to the three lowest dissociation limits of RbSr (turning into six limits when spin-orbit interaction is included), listed with increasing energy: Rb $(5s\\,^2S)$ + Sr ($5s^2\\,^1S$), Rb $(5p\\,^2P)$ + Sr ($5s^2\\,^1S$) and Rb $(5s\\,^2S)$ + Sr ($5s5p\\,^3P$). They give rise to three sets of electronic states (labeled in Hund's case a notation $(N)^{2S+1}\\Lambda$), namely ($(1)^2\\Sigma^+$, or more commonly $X^2\\Sigma^+$), ($(2)^2\\Sigma^+, (1)^2\\Pi$), and ($(3)^2\\Sigma^+, (1)^2\\Pi, (1)^4\\Sigma^+, (1)^4\\Pi$), respectively. The corresponding potential energy curves (PECs), and transition dipole moments (TDMs) between the $X^2\\Sigma^+$ ground state and several excited electronic states are displayed in Fig. \\ref{fig:pot}. For the calculations in the next sections, we have selected full configuration-interaction (FCI) calculation performed on the three valence electrons moving in the field of relativistic large effective core potentials (ECPs), including core-polarization potentials (CPP), and extrapolated to large distances, which were reported in our previous work \\cite{Zuchowski_RbSr}. \n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=8cm]{Fig1_PEC+TDM.pdf}\n\t\\caption{Upper panel: potential energy curves of the electronic states relevant for the present study, restricted to distances smaller than 20~a.u. for clarity. Lower panel: transition dipole moment between the $(1)^2\\Sigma^+$ ground state and the $^2\\Sigma^+$ and $^2\\Pi$ states of the upper panel \\cite{Zuchowski_RbSr} (solid curves), compared to those of Ref.\\cite{pototschnig2014} (dashed lines).  We note that our computed transition dipole moments (TDMs) properly match the atomic TDMs at large distances, while those of Ref.\\cite{pototschnig2014} are overestimated by about 10\\% at this limit.}\n\t\\label{fig:pot}\n\\end{figure}\nSeveral features of the PECs are important to notice for the following. First, except for the (1)$^4\\Sigma^+$ state, the equilibrium distances of excited-state PECs are significantly smaller than the one of the ground-state PEC. Second, two  curve crossings are visible, between the (2)$^2\\Sigma^+$ and (1)$^2\\Pi$ PECs, and between the (3)$^2\\Sigma^+$ and (2)$^2\\Pi$ PECs. These features are also present in two other available calculations displayed in Fig. \\ref{fig:PEC-comparaison} from very different methods, namely the EOM-CCSD (equation-of-motion coupled-cluster method limited to singly and doubly excited configurations) method employed in Ref.\\cite{Zuchowski_RbSr,RbSr_Ernst}, and the MCSCF-MRCI (MultiConfigurational Self-Consistent Field-Multi-Reference Configuration Interaction) method of Ref.\\cite{RbSr_Ernst}. Figure \\ref{fig:PEC-comparaison} reveals a good overall agreement among all the results, recalling however that the EOM-CCSD results for the excited electronic states are probably less accurate than the other results, as already discussed in Ref.\\cite{Zuchowski_RbSr}. We also note in the upper panel, that the PECs from Ref.\\cite{RbSr_Ernst} converge to a dissociation energy larger by 107~cm$^{-1}$  than ours. This is related to the excitation energy of the Sr($5s5p\\,^3P$) level, found 20~cm$^{-1}$ above (resp. 87~cm$^{-1}$ below) the experimental one, in the  MCSCF-MRCI calculations (resp. FCI and EOM-CCSD calculations \\cite{Zuchowski_RbSr,guerout2010}). In order to illustrate in a complementary way the above results, we display in Table \\ref{tab:Spectro_param} the spectroscopic constants of these PECs, as well as the coefficient $C_6$ of the leading-order term of the long-range van der Waals interaction between Rb and Sr \\cite{jiang2013}. For the ground state, the calculations of Chen \\textit{et al.}  \\cite{Chen_RbSr} is also included. Deeply-bound spectra with thermoluminescence and Laser induced fluorescence between (X)$^2\\Sigma^+$ and (2)$^2\\Sigma^+$ states have been made by A. Ciamei et al. \\cite{ciamei2018}. They simulated fluorescence spectra using PECs from FCI ECP+CPP, EOM-CCSD and MSCF-MRC calculations and compared with the experimental one. For all three calculations, only few experimental band heads can be identified unambiguously. The predicted wavenumber for the 0-0 band head is very close in the case of the MSCF-MRCI calculation. The difference with the FCI ECP+CPP and EOM-CCSD calculation are respectively 25 cm$^{-1}$ and 400 cm$^{-1}$. On the other hands, the FCI-ECP and EOM CCSD calculations can predict more bands than the MSCF-MRCI calculation and the shape of the simulated spectrum is closer to the experimental one. \n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=8cm]{Fig2_PEC-comparaison.pdf}\n\t\\caption{Comparison of potential energy curves of the lowest excited states of RbSr calculated with FCI ECP+CPP (solid lines), EOM-CC (dashed lines) \\cite{Zuchowski_RbSr}, and MCSCF-MRCI (dotted lines) methods \\cite{RbSr_Ernst}.} \n\t\\label{fig:PEC-comparaison}\n\\end{figure}\n\n\\begin{table}\n\t\\centering\n\t\\caption{Main spectroscopic constants of the $^{87}$Rb$^{84}$Sr electronic states correlated to the three lowest dissociation limits. The labels ''a'' and  ''b'' for Ref.\\cite{Zuchowski_RbSr} refer to the calculations with the FCI-ECP method, and the EOM-CCSD method, respectively (see text).}\n\t\\begin{tabular}{ccccccc}\n \\hline \\hline\n State           & $R_e$ & $D_e$     &$\\omega_e$ &$C_6$ \\cite{jiang2013}&Limit& \\\\\n                 &($a_0$)&(cm$^{-1}$)&(cm$^{-1}$)&(a.u.)&&\\\\  \\hline\n (X)$^2\\Sigma^+$ &8.69   &1073.3     &38.98      &3699 &Rb$(5s\\,^2S)$+Sr($5s^2\\,^1S$)&\\cite{Zuchowski_RbSr}a\\\\\n                 &8.82   &1040.5     &38.09      &     &                             &\\cite{Zuchowski_RbSr}b\\\\\n\t\t\t\t\t\t\t\t &8.65   &1283.5     &42.1       &     &                             &\\cite{RbSr_Ernst}\\\\\n\t\t\t\t\t\t\t\t &8.827  &1017.58    &36.017     &     &                             &\\cite{Chen_RbSr}\\\\\n (2)$^2\\Sigma^+$ &8.40   &4982.9     &58.37      &23324&Rb$(5p\\,^2P)$+Sr($5s^2\\,^1S$)&\\cite{Zuchowski_RbSr}a\\\\\n                 &8.51   &4609.6     &60.20      &     &                             &\\cite{Zuchowski_RbSr}b\\\\\n                 &8.54   &5144.3     &58.9       &     &                             &\\cite{RbSr_Ernst}\\\\\n (1)$^2\\Pi$      &7.31   &8439.8     &79.50      &8436 &                             &\\cite{Zuchowski_RbSr}a\\\\\n                 &7.42   &8038.6     &83.19      &     &                             &\\cite{Zuchowski_RbSr}b\\\\\n                 &7.39   &8770.2     &79.5       &     &                             &\\cite{RbSr_Ernst}\\\\\n (3)$^2\\Sigma^+$ &7.67   &3828.0     &65.26      &8929 &Rb$(5s\\,^2S)$+Sr($5s5p\\,^3P$)&\\cite{Zuchowski_RbSr}a\\\\\n                 &7.81   &2892.4     &62.48      &     &                             &\\cite{Zuchowski_RbSr}b\\\\\n                 &7.84   &3677.8     &57.4       &     &                             &\\cite{RbSr_Ernst}\\\\\n (2)$^2\\Pi$      &7.65   &4421.2     &67.60      &5716 &                             &\\cite{Zuchowski_RbSr}a\\\\\n                 &7.88   &3303.5     &63.37      &     &                             &\\cite{Zuchowski_RbSr}b\\\\      \n                 &7.80   &4450.3     &65.8       &     &                             &\\cite{RbSr_Ernst}\\\\   \n (1)$^4\\Sigma^+$ &11.63  &336.3      &15.42      &8929 &                             &\\cite{Zuchowski_RbSr}a\\\\\n                 &11.81  &329.2      &15.03      &     &                             &\\cite{Zuchowski_RbSr}b\\\\\n                 &11.64  &396.7      &16.        &     &                             &\\cite{RbSr_Ernst}\\\\\n (1)$^4\\Pi$      &8.06   &2838.1     &56.98      &5716 &                             &\\cite{Zuchowski_RbSr}a\\\\ \n                 &8.24   &2655.7     &54.95      &     &                             &\\cite{Zuchowski_RbSr}b\\\\ \n                 &8.16   &3053.9     &57.6       &     &                             &\\cite{RbSr_Ernst}\\\\ \\hline \\hline\n\t\\end{tabular}\n\t\\label{tab:Spectro_param}\n\\end{table}\n\nIt is well known that a model for a PA spectrum depends on two crucial inputs: the long-range behavior of the PECs of the relevant states, and the scattering length of the ground state. Therefore the (X)$^2\\Sigma^+$ ground state PEC has been smoothly matched at 15$a_0$ to an asymptotic expansion expressed as $-C_6\/R^6-C_8\/R^8-C_{10}\/R^{10}$, with $C_8=4.609\\times 10^{5}$~a.u. and $C_{10}=5.833\\times 10^7$~a.u. \\cite{jiang2013}.  In contrast, the long-range expansion of the excited-state PECs has been restricted to the $-C_6\/R^{-6}$ term (Table \\ref{tab:Spectro_param}). \n\nIn the absence of a global PEC determined spectroscopically for the ground state, one cannot rely on the scattering length provided by the computed PEC. However the binding energies of the two uppermost vibrational levels of the $^{87}$Rb$^{84}$Sr molecule relative to the Rb $(5s\\,^2S, F=1)$ + Sr ($5s^2\\,^1S$) limit (where $F$ denotes the total angular momentum of the Rb atom accounting for the nuclear spin) have been recently measured by A. Ciamei and coworkers \\cite{ciamei2018} in a two-photon photoassociation experiment leading to $9.67\\times10^{-4}$~cm$^{-1}$ and $2.49$~cm$^{-1}$. Therefore we have slightly modified the position of the repulsive wall of the PEC around the dissociation limit  in order to match these experimental energies. The calculations of ground state eigenenergies were performed with the Mapped Fourier Grid Hamiltonian method (MFGH) \\cite{MFGH_1,MFGH_2,MFGH_3,MFGH_4}. The adjustment was constrained to the condition that the spectroscopic data of Table~\\ref{tab:Spectro_param} remain unchanged (i.e. the bottom of the PEC is unchanged). The best agreement was found when moving the inner turning point by 0.042~$a_0$ toward smaller distances. After this adjustment, the scattering length has a value of 89.3 a.u. which is close to the experimental one (92.7 a.u.) \\cite{ciamei2018} .\n\nImportant features of TDMs can be pointed out, and may have a strong influence on the optical response of RbSr molecules. First, the spin selection rule forbids transitions between doublet and quartet states (the latter are not reported in Fig.\\ref{fig:pot}). Second, the atomic transition Rb $(5s\\,^2S)$ $\\rightarrow$ Rb $(5p\\,^2P)$ is allowed while the atomic transition Sr ($5s^2\\,^1S$) $\\rightarrow$ Sr ($5s^2\\,^3P$) is spin-forbidden. This atomic selection rule is visible in the long-range part of the TDMs on Fig.\\ref{fig:pot} b). Finally, at short-range, the TDM with the (2) $^2\\Sigma^+$ state is  large, while the TDM with the (1) $^2\\Pi$ state is close to zero at short range. \n\nSpin-orbit (SO) splitting being large for the lowest excited states of both atoms ($\\Delta_{\\textrm{fs}}^{Rb}=237.1$~cm$^{-1}$ and $\\Delta_{\\textrm{fs}}^{Sr}=581.1$~cm$^{-1}$), it must be taken into account to model the PA spectrum. We used the same approach as in Ref.\\cite{Zuchowski_RbSr}: first the two sets of PECs correlated to the Rb $(5p\\,^2P)$ + Sr ($5s^2\\,^1S$) and Rb $(5s\\,^2S)$ + Sr ($5s5p\\,^3P$) dissociation limits are considered independently, and the atomic SO operators $\\hat{W}_{\\textrm{so}}^{Rb}=A^{Rb}\\vec{\\ell}^{Rb} . \\vec{s}^{Rb}$ and $\\hat{W}_{\\textrm{so}}^{Sr}=A^{Sr}(\\vec{\\ell}_1^{Sr} . \\vec{s}_1^{Sr}+\\vec{\\ell}_2^{Sr} . \\vec{s}_2^{Sr}$), respectively, are used as perturbations to the Hamiltonian containing the kinetic operator and the electrostatic interactions. The states including SO are labeled according to the projection $|\\Omega|$ of the total electronic angular momentum on the molecular axis (Hund case (c)). For the former asymptote, the $|\\Omega|=3\/2$ Hamiltonian matrix (including electrostatic interaction and SO) reduces to a single element $W_{\\textrm{so}}^{Rb}(|\\Omega|=3\/2)= V((1)^2\\Pi)+2 A^{Rb}$, where $A^{Rb}=\\Delta_{\\textrm{fs}}^{Rb}\/3$, while the matrix for $|\\Omega|=1\/2$ reads\n\\begin{equation}\nW_{\\textrm{so}}^{Rb}\\left(|\\Omega|=\\frac{1}{2}\\right)=\\begin{pmatrix}\nV((2)^2\\Sigma^+) & \\sqrt{2} A^{Rb} \\\\\n\\sqrt{2} A^{Rb} & V((1)^2\\Pi)+A^{Rb}\n\\end{pmatrix}.\n\\label{eq:Rb-omega12}\n\\end{equation}\nFor the latter dissociation limit, defining $A^{Sr}=\\Delta_{\\textrm{fs}}^{Sr}\/3$, the maximal value of $|\\Omega|$ is 5\/2, with a single matrix element  $W_{\\textrm{so}}^{Sr}(|\\Omega|=5\/2)= V((1)^4\\Pi)+A^{Sr}$, while the matrices are, for $|\\Omega|=3\/2$ and $|\\Omega|=1\/2$ respectively,\n\\begin{equation}\nW_{\\textrm{so}}^{Sr}\\left(|\\Omega|=\\frac{3}{2}\\right)=\\begin{pmatrix}\nV((2)^2\\Pi)+\\frac{2}{3} A^{Sr} & \\sqrt{\\frac{1}{3}}A^{Sr} & -\\frac{\\sqrt{2}}{3} A^{Sr} \\\\\n\\sqrt{\\frac{1}{3}} A^{Sr} & V((1)^4\\Sigma^+) & \\sqrt{\\frac{2}{3}} A^{Sr} \\\\\n-\\frac{\\sqrt{2}}{3} A^{Sr} & \\sqrt{\\frac{2}{3}} A^{Sr} & V((1)^4\\Pi)+\\frac{1}{3} A^{Sr}\n\\end{pmatrix}\n\\label{eq:Sr-omega12}\n\\end{equation}\nand \n\\begin{equation}\nW_{\\textrm{so}}^{Sr}\\left(|\\Omega|=\\frac{1}{2}\\right)=\\begin{pmatrix}\nV((3)^2\\Sigma^+) & \\sqrt{\\frac{8}{9}}A^{Sr} & 0 & -\\frac{1}{3} A^{Sr} & \\sqrt{\\frac{1}{3}}A^{Sr} \\\\\n\\sqrt{\\frac{8}{9}}A^{Sr} & V((2)^2\\Pi)-\\frac{2}{3}A^{Sr} & \\frac{1}{3} A^{Sr} & -\\frac{\\sqrt{2}}{3} A^{Sr} & 0 \\\\\n0 & \\frac{1}{3} A^{Sr} & V(^4\\Sigma^+) & \\sqrt{\\frac{8}{9}}A^{Sr} & \\sqrt{\\frac{2}{3}}A^{Sr} \\\\\n-\\frac{1}{3} A^{Sr} & -\\frac{\\sqrt{2}}{3} A^{Sr} & \\sqrt{\\frac{8}{9}}A^{Sr} & V((1)^4\\Pi)-\\frac{1}{3} A^{Sr} & 0 \\\\\n\\sqrt{\\frac{1}{3}} & 0 & \\sqrt{\\frac{2}{3}}A^{Sr} & 0 & V((1)^4\\Pi)-A^{Sr}\n\\end{pmatrix}\n\\label{eq:Sr-omega32}\n\\end{equation} \n\nThe Hund's case c PECs (N)$\\Omega$ including SO interaction are straightforwardly obtained by diagonalization of the full hamiltonian involving these matrices at each fixed R value (Fig.\\ref{fig:PEC_SO}). The asymptote Rb $(5p\\,^2P)$ + Sr ($5s^2\\,^1S$) is split in Rb $(5p\\,^2P_{1\/2})$ + Sr ($5s^2\\,^1S$) and Rb $(5p\\,^2P_{3\/2})$ + Sr ($5s^2\\,^1S$) while the asymptote Rb $(5s\\,^2S)$ + Sr ($5s5p\\,^3P$) is split in Rb $(5s\\,^2S)$ + Sr ($5s5p\\,^3P_0$), Rb $(5s\\,^2S)$ + Sr ($5s5p\\,^3P_1$) and Rb $(5s\\,^2S)$ + Sr ($5s5p\\,^3P_2$) (hereafter referred to as the  $^3P_0$,  $^3P_1$ and $^3P_2$ asymptotes, in short). A single $\\Omega=1\/2$ PEC is correlated to the $^3P_0$ asymptote, while two such PECs match  the $^3P_1$ and $^3P_2$ asymptotes. Similarly, a single $\\Omega=3\/2$ PEC is correlated to the $^3P_1$ asymptote and two PECs to the  $^3P_2$ asymptote. The equilibrium distances for excited states PECs are still smaller than for the ground state, except for the $(8) 1\/2$ and $(3) 3\/2$ states composed mainly of (1)$^4\\Sigma^+$ state. The crossings in the Hund case (a) PECs become avoide crossings in the Hund case (c).  \n\nWe display in Table \\ref{Spectro_param_comparaison_SO} the corresponding fundamental spectroscopic constants, as they are provided in several other publications \\cite{RbSr_Ernst,Chen_RbSr}. As expected from Fig.\\ref{fig:PEC-comparaison}, the equilibrium distances $R_e$ and the harmonic constants $\\omega_e$ are those of the states without SO, as the avoided crossings occur far from $R_e$. The dissociation energies are significantly changed, reflecting the magnitude of the atomic SO splittings. As already noted, our results are in good agreement with those of Ref. \\cite{RbSr_Ernst}. In contrast, significant differences are found with the work of Ref. \\cite{Chen_RbSr}, in particular for the well depth and for the $\\omega_e$ constant. In the latter work, the authors used the relativistic Dirac-Coulomb Hamiltonian where the electronic spin and consequently the related $R$-dependent relativistic interactions are explicitly accounted for in the Hamiltonian under a four-component framework, as initially developed in the approach of Ref.\\cite{sorensen2009}, so that the spin-orbit interaction is included in a non-perturbative way. But they used a basis set which is significantly smaller than the one of Ref.\\cite{RbSr_Ernst}, which thus may not be fully appropriate for excited states. Such differences in the PECs may also indicate a noticeable variation of the molecular SO coupling with the internuclear distance. Note that we have proved for the heaviest alkali atom Fr \\cite{aymar2006} that an electronic structure calculation including only the scalar relativistic term (as performed here) yields satisfactory electronic atomic orbitals even for such a heavy species.  \n\nIn Fig.\\ref{fig:PEC_SO} we also displayed the $R$-dependent quantity labeled with effective $\\mathrm{TDM}^2$, representing, for each molecular state $(N)\\Omega$ including SO, the linear combination of the squared $R$-dependent TDMs of Fig.\\ref{fig:pot} associated to the transitions from the ground state toward the states involved in $(N)\\Omega$. The weights of this combination are the squared components of the eigenvector associated to $(N)\\Omega$. It should be noted that this quantity is not the one to consider for actually computing transition probabilities, \\textit{i.e.} it should not be used in an integration over $R$ weighted by a pair of radial vibrational wave functions. Therefore the $\\mathrm{TDM}^2$ quantities, referred to as ''effective squared TDM'' for convenience, only provide a qualitative description of the actual mixture of states without SO composing $(N)\\Omega$, by comparison between Fig.\\ref{fig:PEC_SO} and Fig.\\ref{fig:pot}. As expected, our results for the $\\mathrm{TDM}^2$ quantities are very similar to those of Ref.\\cite{RbSr_Ernst}. This representation of the TDMs allows us to point some features that have an impact on PA. First, the avoided crossings between PECs induce crossings between TDM curves due to the change of the nature of the electronic states. At short range, the (2) $1\/2$ state (resp. the (3) $1\/2$ state) is mainly composed of (2) $^2\\Pi$ state (resp. (1) $^2\\Sigma^+$ state). This results in a high value for the former, and a low value for the latter. The $^2\\Pi$ state composition of the (1) $\\Omega=3\/2$ state is also clear. For the TDMs for the states correlated to Rb$(5s\\,^2S)$+Sr($5s5p\\,^3P_{0,1,2}$), the main characteristic is the non-zero value for (6),(7) and (8) $\\Omega=1\/2$ states. Even if they are constituted of quartet state at short-range, the mixture with doublet states due to the spin-orbit interaction implies non-zero values at larger distance. At short-range, we can also notice that the (4) $\\Omega=1\/2$ and (2) $\\Omega=3\/2$ states are mainly composed by (2) $^2\\Pi$ state while the (5) $\\Omega=1\/2$ state is mainly constituted by (3) $^2\\Sigma^+$ state.  \n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=16cm]{Fig3_PEC-SO.pdf}\n\t\\caption{RbSr potential energy curves of excited states including spin-orbit interaction as described in the text, and corresponding effective squared transition dipole moments from the $X^2\\Sigma^+$ ground state towards the states correlated to Rb$(5p\\,^2P_{1\/2,3\/2})$+Sr($5s^2\\,^1S$) (left column), and to Rb$(5s\\,^2S)$+Sr($5s5p\\,^3P_{0,1,2}$) (right column).}\n\t\\label{fig:PEC_SO}\n\\end{figure}\n\n\\begin{table}\n\t\\caption{Main spectroscopic constants of the $^{87}$Rb$^{84}$Sr electronic states including spin-orbit interaction, as compared to other published data. The corresponding dissociation limits are also indicated for clarity sake.}\n\t\\resizebox{17cm}{3.3cm}{\n\t\t\\centering\n\t\t\\begin{tabular}{c|ccc|ccc|ccc|c}\n\t\t\\hline\n\t\t\\hline\n\t\t& \\multicolumn{3}{c}{FCI ECP+CPP \\cite{Zuchowski_RbSr} } & \\multicolumn{3}{c}{KR-MRCI \\cite{Chen_RbSr}} & \\multicolumn{3}{c}{MCSCF-MRCI \\cite{RbSr_Ernst}}& Asymptotes  \\\\\n\t\tState & $R_e$ ($a_0$) & $D_e$ ($cm^{-1}$) & $\\omega_e$ ($cm^{-1}$) & $R_e$ ($a_0$) & $D_e$ ($cm^{-1}$) & $\\omega_e$ ($cm^{-1}$) & $R_e$ ($a_0$) & $D_e$ ($cm^{-1}$) & $\\omega_e$ ($cm^{-1}$) \\\\\n\t\t\\hline\n\t\t(2) $\\Omega=\\frac{1}{2}$ &7.30 &8283.9 &80.12&7.27&7883.09&85.73&7.43&8569.0&79.6&Rb$(5p\\,^2P_{1\/2})$+Sr($5s^2\\,^1S$) \\\\\n\t\t(3) $\\Omega=\\frac{1}{2}$ &8.39 &5136.41&59.04&8.39&4683.56&58.43&8.56&5252.3&58.7&Rb$(5p\\,^2P_{3\/2})$+Sr($5s^2\\,^1S$) \\\\\n\t\t(1) $\\Omega=\\frac{3}{2}$ &7.31 &8439.8 &79.50&7.29&7957.31&87.18&7.41&8727.6&80.4&\\\\\n\t\t(4) $\\Omega=\\frac{1}{2}$ &7.66 &4234.40&68.85&    &       &     &7.82&4202.3&64.1&Rb$(5s\\,^2S)$+Sr($5s5p\\,^3P_0$)\\\\\n\t\t(5) $\\Omega=\\frac{1}{2}$ &7.69 &3635.19&65.77&    &       &     &7.94&3436.2&52.4&Rb$(5s\\,^2S)$+Sr($5s5p\\,^3P_1$)\\\\\n\t\t(6) $\\Omega=\\frac{1}{2}$ &8.06 &2851.68&57.41&    &       &     &    &      &    &\\\\\n\t\t(7) $\\Omega=\\frac{1}{2}$ &8.03 &3112.45&57.63&    &       &     &    &      &    &Rb$(5s\\,^2S)$+Sr($5s5p\\,^3P_2$)\\\\\n\t\t(8) $\\Omega=\\frac{1}{2}$ &11.27&476.41 &20.57&    &       &     &    &      &    &\\\\\n\t\t(2) $\\Omega=\\frac{3}{2}$ &7.66 &4129.30&68.34&    &       &     &7.82&4107.4&64.5&Rb$(5s\\,^2S)$+Sr($5s5p\\,^3P_1$)\\\\\n\t\t(3) $\\Omega=\\frac{3}{2}$ &8.06 &2990.04&57.64&    &       &     &    &      &    &Rb$(5s\\,^2S)$+Sr($5s5p\\,^3P_2$)\\\\\n\t\t(4) $\\Omega=\\frac{3}{2}$ &11.30&501.73 &21.05&    &       &     &    &      &    &\\\\\n\t\t(1) $\\Omega=\\frac{5}{2}$ &8.06 &2990.04&56.98&    &       &     &    &      &    &\\\\\n\t\t\\hline\n\t\t\\hline\n\t\\end{tabular}\n\t}\n\t\\label{Spectro_param_comparaison_SO}\n\\end{table}\n\n\\section{Photoassociation of $^{87}$Rb$^{84}$Sr molecules}\n\\label{sec:PA}\n\n\\subsection{Methodology}\n\nThe photoassociation rate toward a vibrational level $v'$ of an excited electronic state $e$, at low laser intensity $I$, of a $^{87}$Rb$^{84}$Sr pair colliding in the the $X$ ground state with relative energy $E=k_BT$, is computed according to the perturbative approach reported in Ref. \\cite{Pillet_Photoassociation}: \n\\begin{equation}\nR_{PA}(X \\rightarrow e, v';T,I)=\\frac{2\\pi^{1\/2}h}{c} \\left(\\frac{3}{2}\\right)^{3\/2} \\lambda_{th} I A|\\braket{\\phi_{v'}^e|\\mu_{eX}^{el}|u_{\\ell}^X(E)}|^2 ,\n\\label{eq:pa-rate}\n\\end{equation}\nwhere $\\lambda_{th}=\\sqrt{\\frac{h^2}{3\\mu k_BT}}$ is the de Broglie thermal wavelength of the atom pair with reduced mass $\\mu$, $I$ is the intensity of photoassociation laser, $A$ is an angular factor, $\\phi_{v'}^e(R)$ is the vibrational wave function of the photoassociated level $v'$, $u_{\\ell}^X(k_BT,R)$ is the continuum wave function of the colliding pair assuming a rotational quantum number (partial wave) $\\ell$, and $\\mu_{eg}^{el}(R)$ is the $R$-dependent transition dipole moment between the X and $e$ electronic states. For simplicity, we have considered an $s$-wave collisional regime ($\\ell=0$), and we have taken $A=1$, thus ignoring the dependence on the light polarization and on the internal states of the colliding atoms. The values of $I=10$~W\/cm$^2$ and $T=5.5$~$\\mu$K are typical of the ongoing experiment \\cite{ciamei2018}, and are within the limits of validity of Eq. (\\ref{eq:pa-rate}), namely in the linear regime for $I$, and  a non-degenerate quantum gas.\n\nThe radial wave functions above are calculated with the Mapped Fourier Grid Hamiltonian (MFGH) method \\cite{MFGH_1}\\cite{MFGH_2}\\cite{MFGH_3}\\cite{MFGH_4}, using a grid extending from $R_{min}=5 a_0$ to $R_{max}=2000 a_0$, containing up to 1551 points. We have checked the convergence of the calculations with the size of the grid. The continuum and vibrational levels of the $X^2\\Sigma^+$ ground state are described in a single-channel representation, the wave functions (normalized to unity) discretized levels of the continuum being renormalized in energy at the end of the calculation by dividing the wave function by the square root of the level density \\cite{luc-koenig2004}. The vibrational wave functions of the $^{87}$Rb$^{84}$Sr excited electronic states coupled by SO interaction are obtained from a multichannel representation according to Equations (\\ref{eq:Rb-omega12}-\\ref{eq:Sr-omega32}), and are therefore linear combinations of the related Hund's case a electronic states weighted by the radial wave functions $\\phi_{v'^2\\Sigma}^e$ and $\\phi_{v'^2\\Pi}^e$. The squared matrix elements of the transition dipole moment $\\mu_{eg}^{el}(R)$ involve the contributions of the $^2\\Sigma^+$ and $^2\\Pi$ components of the coupled excited electronic states as\n\\begin{equation}\n |\\braket{\\phi_{v'}^e|\\mu_{eg}|u_{0}^X(E)}|^2=|\\braket{\\phi_{v'^2\\Sigma}^e|\\mu_{^2\\Sigma X}|u_{0}^X(E)}|^2+|\\braket{\\phi_{v'^2\\Pi}^e|\\mu_{^2\\Pi X}|u_{0}^X(E)}|^2,\n\\label{eq:squared-tdm}\n \\end{equation}\nwhere the TDM functions $\\mu_{^2\\Sigma X}(R)$ and $\\mu_{^2\\Pi X}(R)$ are those displayed in Fig.\\ref{fig:pot}. This equation is valid in the case of unpolarized light in the laboratory frame.\n\nTwo cases are of relevance for our study. First, the Sr intercombination (spin-forbidden) transition is used in ongoing experiments devoted to the formation of the quantum degenerate mixture of strontium and rubidium atoms \\cite{Mixture_BEC_Sr_Rb}. Therefore we have studied the photoassociation close to the $^{87}$Rb$(5s\\,^2S)$+$^{84}$Sr($5s5p\\,^3P_{0,1,2}$) asymptotes. However PA will proceed in a very different way if it is implemented for laser frequencies close to the Rb dipole-allowed transition, namely, exploring bound levels close to the $^{87}$Rb$(5p\\,^2P_{1\/2,3\/2})$+$^{84}$Sr($5s^2\\,^1S$) asymptotes, as investigated theoretically in Ref. \\cite{Chen_RbSr}. \n\nBefore the presentation of our systematic results for these situations, it is worthwhile to emphasize on the importance of accounting for the $R$ dependence of the TDM functions in the PA rates, which is not considered in Ref. \\cite{Chen_RbSr}. For this purpose it is convenient to decompose the squared TDMs of Eq. (\\ref{eq:squared-tdm}) as the product of the squared overlap between radial wave functions (or Franck-Condon factors, FCF), and the values of electronic transition dipole moment at the outer turning point $R_C$ of $\\phi_{v'}^e$ for the corresponding excited electronic state :\n\\begin{equation}\n|\\braket{\\phi_{v'}^e|\\mu_{eg}|u_{0}^X(E)}|^2=\\mu_{eg}^2(R_C)|\\braket{\\phi_{v'}^e|u_{0}^X(E)}|^2.\n\\label{eq:FC}\n\\end{equation}\nThe general trend of the squared TDMs is illustrated in Fig.\\ref{fig:squaredTDME} with the (4) $\\frac{1}{2}$ and the (2) $\\frac{1}{2}$ states, respectively correlated to $^{87}$Rb$(5s\\,^2S)$+$^{84}$Sr($5s5p\\,^3P_{0}$ and $^{87}$Rb$(5p\\,^2P_{1\/2})$+$^{84}$Sr($5s^2\\,^1S$). The squared TDMs have similar magnitude over most of the energy range of the potential wells, like the corresponding transition dipole moments in the molecular range ($R<12$~a.u. typically, see Fig. \\ref{fig:pot}). Their behaviors are very similar to the FCF's ones. Close to the $^{87}$Rb$(5s\\,^2S)$+$^{84}$Sr($5s5p\\,^3P_{0}$) asymptote, the squared TDM vanishes as $\\mu_{eg}^2(R_C)$ while the FCFs increase. Therefore the FCFs are not anymore representative of the behavior of the squared TDMs. This is in striking contrast with PA close to the $^{87}$Rb$(5p\\,^2P_{1\/2})$+$^{84}$Sr($5s^2\\,^1S$) limit, for which the squared TDMs and the squared FCFs have similar behaviors. This decomposition of squared TDMs will also be useful for analyzing the PA spectra in the next section. \n\n\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=12cm]{Fig4_squaredTDME.pdf}\n\t\t\\caption{Squared dipole matrix element for (a) the (4) $\\frac{1}{2}$ levels correlated to $^{87}$Rb$(5s\\,^2S)$+$^{84}$Sr($5s5p\\,^3P_0$), and (b) the (2) $\\frac{1}{2}$ levels correlated to $^{87}$Rb$(5p\\,^2P_{1\/2})$+$^{84}$Sr($5s^2\\,^1S$), as a function of the PA laser wavenumber. Black lines: squared TDM from Eq.(\\ref{eq:squared-tdm}). Red lines: squared overlap (FCF) between the radial wave functions involved in the PA rate. Blue lines: value of squared TDM at the outer turning point $R_C$ of the radial wave function of the photoassociated level.}\n\t\t\\label{fig:squaredTDME}\n\\end{figure}\n\nIn this Section we consider the formation of ultracold molecules by RE of the photoassociated molecules. For simplicity we assume, like in Ref.\\cite{Pillet_Photoassociation}, that the RE probability is given by $\\sum_{v''}|\\braket{\\phi_{v'}^e(R)|\\phi_{v''}^g(R)}|^2$, such that the resulting ultracold molecule formation (UMF) rate per atom is written as\n\\begin{equation}\nR_{mol}^{v'}=R_{PA}^{v'}\\sum_{v''}{|\\braket{\\phi_{v'}^e|\\phi_{v''}^g}|^2}.\n\\label{eq:CMrate}\n\\end{equation}\nNote that Eq. (\\ref{eq:CMrate}) should be employed with care in the case of PA spectra close to the $^{87}$Rb$(5s\\,^2S)$+$^{84}$Sr($5s5p\\,^3P_{0,1,2}$) asymptotes as the TDM vanishes at large distances. It is actually more interesting to focus on the vibrational distribution in the ground state levels after RE, which is more sensitive to the $R$-variation of the electronic TDMs than the total UMF rate. We express the probability $P(v''\\leftarrow v')$ of a ground-state level $v''$ to be occupied after RE from the photoassociated level $v'$ as \n\\begin{equation}\nP(v''\\leftarrow v')=\\frac{|\\braket{\\phi_{v'}^e|\\mu_{eg}|\\phi_{v''}^g}|^2}{\\sum_{v''}{|\\braket{\\phi_{v'}^e|\\mu_{eg}|\\phi_{v''}^g}|^2}}.\n\\label{eq:vibdist}\n\\end{equation}\n\n\\subsection{The $^{87}$Rb$(5s\\,^2S)$+$^{84}$Sr($5s5p\\,^3P_{0,1,2}$) dissociation limit}\n\nPA rates for the five $\\Omega=\\frac{1}{2}$ states are shown on Fig. \\ref{fig:forbiddenPA}. For each potential, the analysis of these PA spectra could be divided in three parts : deeply levels, intermediate levels and levels close to the asymptotes. For the deeply levels, the PA rate depends on their main composition of Hund case (a) states. The most important values are obtained for (4) $\\frac{1}{2}$ and (5) $\\frac{1}{2}$ that are composed by respectively (2) $^2\\Pi$ and (3) $^2\\Sigma$ states. Some of these deeply levels of (4) $\\frac{1}{2}$ and (5) $\\frac{1}{2}$ have a PA rates with only one order of magnitude lesser than the largest ones. These high values can be explained by the relative position of the (4) $\\frac{1}{2}$ and (5) $\\frac{1}{2}$ potential wells  with respect to the ground state one that favors the contribution  of the distances at the inner turning point of the ground state in the PA rate.\n\nThe intermediate levels have the largest PA rates. This comes from the competition between the FC factors and the value of TDMs at the outer turning point as illustrated on Fig.\\ref{fig:squaredTDME}. Due to their most important composition in (2) $^2\\Pi$ and (3) $^2\\Sigma$ states, the largest rates are obtained with the states (4) $\\frac{1}{2}$ and (5) $\\frac{1}{2}$ : $1.37\\times 10^{-15}$ ~cm$^{-3}$s$^{-1}$ at 12295.1~cm$^{-1}$ ($v'=44$), and $5.37\\times 10^{-16}$  ~cm$^{-3}$s$^{-1}$ at 12601.3~cm$^{-1}$ ($v'=36$), respectively. For the (6),(7) and (8) $\\Omega=\\frac{1}{2}$ states, the PA rates significantly increase, due to the admixture of doublet states in addition to the quartet component at large interatomic separation, but remain smaller. \n\nFinally, close to the asymptotes, the PA rates significantly drop down for all states, for example at a detuning around 0.15~cm$^{-1}$, the PA rates take the values : $7.2\\times 10^{-20}$~cm$^{-3}$s$^{-1}$ and $2.7\\times 10^{-22}$ ~cm$^{-3}$s$^{-1}$ for respectively the (4) $\\frac{1}{2}$ and (5) $\\frac{1}{2}$ states. Therefore PA already appears as a challenge close to the asymptote. New results for PA in RbSr are available \\cite{ciamei2018}. In particular, the authors report difficulties to observe PA close to the $^{87}$Rb$(5s\\,^2S)$+$^{84,87}$Sr($5s5p\\,^3P_{1}$) asymptote, and measure small photoassociation rates. Such low rates are consistent with the low rates of our calculations.  \n\n\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=10cm]{Fig5_forbiddenPA.pdf}\n\t\t\\caption{Photoassociation rates for $^{87}$Rb$^{84}$Sr levels as a function of the PA laser wavenumber, for states correlated to the $^{87}$Rb$(5s\\,^2S)$+$^{84}$Sr($5s5p\\,^3P_{0,1,2}$) dissociation limit. (a) (7) $\\frac{1}{2}$ in black, and (8) $\\frac{1}{2}$ in red. (b) (5) $\\frac{1}{2}$ in black, and (6) $\\frac{1}{2}$ in red. (c) (4) $\\frac{1}{2}$ in black. (d) (2) $\\frac{3}{2}$ in black, (3) $\\frac{3}{2}$ in red, (4) $\\frac{3}{2}$ in blue.}\n\t\t\\label{fig:forbiddenPA}\n\t\\end{figure}\n\nThe results for the $\\frac{3}{2}$ are rather similar (Fig. \\ref{fig:forbiddenPA}d). The maximal PA rate ($4.7\\times 10^{-16}$~cm$^{-3}$s$^{-1}$ for a detuning of 13018.32~cm$^{-1}$ ($v'=59$)) is found for the (2) $\\frac{3}{2}$, mainly composed of doublet states. Close to the asymptote, they also become rather low, remaining for the (4) $\\frac{3}{2}$ as large as the $\\frac{1}{2}$ states.\n\nThe computed energy variations of the UMF rates are found very similar to those of the PA rates, and are displayed in the Appendix for the sake of conciseness. Using Eq. (\\ref{eq:CMrate}), we find that  the sum of squared overlap in Eq. (\\ref{eq:CMrate}) decreases for (4) $\\frac{1}{2}$ and (5) $\\frac{1}{2}$ while increases for (6) $\\frac{1}{2}$ and (7) $\\frac{1}{2}$ due to the admixtures between doublet and quartet states as already explained for the PA rates. However, over the entire range, it is always in the same order of magnitude and does not induce a change of the spectra of UMF rates with respect to the PA one. The vibrational distributions in the ground-state levels (see Eq. (\\ref{eq:vibdist})) are displayed in Fig. \\ref{fig:forbiddenvibdist} for few typical photoassociated levels $v'$ : deeply levels ($v'=5$ of (4) $\\frac{1}{2}$, $v'=10$ of (5) $\\frac{1}{2}$ and $v'=11$ of (2)$\\frac{3}{2}$), intermediate levels ($v'=43$ of (4) $\\frac{1}{2}$ and $v'=59$ of (2) $\\frac{3}{2}$) that have the largest PA rates, and the last bound levels ($v'=129$ of (4) $\\frac{1}{2}$ and $v'=124$ of (2) $\\frac{3}{2}$). The deeply-bound levels can yield a main fraction of the population in the ground state $v''=0$ level, as it can be expected from the favorable relative position of the minimum of the corresponding potential well in the excited and in the ground state. The levels with the largest PA rates (in the (4)$\\frac{1}{2}$ and the (2)$\\frac{3}{2}$ states) induce a population spread over numerous ground state vibrational levels, with no specific emergence of a particular level (a maximum population being found however for respectively level $v''$=16 ($E_{bind}$=-541~cm$^{-1}$) and $v''$=23 ($E_{bind}$=-372~cm$^{-1}$)). Finally, for the last bound levels, different levels with binding energies below 200~cm$^{-1}$ are populated. The population goes to zero for the last bound levels of ground state and it is a consequence of the vanishing TDM at large distance.\n\n\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=12cm]{Fig6_forbideenvibdist}\n\t\t\\caption{Distribution of vibrational populations (displayed in percentage) of the ultracold $^{87}$Rb$^{84}$Sr molecules created after spontaneous emission from several photoassociated levels belonging to states converging toward the $^3P_0$ ((4) $\\frac{1}{2}$ ) and $^3P_1$ ((5)$\\frac{1}{2}$ and (2) $\\frac{3}{2}$ ) limits . (a) the $v'=43$ (black line), $v'=10$ (red line)  and $v'=129$ (pink line)  levels of (4)$\\frac{1}{2}$ and the $v'=5$ (blue line) level of (5)$\\frac{1}{2}$. (b) the $v'=59$ (black line), $v'=11$ (red line) and $v'=124$ (blue line) levels of (2) $\\frac{3}{2}$.}\n\t\t\\label{fig:forbiddenvibdist}\n\\end{figure}\n\t\n\\subsection{The Rb$(5p\\,^2P_{1\/2,3\/2})$+Sr($5s^2\\,^1S$) dissociation limit}\n\\label{ssec:allowed}\n\nThe computed PA rates for both $\\Omega=\\frac{1}{2}$ and $\\Omega=\\frac{3}{2}$ states are displayed in Fig. \\ref{fig:allowedPA}. The analysis of these PA spectra could be also divided for deeply-bound levels on one hand, and for levels close to the asymptotes on the other hands. For deeply levels, the PA rates are more important for (2) $\\Omega=\\frac{1}{2}$ than for (3) $\\Omega=\\frac{1}{2}$. This striking difference is at first sight counter-intuitive. The levels from (2) $\\frac{1}{2}$ state composed mainly of $^2\\Pi$ (low value at short-range) state have larger PA rates than levels from (3) $\\frac{1}{2}$ state composed mainly of $^2\\Sigma^+$ (high value at short-range). A deeper study show that the main parameter is the overlaps and so the FCFs that are specially weak for the deeply-bound levels of (3) $\\frac{1}{2}$ state. Close to the asymptotes, PA rates quickly rise. The results are very similar to those obtained for the alkali-metal dimers, as the dominant role is given to the dipole-allowed Rb transition. In a classical view, PA takes place mostly at large interatomic distances, where the electronic TDM is large, and where the overlap between the relevant radial wave functions is favored. This could be seen on Fig.\\ref{fig:squaredTDME} b). The rate magnitude is found similar to the one for Rb photoassociation \\cite{Rb2_Fioretti}. The maximal values ($10^{-10}-10^{-11}$~cm$^{-3}$s$^{-1}$) close to the dissociation limits are tedious to compare to experiment, as the cloud of cold atoms is strongly perturbed if the PA laser is tuned too close to the atomic resonance. \n\n\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=12cm]{Fig7_allowedPA.pdf}\n\t\t\\caption{Photoassociation rates for the bound levels of (a) the (2) $\\frac{1}{2}$, (3)$\\frac{1}{2}$ states, and (b) of the (1)$\\frac{3}{2}$ state, correlated to the Rb$(5p\\,^2P_{1\/2,3\/2})$+Sr($5s^2\\,^1S$) dissociation limit and identified by the laser wavenumber.}\n\t\t\\label{fig:allowedPA}\n\\end{figure}\n\nAs previously, the UCM rate variations with the PA laser frequency are very similar to the ones of the PA rates, and are displayed in the Appendix. The computed vibrational distributions generated by the photoassociated levels are actually quite remarkable, as illustrated in Fig.\\ref{fig:allowedvibdist} for the uppermost PA levels. They reflect an almost diagonal Franck-Condon matrix, namely each photoassociated level almost populates a single ground state level: for instance, the last-but-one ground-state level (noted $v''=-1$ for convenience) is populated at 97.75\\% by the last-but one (also noted $v'=-1$ for convenience) level of (2) $\\frac{1}{2}$, at 99.98 \\% with the $v'=-1$ level of (3) $\\frac{1}{2}$, and at 99.94\\% of (1) $\\frac{3}{2}$. The purity of the relaxation process is worse when the detuning is increased. But in any case, only weakly-bound ground state molecules could be efficiently created.  Note that this situation has been also investigated in Ref. \\cite{Chen_RbSr} based only on the Franck-Condon factors, yielding results similar to ours.\n\n\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=8cm]{Fig8_allowedvibdist.pdf}\n\t\t\\caption{Distribution of vibrational populations (displayed in percentage) of the five uppermost vibrational levels (labeled as $v=-1, -2, -3, -4, -5$ for convenience) of the $^{87}$Rb$^{84}Sr$ ground state, generated after spontaneous emission from the photoassociated levels of the (2) $\\frac{1}{2}$, (3) $\\frac{1}{2}$, and (1) $\\frac{3}{2}$ states (panels (a), (b) and (c), respectively) identified by the PA laser wavenumber. }\n\t\t\\label{fig:allowedvibdist}\n\\end{figure}\n\t\n\\section{Formation of ultracold weakly-bound molecules by a STIRAP transfer in a tight optical trap}\n\\label{sec:STIRAPtrap}\n\nIn the previous section, we have shown that the formation of RbSr ground-state molecules could be achieved by PA. However, the spontaneous emission induces two well-known drawbacks: the loss of atoms from the trap, and a broad distribution of occupied vibrational levels. Even if this last point is minimized when PA is implemented with a laser frequency close to the dipole-allowed Rb transition (see Section \\ref{ssec:allowed}), employing a coherent method avoiding spontaneous emission would decrease the loss of atoms. The most efficient method of coherent population transfer is the STimulated Rapid Adiabatic Passage (STIRAP) method \\cite{STIRAP_1} relying on a proper choice of three energy levels, refereed to as a $\\Lambda$ system. It allows the coherent population transfer between an initial level $\\ket{i}$ and a final level $\\ket{f}$ through a dark state involving an excited level $\\ket{e}$. When the two-photon detuning $\\delta$ is zero, one of the instantaneous eigenstates of the Hamiltonian including the light is a coherent superposition of only $\\ket{i}$ and $\\ket{f}$ and is the so-called dark state. The transfer could be made without populating the intermediate level $\\ket{e}$, thereby avoiding loss by spontaneous emission. The complete population transfer from $\\ket{i}$ to $\\ket{f}$ is achieved by the application of a pair of pump and dump pulses in a counter-intuitive order. As the system must remain in the dark state during all the process, the dynamics has to be adiabatic, thus imposing constraints on the corresponding Rabi frequencies $\\Omega_{\\textrm{pump}}$ and $\\Omega_{\\textrm{dump}}$: they must have the same order of magnitude, and they must be sufficiently large to satisfy the condition $\\Omega_{\\textrm{pump,dump}} T>>\\pi$ where $T$ is the duration of the pulses.\n\nThe use of STIRAP in a PA experiment (\\textit{i.e.} without relying on Feshbach resonances) was previously investigated \\cite{STIRAP_photoassociation_1}, revealing some difficulties due to the fact that the initial level belongs to a dissociation continuum, preventing the perfect creation of the dark state. One can overcome this drawback by placing the initial cold atoms in a tight optical trap (say, at a typical wavelength of 1064~nm), such that the motional states of the atom pair become quantized \\cite{jaksch2002}. At ultracold temperature, the atoms occupy the lowest motional level of the trap. Therefore, the radial wave function of the atom pair should be  localized at shorter distance, and the Franck-Condon factors with the bound levels of excited electronic states should increase. \n\nThe Hamiltonian describing two non-identical atoms of mass $m_1$ and $m_2$ at positions $\\vec{r}_1$ and $\\vec{r}_2$ in an optical anharmonic trap with harmonic frequencies $\\omega_1$ and $\\omega_2$ felt by each atomic species, is \\cite{Tight_trap_1}\n\\begin{equation}\nH_{\\textrm{trap}}=-\\frac{\\hbar^2}{2M}\\Delta_{\\textrm{com}}+\\frac{1}{2} M \\omega^2_{\\textrm{com}} R_{\\textrm{com}}^2-\\frac{\\hbar^2}{2\\mu}\\Delta_{R}+\\frac{1}{2}\\mu\\omega^2_{R}R^2+V(R)+\\mu\\Delta\\omega\\vec{R}_{\\textrm{com}}.\\vec{R}+V_{\\textrm{anharm}}\n\\label{eq:Htrap}\n\\end{equation}\nwith the total mass $M=m_1+m_2$, the reduced mass $\\mu=\\frac{m_1m_2}{M}$, the position of the center-of-mass $\\vec{R}_{\\textrm{com}}=\\frac{m_1\\vec{r}_1+m_2\\vec{r}_2}{M}$, the relative position vector $\\vec{R}=\\vec{r}_1-\\vec{r}_2$, and $\\Delta\\omega=\\sqrt{\\omega_1^2-\\omega_2^2}$. The first two terms represent the center-of-mass motion in the trap, with frequency $\\omega_{\\textrm{com}}=\\sqrt{\\frac{m_1\\omega_1^2+m_2\\omega_2^2}{m_1+m_2}}$. The next three terms describe the relative motion of the atom pair interacting through the potential $V(R)$, in the presence of a trapping potential of frequency $\\omega_{R}=\\sqrt{\\frac{m_2\\omega_1^2+m_1\\omega_2^2}{m_1+m_2}}$. These two motions are in principle coupled by the anharmonic terms $V_{\\textrm{anharm}}$ of the trapping potential, and by a dynamical term proportional to $\\vec{R_{\\textrm{com}}}.\\vec{R}$. The former can be safely neglected if we assume that the atoms are trapped in the lowest motional level. The latter depends on the differences of masses and polarizabilities that are almost the same in our case. In our calculation, we have therefore neglected the coupling between the two motions, and worked with relative coordinate. We have taken the experimental trapping frequencies $2\\pi\\times 65$~kHz for $^{84}$Sr and $2\\pi \\times 110$~kHz for $^{87}$Rb \\cite{Schreck_pc}. The characteristic length of the relative motion in the trap is $a_{\\omega}=\\sqrt{\\hbar\/\\mu\\omega_{rel}}=969$ a.u. which is much larger than the scattering length. Therefore, the tight trap does not induce any significant modification of bound levels of the ground and excited molecular states. The eigenstates of $H_{\\textrm{trap}}$ for the ground and excited states of Fig.\\ref{fig:PEC_SO} as well as the transition matrix elements are computed with the same procedure than in the previous sections. The main difference is that the radial wave functions of the trap states are now normalized to unity, as they are no longer continuum states. \n\nThe initial level $\\ket{i}$ is taken as the first trap state. For the final level $\\ket{f}$, we have first chosen the vibronic ground-state level ($v''=0$). In addition, we have examined the possibility to improve the STIRAP process toward another final level. The crucial element of the model is the choice of the best possible intermediate level $\\ket{e}$ belonging to an excited electronic state. Two requirements  must be considered, independently of the experimental laser intensities used in the experiment: the squared matrix elements of the transition dipole moment (squared TDMEs in short) for the pump and dump transitions must be of the same order of magnitude, and be sufficiently high (typically more than $10^{-6}$ a.u., see for instance Ref.\\cite{borsalino2014}). \n\nFor the transfer toward $\\ket{f} \\equiv \\ket{v''=0}$ via the states $\\ket{e}$ correlated to $^{87}$Rb$(5s\\,^2S)$+$^{84}$Sr($5s5p\\,^3P_{0,1,2}$) (Fig. \\ref{fig:STIRAPtrap}a), the squared TDMEs curves for the pump and dump transitions cross twice each other, around 12000~cm$^{-1}$ and 14500~cm$^{-1}$ with very weak magnitudes ($10^{-9}-10^{-10}$~a.u.). A similar conclusion holds for the transfer via the levels $\\ket{e}$ belonging to states correlated to Rb$(5p\\,^2P_{1\/2,3\/2})$+Sr($5s^2\\,^1S$), where the two curves cross once around 8500~cm$^{-1}$ (Fig. \\ref{fig:STIRAPtrap}c) with a low magnitude ($10^{-9}$~a.u.). These statements actually reflect the behavior of the corresponding PA rates of Figs. \\ref{fig:forbiddenPA} and \\ref{fig:allowedPA}. While providing a discrete level for the initial state for STIRAP, the choice of a confined trap level for the pump step does not significantly improve the magnitude of the squared TDMEs compared to the conventional PA starting from a real continuum state.\n\nPanels (b) and (d) in Fig. \\ref{fig:STIRAPtrap} illustrate another possible way to progress on the way to the creation of ultracold RbSr molecules. We have calculated the TDMEs involved in the transfer from the initial trap state toward the $v=-3$ level of the RbSr ground state, leading to a contrasted result: while the STIRAP transfer does not seem to be possible via $\\ket{e}$ levels belonging to states correlated to $^{87}$Rb$(5s\\,^2S)$+$^{84}$Sr($5s5p\\,^3P_{0,1,2}$) (Fig. \\ref{fig:STIRAPtrap}b)), it appears doable via $\\ket{e}$ levels close to the Rb$(5p\\,^2P_{1\/2,3\/2})$+Sr($5s^2\\,^1S$) dissociation limit (see the extreme right part of the Fig. \\ref{fig:STIRAPtrap} d), for which the squared TDMEs for the pump and dump transitions can reach similar values up to 10$^{-5}$~a.u. In fact, the last five bound levels could be populated by a STIRAP with an intermediate level close to the asymptotes $^2P_{1\/2}$ or $^2P_{3\/2}$ (see table \\ref{STIRAP_trap_allo_trans}). As expected, the STIRAP is more tedious for deeper final levels. In conclusion, the STIRAP method in a tight trap can create only weakly-bound $^{87}$Rb$^{84}$Sr ground state molecules, just like PA or MFR.\n\n\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=8cm]{Figure_9.pdf}\n\t\t\\caption{Squared matrix elements of the transition dipole moment (squared TDMEs in short) relevant for the formation of $^{87}$Rb$^{84}$Sr molecules with STIRAP, starting from an atom pair confined in the lowest motional level of a tight optical trap (see text for details), as a function of the excitation energy of the chosen intermediate level $\\ket{e}$. Pump transitions: black lines; dump transitions: red lines. Panels (a) and (c) (rep. (b) and (d)) correspond to the final level $\\ket{f} \\equiv v''=0$ (resp. $\\ket{f} \\equiv v''=-3$) of the electronic ground state. The levels $\\ket{e}$ belong to all electronic states $\\Omega=\\frac{1}{2}$ correlated to $^{87}$Rb$(5s\\,^2S)$+$^{84}$Sr($5s5p\\,^3P_{0,1,2}$) (panels (a) and (b)), and  to Rb$(5p\\,^2P_{1\/2})$+Sr($5s^2\\,^1S$)) (panels (c) and (d)).}\n\t\t\\label{fig:STIRAPtrap}\n\\end{figure}\n\n \n\\begin{table}\n\\centering\n\\caption{Characteristics of the optimal transitions for the STIRAP scheme in a tight trap via an intermediate level close to Rb$(5p\\,^2P_{1\/2})$+Sr($5s^2\\,^1S$)). The initial level is the first trap state, with an energy of $5.10^{-6}$~cm$^{-1}$ (or about 150~kHz) above Rb$(5s\\,^2S_{1\/2})$+Sr($5s^2\\,^1S$). The final level of the ground state is labeled with negative index $\\tilde{v}_f$ starting from the Rb$(5s\\,^2S_{1\/2})$+Sr($5s^2\\,^1S$) asymptote, with a binding energy $E_f$. The vibrational index $v_e$ and binding energy $E_{e}$ of several optimal intermediate levels are displayed. The energies $E_{\\textrm{pump}}$ and $E_{\\textrm{dump}}$, and the related squared transition dipole moments $|d_{ie}|^2$ and $|d_{ef}|^2$ of the pump and dump transitions are also reported. Numbers in parenthesis hold for powers of 10.}\n\\begin{tabular}{cccccc}\n\\hline\n\\hline\n$\\tilde{v}_f$      & -1             & -2             & -3             & -4             & -5 \\\\\n$E_{f}$ ($cm^{-1}$)& 1.3(-3)        & 2.58(-2)       & 1.147(-1)      & 3.112 (-1)     & 6.573 (-1) \\\\\\hline\n$v_e$              & 161            & 201            & 199            & 198            & 197  \\\\\n                   &3($\\frac{1}{2}$)&2($\\frac{1}{2}$)&2($\\frac{1}{2}$)&2($\\frac{1}{2}$)&2($\\frac{1}{2}$) \\\\\n$E_{e}$ ($cm^{-1}$)& 2.9 (-3)       & 2.0(-3)        & 7.16(-2)       & 1.755 (-1)     & 3.494 (-1) \\\\\\hline\n$E_{\\textrm{pump}}$ (cm$^{-1}$)&12895.4000&12657.7979& 12657.7279     & 12657.6238     & 12657.4490 \\\\\n$|d_{ie}|^2$ (a.u.)             & 6.2 (-2)  & 4.5 (-4)  & 9.3 (-5)       & 2.8 (-6)       & 2.8 (-5) \\\\\n$E_{\\textrm{dump}}$ (cm$^{-1}$)&12895.4012&12657.8238& 12657.8426     & 12657.9350     & 12658.1063 \\\\\n$|d_{ef}|^2$ (a.u.)             & 1.2 (0)  & 7.4 (-4) & 2.3 (-4)       & 7.1 (-6)       & 2.5 (-5) \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\label{STIRAP_trap_allo_trans}\n\\end{table}\n\n\\section{Population transfer from weakly-bound RbSr ground-state molecules to the rovibrationnal ground state}\n\\label{sec:STIRAPgroundstate}\n\nIn the two last sections, we have shown that the formation of weakly-bound $^{87}$Rb$^{84}$Sr ground-state molecules is achievable by PA and by STIRAP in a tight trap. A second STIRAP step could then be implemented to transfer these molecules into the lowest level of the ground state. Such a double STIRAP sequence has already been applied for ultracold Cs$_2$ molecules \\cite{danzl2010}. We have looked for an optimal STIRAP transfer starting from the five uppermost ground state levels above, now labeled as $v_i=-1, -2, -3, -4, -5$.\n\nWe have identified three efficient STIRAP paths in three different spectral zones, based upon the same criterion than above of the equality of the squared TDMs for the pump and dump transitions:\n\\begin{itemize}\n\t\n\t\\item the first scheme relies on intermediate levels of the 2($\\frac{1}{2}$) state correlated to Rb$(5p\\,^2P_{1\/2})$+Sr($5s^2\\,^1S$)), corresponding to $E_{\\textrm{pump}}$ in the 4570-4890~cm$^{-1}$ range, and $E_{\\textrm{dump}}$ in the 5625-5945~cm$^{-1}$ range (Table \\ref{STIRAP_weakly_schema_1}, and Fig\\ref{fig:STIRAPweak}a). Despite a strong magnitude of the corresponding squared TDMs, that may not be the most practical frequencies to implement experimentally. As already noticed before, the possibility to use the lowest bound levels of the intermediate state comes from the relative position of PECs, and from the position of the inner turning point of the initial weakly-bound vibrational wave function, located close to the equilibrium distance of excited states. \n\t\n\t\\item The second scheme relies on the same intermediate state, with levels that can be reached with $E_{\\textrm{pump}}$ in the 7175-7590~cm$^{-1}$ range, inducing $E_{\\textrm{dump}}$ located in the 8230-8640~cm$^{-1}$ range (Table \\ref{STIRAP_weakly_schema_2}, and Fig\\ref{fig:STIRAPweak}a). This scheme is expected to be slightly less efficient than the previous one, but in a more accessible frequency domain for the STIRAP lasers. This solution involves the vibrational levels close to the avoided crossing between the $^2\\Sigma^+$ and $^2\\Pi$ states (see fig. \\ref{fig:STIRAPtrap} ).\n\t\n\t\\item The third scheme involves levels of the $4 (\\frac{1}{2})$ state correlated to Rb$(5s\\,^2S)$+Sr($5s5p\\,^3P_2$) (Table \\ref{STIRAP_weakly_schema_3}, and Fig\\ref{fig:STIRAPweak}b), with $E_{\\textrm{pump}}$ in the 11200-11360~cm$^{-1}$ range, and $E_{\\textrm{dump}}$ in the 12255-12415~cm$^{-1}$ range. This corresponds to levels with an energy close to the avoided crossing visible in Fig. \\ref{fig:STIRAPtrap}. The efficiency of this STIRAP path seems to be the best of the three presented in this work. The advantage of this path is that a laser Ti:sapphire could be used. \n\\end{itemize} \n\nChen \\textit{et al.} \\cite{Chen_RbSr} have proposed another STIRAP path using the $v'=21$ level of the (2) $\\frac{1}{2}$ state as the intermediate level, relying on a hypothesis different than ours:  the selected intermediate level should be the one with the largest value for the product of the squared TDMs for the pump and dump transitions (actually reduced to FCF in their paper). The drawback of such a methodology is that the squared TDMs for the pump and the dump transitions could be vastly different, thus implying very different laser intensities. It is indeed the case here, as there are 4 orders of magnitude of difference for squared TDM. The advantage of STIRAP path presented in our work is that the intensities for the two transitions would be similar.  \n\n\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=12cm]{Figure_10.pdf}\n\t\t\\caption{Squared matrix elements of the transition dipole moment (squared TDMEs in short) relevant for transferring population from v''=-3 to v''=0  with STIRAP as a function of the excitation energy of the chosen intermediate level $\\ket{e}$. Pump transitions: black lines; dump transitions: red lines. Panels (a) (resp. (b)) correspond to STIRAP with intermediate levels belonging to all electronic states correlated to Rb$(5p\\,^2P_{1\/2})$+Sr($5s^2\\,^1S$) (resp. $^{87}$Rb$(5s\\,^2S)$+$^{84}$Sr($5s5p\\,^3P_{0,1,2}$}\n\t\t\\label{fig:STIRAPweak}\n\t\\end{figure}\n\t\n\\begin{table}\n\\centering\n\\caption{Characteristics of the optimal transition for the first STIRAP scheme with an intermediate level belonging to an PEC correlated to the asymptotes $^2P_{1\/2}$ and $^2P_{3\/2}$.The final level is the rovibrational ground state ($v''=0$). The vibrational number and binding energy $E_{e}$ are given for the intermediate level. The energies $E_{\\textrm{pump}}$ and $E_{\\textrm{dump}}$ and the related squared transition dipole moments $|d_{ie}|^2$ and $|d_{ef}|^2$ of the pump and stoke transition are also reported. Number in parenthesis hold for powers of 10.}\n\\begin{tabular}{cccccc}\n\\hline\n\\hline\n$v_i$ & -1 & -2 & -3 & -4 & -5 \\\\\n$E_{i}$ (cm$^{-1}$) & 1.3(-3) & 2.58(-2) & 1.147(-1) & 3.112 (-1) & 6.573 (-1) \\\\\n\\hline\n$v_f$ & 0 & 0 & 0 & 0 & 0 \\\\\n$E_{f}$ (cm$^{-1}$) & 1054.3406 & 1054.3406 & 1054.3406 & 1054.3406 & 1054.3406 \\\\\n\\hline\n$v_e$ ((2) $\\Omega=\\frac{1}{2}$) & 4 & 4 & 5 & 6 & 6 \\\\\n$E_{e}$ (cm$^{-1}$) & 7926.4524 & 7926.4524 & 7847.7908 & 7769.4210 & 7769.4210 \\\\\n\\hline\n$E_{\\textrm{pump}}$ (cm$^{-1}$) & 4731.3473 & 4731.3473 & 4810.3193 & 4889.0356 & 4889.0356 \\\\\n$|d_{ie}|^2$ (a.u.) & 2.1 (-7) & 1.7 (-6) & 4.5 (-6) & 1.2 (-5) & 2.0 (-5) \\\\\n$E_{\\textrm{dump}}$ (cm$^{-1}$) & 5785.6882 & 5785.6882 & 5864.3497 & 5942.7196 & 5942.7196 \\\\\n$|d_{ef}|^2$ (a.u.) & 6.1 (-6) & 6.1 (-6) & 2.4 (-5) & 7.6 (-5) & 7.6 (-5) \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\label{STIRAP_weakly_schema_1}\n\\end{table}\n\n\n\\begin{table}\n\\centering\n\\caption{Same as Table \\ref{STIRAP_weakly_schema_1} for the second STIRAP scheme with an intermediate level belonging to a PEC correlated to the asymptotes $^2P_{1\/2}$ and $^2P_{3\/2}$.}\n\\begin{tabular}{cccccc}\n\\hline\n\\hline\n$v_i$ & -1 & -2 & -3 & -4 & -5 \\\\\n$E_{i}$ (cm$^{-1}$) & 1.3(-3) & 2.58(-2) & 1.147(-1) & 3.112 (-1) & 6.573 (-1) \\\\\n\\hline\n$v_f$ & 0 & 0 & 0 & 0 & 0 \\\\\n$E_{f}$ (cm$^{-1}$) & 1054.3406 & 1054.3406 & 1054.3406 & 1054.3406 & 1054.3406 \\\\\n\\hline\n$v_e$ ((2) $\\Omega=\\frac{1}{2}$) & 40 & 40 & 39 & 37 & 37 \\\\\n$E_{e}$ (cm$^{-1}$) & 5275.7563 & 5275.7563 & 5344.3918 & 5482.5236 & 5482.5236 \\\\\n\\hline\n$E_{\\textrm{pump}}$ (cm$^{-1}$) & 7382.0436 & 7382.0687 & 7313.5220 & 7175.5865 & 7175.9330 \\\\\n$|d_{ie}|^2$ (a.u.) & 2.5(-7) & 2.1 (-6) & 1.4 (-6) & 1.1 (-5) & 1.9 (-5) \\\\\n$E_{\\textrm{dump}}$ (cm$^{-1}$) & 8436.3843 & 8436.3843 & 8367.7488 & 8229.6170 & 8229.6170 \\\\\n$|d_{ef}|^2$ (a.u.) & 1.8 (-6) & 1.8 (-6) & 3.8 (-6) & 1.6 (-5) & 1.6 (-5) \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\label{STIRAP_weakly_schema_2}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\caption{Same as Table \\ref{STIRAP_weakly_schema_1} for the third STIRAP scheme with an intermediate level belonging to a PEC correlated to the asymptotes $^3P_{0}$,$^3P_{1}$ and $^3P_{2}$.}\n\\begin{tabular}{cccccc}\n\\hline\n\\hline\n$v_i$ & -1 & -2 & -3 & -4 & -5 \\\\\n$E_{i}$ (cm$^{-1}$) & 1.3(-3) & 2.58(-2) & 1.147(-1) & 3.112 (-1) & 6.573 (-1) \\\\\n\\hline\n$v_f$ & 0 & 0 & 0 & 0 & 0 \\\\\n$E_{f}$ (cm$^{-1}$) & 1054.3406 & 1054.3406 & 1054.3406 & 1054.3406 & 1054.3406 \\\\\n\\hline\n$v_e$ ((5) $\\Omega=\\frac{1}{2}$) & 16 & 15 & 15 & 15 & 15 \\\\\n$E_{e}$ (cm$^{-1}$) & 2746.0037 & 2746.0131 & 2746.0131 & 2746.0131 & 2746.0131 \\\\\n\\hline\n$E_{\\textrm{pump}}$ (cm$^{-1}$) & 11675.2972 & 11626.3119 & 11626.4005 & 11626.5954 & 11626.9390 \\\\\n$|d_{ie}|^2$ (a.u.) & 7.4 (-6) & 2.0 (-5) & 5.5 (-5) & 1.1 (-4) & 1.7 (-4) \\\\\n$E_{\\textrm{dump}}$ (cm$^{-1}$) & 12414.0104 & 12362.3483 & 12362.3483 & 12362.3483 & 12362.3483 \\\\\n$|d_{ef}|^2$ (a.u.) & 1.8 (-5) & 9.8 (-5) & 9.8(-5) & 9.8 (-5) & 9.8 (-5) \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\label{STIRAP_weakly_schema_3}\n\\end{table}\n\n\n\\section{Conclusion}\nIn this work, we have made a complete investigation about ways to create ultracold $^{87}$Rb$^{84}$Sr bosonic molecules in their rovibronic absolute ground state by all-optical methods. We have modeled the photoassociation of ($^{87}$Rb,$^{84}$Sr) atom pairs close to two atomic transitions: the allowed $5s^2S_{1\/2} \\rightarrow 5p\\,^2P_{1\/2,3\/2}$ Rb transition, and the $5s^2\\,^1S \\rightarrow 5s5p\\,^3P_{0,1,2}$ intercombination in strontium. As expected the photoassociation spectra show opposite behaviors. In the former case, the photoassociation rates are very high close to the asymptote. In the latter case, the photoassociation rates are very low close to the asymptotes. The distributions of ground-state vibrational levels after spontaneous emission are also different. Mainly one vibrational level is populated in the former case, but this level is highly excited. In the latter case, the lowest rovibrational level of the ground state could be populated, but many other vibrational levels as well. Therefore a further step of internal cooling is necessary to achieve a significant creation of ultracold RbSr molecules in their lowest rovibrational level.\n\nWe have then proposed to implement the formation of ultracold $^{87}$Rb$^{84}$Sr molecules by a STIRAP method in a tight trap. We found that a single STIRAP sequence to reach the lowest rovibrational ground-state level is tedious with moderate laser intensity. However, with an intermediate level close to the allowed $5s^2S_{1\/2} \\rightarrow 5p\\,^2P_{1\/2,3\/2}$ Rb transition, a STIRAP schema is possible for populating one of the five last vibrational levels of the ground state. We then completed our study by modeling a further STIRAP sequence to efficiently transfer the population from these uppermost levels toward the lowest rovibrational ground-state level. Three STIRAP schemes have been identified in three different spectral zones. \n\nTogether with the recent spectacular experimental achievements of the Amsterdam group \\cite{barbe2017,ciamei2018} revealing magnetic Feshbach resonances in RbSr and a novel description of the entire PEC of the RbSr ground state, the present work should help to progress toward the realization of a molecular sample of ultracold RbSr polar molecules. From our investigation it appears that in contrast to the ongoing experiment, considering the possibility to use lasers close to the allowed $5s^2S_{1\/2} \\rightarrow 5p\\,^2P_{1\/2,3\/2}$ Rb transition, would probably be necessary to reach this objective.\n\n\\section*{Acknowledgements}\nThe authors are grateful to Alessio Ciamei, Florian Schreck, and the Amsterdam group, for providing us with experimental results prior to publication. A.D. and O.D. acknowledge the support of the ANR BLUESHIELD (Grant No. ANR-ANR-14-CE34-0006).\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Cabibbo-Kobayashi-Maskawa (CKM) matrix describes quark flavour mixing\nin the Standard Model (SM).\nThe unitarity relation imposes for the first row\n\\be\n\\label{eq:unit}\n|V_{\\rm ud}|^2 + |V_{\\rm us}|^2 + |V_{\\rm ub}|^2  =  1  \\;.\n\\ee\nThe values given by the PDG 2012 read \n\\be\nV_{\\rm ud} = 0.97427(15)\\,, \\quad V_{\\rm us} = 0.22534(65)\\,,  \\quad V_{\\rm ub} = 0.00351(15) \\;.\n\\ee\nWith these numerical values, \none clearly sees why finding a deviation of Eq.(\\ref{eq:unit}) \nis a difficult task, but with the constant improvement on both the experimental\nand the theoretical side, the first row is a very good framework for performing \nprecise tests of the SM.\nWe note that the  value of $ |V_{\\rm ub}|^2$ is an order of magnitude smaller than the current uncertainty on \nthe $|V_{\\rm ud}|^2$ and $ |V_{\\rm us}|^2 $, which are of the same order.\\\\\n\nThere is currently a huge effort in the lattice community to \nimprove the determination of $V_{\\rm ud}$ and $V_{\\rm us}$. \nWe refer the reader to FLAG~\\cite{Aoki:2013ldr} for a comprehensive review.\nIn this proceeding, I present the recent ideas and highlight the \nnewest computations.\n\n\\section{Theoretical Framework - Lattice Computation}\n\nThe basis idea is that since $ |V_{\\rm us} f_+(0)| $ \nand $|V_{\\rm us} \/V_{\\rm ud} | f_{K^\\pm} \/ f_{\\pi^\\pm}$ are experimentally well measured \n(the numbers are taken from ~\\cite{Aoki:2013ldr})\n\\bean\n|V_{us} f_+(0)| &=&  0.2163(5) \\\\\n\\left| \\frac {V_{us}}{V_{ud}} \\right| \\frac{f_{K^\\pm}}{f_{\\pi^\\pm} } &=&  0.2758(5) \\;,\n\\eean\none can compute $ f_+(0)$ and $ {f_{K^\\pm}}\/{f_{\\pi^\\pm}}$ on the lattice \nand extract $ V_{\\rm us}$ and $V_{\\rm ud}$. \n(In this report we only consider QCD in the isopsin limit $m_u=m_d$,\nand therefore do not write the charge explicitly, eg $f_{p^+}=f_{p^-}$,\nbut electromagnetic corrections are applied~\\cite{Aoki:2013ldr}.)\nWe start with some basic definitions of the relevant form factors, \nfirst the decay constant \n\\be\n\\label{eq:defpi}\n\\la 0 | A_\\mu | P(p) \\ra = ip_\\mu f_P \\;, \n\\qquad \\mbox{ where }\nA_\\mu =  \\bar \\psi_1 \\gamma_\\mu \\gamma_5 \\psi_2 \\;.\n\\ee \nHere $P=\\bar \\psi_1 \\gamma_5 \\psi_2$ is either a pion or a kaon, hence\nEq.~(\\ref{eq:defpi}) defines $f_\\pi$ and $f_K$. \nFrom the vector current $V_\\mu =  \\bar \\psi_1 \\gamma_\\mu \\psi_2$ we define\nthe form factors $f_+$ and $f_-$\n\\be\n\\label{eq:Vform}\n\\la \\pi (p') | V_\\mu | K(p) \\ra = (p + p')_\\mu f_+(q^2) + (p - p')_\\mu f_-(q^2) \\;,\n\\ee\nwhere $q=p'-p$ is the momentum transfer. Finally we also introduce the scalar form factor $f_0$ defined by\n($S = \\bar \\psi_1 \\psi_2$ )\n\\be\n\\label{eq:Sform}\n\\la \\pi(p') | S | K(p) \\ra = \\frac{m_K^2 - m_\\pi^2}{m_s-m_l} f_0(q^2)  \\;.\n\\ee\nThe vector Ward Identity implies a relation between the vector current and the scalar density (for non-flavour singlet)\n$\\partial^\\mu V_\\mu = (m_2 - m_1) S$. In particular, this gives\n\\be \nf_0(q^2) = f_+(q^2) + {q^2 \\over m_K^2-m_\\pi^2} f_-(q^2) \\;.\n\\ee\nIn particular $f_0(0) = f_+(0)$,\nhence at zero-momentum transfer the form factor can either be extracted from the vector current, Eq.~(\\ref{eq:Vform}),\nor from the scalar density~\\cite{Na:2010uf}, see Eq.~(\\ref{eq:Sform}).\nA standard method introduced by~\\cite{Hashimoto:1999yp} is to compute a ratio such as \n\\be\n\\label{eq:ratio}\n\\frac{\\la \\pi | \\overline s \\gamma_0 l | K   \\ra}{\\la \\pi | \\overline l \\gamma_0 l | \\pi \\ra}\n\\frac{\\la K   | \\overline l \\gamma_0 s | \\pi \\ra}{\\la K   | \\overline s \\gamma_0 s | K \\ra}\n=\n{(f_0(q_{\\rm max}^2))}^2 \\, \\frac{(m_K+m_\\pi)^2}{4 m_K m_\\pi}\n\\ee\nwhere all the hadronic states are taken at rest and $q^2_{\\rm max} = (m_K-m_\\pi)^2$. \nThis ratio can be numerically very well determined (most of the systematics cancel out \nand the statistical precision is better at zero-momentum).\nIn addition the same ratio can also be evaluated with non-vanishing momenta\n(for either the pion, the kaon or both)\nand the zero-momentum transfer form factor can be obtained by an interpolation (see for example \n~\\cite{Becirevic:2004ya,Boyle:2007qe}).\n\nSimulating light quark masses is numerically expensive,\nand even if nowadays physical pion masses are accessible, \none would like to take advantage of  un-physical heavier \nquark since they are statistically\nmore precise. The Ademollo-Gatto theorem plays a central  \nrole here: the form factor $f_+(0)$ is exactly one in the $SU(3)$ flavour \nlimit and the first correction is parametrised by a known function $f_2$.\nIn practise, one can use an Ansatz of the form:\n\\be\n\\label{eq:chipt}\nf_+(0) = 1 + f_2(f,m_\\pi^2,m_K^2) + \\mbox{higher order}\n\\ee\n\nEnormous progress have been made recently on the lattice side, \ndevelopment of new ideas, algorithms, discretisation of the Lagrangian,\nand of course hardware improvement, too numerous to be explained in detail in \nthis report. Instead, I highlight some important improvements developed in the last years \nrelevant for the lattice computation of $V_{\\rm us}$\n\\vspace{0.5cm}\\\\\n{\\em Theoretical developments}\n\\begin{itemize}\n\\item \nThanks to partially twisted boundary conditions, \nthe momenta are not restricted to the Fourier modes and the form factor can be computed \ndirectly a zero-momentum transfer~\\cite{Boyle:2010bh}. \nNo interpolation in momenta is required, avoiding a possible model-dependence\nAnsatz.\\\\\n\\item \nThe use of the scalar density (instead of the vector current) \nto extract $f_0(0)=f_+(0)$. \nOne advantage is that  \nin Eq~(\\ref{eq:Sform}) the quantity $(m_2-m_1) S$ \nis protected by a Ward Identity and hence no renormalisation is required.\n\\end{itemize}\n{ \\em Lattice improvements}\n\\begin{itemize}\n\\item \nSimulation with physical quark masses: FNAL\/MILC and RBC-UKQCD are computing \n$f_+(0)$ with light quarks down to their physical value~\\cite{Bazavov:2013maa,Juettner:2014ssa}. \nFNAL\/MILC simulates $2+1+1$ dynamical flavours of Highly Improved Staggered Quarks (HISQ)\nand RBC-UKQCD simulates $2+1$ flavours of Domain-Wall (DW) fermions, an action notoriously \nexpensive which preserves chiral-flavour symmetry at finite lattice spacing.\nHence the uncertainty due to the chiral extrapolation (which was the dominant one \nin 2013) is removed, or at least drastically reduced. \n\n\\item Inclusion of dynamical quarks: in 2013, FLAG reported that three\ncollaborations (FNAL\/MILC, JLQCD and RBC\/UKQCD) have computed $f_+(0)$ with $2+1$ flavours, \nie a degenerate light doublet and a strange quark in the sea. \nMore recently, two collaborations (ETM and the FNAL\/MILC collaborations) \nhave also included a dynamical charm: although one does not expect the charm to have \na big effect in this sector, the lattice results are becoming so precise\nthat this should certainly be checked. \nLet us also mention that HPQCD has computed $f_K\/f_\\pi$ on the $2+1+1$ MILC ensemble \nwith physical quark masses~\\cite{Dowdall:2013rya}.\n\\end{itemize}\n\\section{Results: 2014 update }\nIn 2013, the FLAG reported \n\\bea\nf_+(0) &=& 0.9661(32) \\qquad n_f=2+1\\\\\nf_+(0) &=& 0.9560(57)(62)  \\qquad n_f=2\n\\eea\nand noted that the major source of error came for the chiral extrapolation.\nWe refer the reader to the original publications for more details\n\\cite{Boyle:2013gsa,Bazavov:2012cd,Boyle:2010bh,Boyle:2007qe,Lubicz:2009ht,Dawson:2006qc}.\n\nThese averages do not include the most recent results which are given in \nTable~\\ref{tab:details1}, together with some important features \nof the simulations.  \nThe action denotes the type of \ndiscretisation used for the Dirac operators. Even if the results \nshould converge to the same continuum limit, at finite lattice \nspacing the theory suffers from distortion which are action-dependent. \nIt is important to note that $f_+(0)$ is now being computed with\nphysical quark masses and that $2+1+1$ results are also available.\n\\begin{center}\n\\begin{table}[!h]\n\\begin{tabular}{ccccccl}\nCollaboration & Action & $m_\\pi$ (MeV) & $a$ (fm) &  $N_f$ & $f_+(0)$ & \\qquad $|V_{\\rm us}|$ \\\\\n\\hline\nFNAL\/MILC \\cite{Bazavov:2013maa}  &  HISQ & $130$ & $0.06$ & $2+1+1$ & $0.9704(32)$ & $0.22290(74)(52)$ \\\\ \nETM       \\cite{Carrasco:2014uda} &  OS   & $210$ & $0.06$ & $2+1+1$ & $0.9683(65)$ & $0.2234(16)$\\\\\n\\end{tabular}\n\\caption{Summary of results for the most recent computations of $f_+(0)$, not included in the FLAG average yet.\nFor each computation we give the lightest simulated pion mass, the finest lattice spacing\nand the number of quark flavours included in the sea. \nNote that the lightest pions mass is not necessarily simulated on the finest ensemble.\nThe column ``action'' corresponds to the discretisation of the Dirac operator,\nsee the original references for more details.\n}\n\\label{tab:details1}\n\\end{table}\n\\end{center}\nWe now turn to the ratios of decay constant $f_K\/f_\\pi$. In their 2013 report, FLAG quoted\n\\bean\nf_K\/f_\\pi &=& 1.194(5) \\qquad n_f=2+1+1\\\\\nf_K\/f_\\pi &=& 1.192(5) \\qquad n_f=2+1\\\\\nf_K\/f_\\pi &=& 1.205(6)(17) \\qquad n_f=2\n\\eean\nand again we refer the reader to the original work for more details\n\\cite{Aoki:2010dy,Aoki:2009ix,Durr:2010hr,Bazavov:2009bb,Aoki:2008sm,Allton:2008pn,Follana:2007uv,Beane:2006kx,Aubin:2004fs,Engel:2011aa,Blossier:2009bx}.\nNote that some of these results were obtained with \nphysical quark masses. \nAt Lattice 2014, both the FNAL\/MILC and the RBC-UKQCD collaborations have reported \ntheir new results, see Table~\\ref{tab:details2}. \n\\begin{table}[!h]\n\\begin{center}\n\\begin{tabular}{cccccl}\nCollaboration & Action & $m_\\pi$ (MeV) & $a$ (fm) &  $N_f$ & \\qquad $f_K\/f_\\pi$ \n\\\\\n\\hline\nFNAL\/MILC~\\cite{Bazavov:2014wgs}  &  HISQ & $130$ & $0.06$ & $2+1+1$ & $1.1956 (10)\\left(^{+26}_{-18}\\right)$ \n\\\\\nRBC-UKQCD~\\cite{RBC:2014tka} &  DW & $139$ & $0.08$ & $2+1$ & $1.1945(45)$  \n\\\\\n\\end{tabular}\n\\caption{2014 Update for $f_K\/f_\\pi$. The details are the same as in\nTable~\\ref{tab:details1}. The precision is to be compared to the FLAG13 average. \n}\n\\label{tab:details2}\n\\end{center}\n\\end{table}\n\nIt is interesting to look at the errors in more details. For example,\nfor $V_{\\rm us}\/V_{\\rm ud}$~\\cite{Bazavov:2014wgs}\n\\be\n|V_{\\rm us}\/V_{\\rm ud}| = 0.23081(52)_{\\rm LQCD}(29)_{\\rm BR(K_{l2})}(21)_{\\rm EM}\n\\qquad \\mbox{ Fermilab Lattices\/MILC 2014}\n\\ee\nEven if the lattice errors still dominate, they are clearly becoming competitive\n\n\n\\section{Conclusions - Outlook}\n\nThe latest lattice simulations are truly impressive, \ntremendous progress have been made this year, in particular \nregarding the extraction of $f_+(0)$: the lattice simulations are now reaching the physical quark\nmasses and include three or four flavour of dynamical quarks. \nThe errors are usually dominated by the continuum extrapolation Ansatz.\nImproving this error by brute force (going to finer lattices) \nis a real challenge as it requires solving some theoretical issues (see for example~\\cite{Luscher:2011kk}). \nTherefore it is very important to perform these computations with \nimproved lattice actions (which have genuinely smaller lattice artifacts).\nAnother challenge to face is that most of the lattice simulations are done \nin ``pure'' QCD in the isospin limit $m_u=m_d$. The results are now \nso precise that the effects of this approximation are becoming visible. \nFor $f_{+}(0)$ or $f_K\/f_\\pi$, a (model dependent) correction is applied {\\em a posteriori} \nto the lattice results \\cite{Aoki:2013ldr}. However, important progress have been made \nrecently in that respect: see~\\cite{Borsanyi:2014jba} for an implementation \nof QCD+QED at order $\\alpha$ and see~\\cite{Portelli:CKM2014} for a review. \n\n\n\\vspace{0.5cm}\n{\\bf Acknowledgements -} I would like to thank the organisers of CKM2014 and in particular\nthe conveners of WG1, Stefan Bae{\\ss}ler, Anze Zupanc and Elvira G\\'amiz\nfor their kind invitation.\nI acknowledge support from STFC under the grant ST\/J000434\/1\nand from the EU Grant Agreement 238353 (ITN STRONGnet).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Braid groups}\n\n\\begin{wrapfigure}[6]{r}{0.2\\textwidth}\n    \\vspace{-20pt}\n    \\includegraphics[height=0.18\\textwidth]{Threedimensional_braid_remastered.png}\n    \\caption{An example braid}\n    \\label{fig-example-braid}\n\\end{wrapfigure}\n\n\\hspace{\\parindent}Fix, once and for all, an arbitrary integer $n\\geq 2$; we shall use the notation $[n] := \\{1, \\dots, n\\}$.\nA \\textit{braid} on $n$ strands is a topological object consisting of $n$ strands in 3-dimensional space whose endpoints are fixed to two distinguished parallel planes, such as in Figure \\ref{fig-example-braid}.\n(We assume, as is common, that the endpoints on each plane are collinear.\nWe also choose one plane to be the ``top'' and number the strands from $1$ to $n$ in their order on this plane; we choose the other to be the ``bottom''.)\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{braid-composition.png}\n    \\caption{Braid multiplication}\n    \\label{fig-braid-multiplication}\n\\end{figure}\n\nBy ``gluing'' two braids on $n$ strands together (i.e., identifying the ``bottom'' plane of the first with the ``top'' plane of the second, as depicted in Figure \\ref{fig-braid-multiplication}), we can create a third braid on $n$ strands.\nThis operation, which we view as composition, endows the set of all braids on $n$ strands up to isotopy (i.e., topological deformation) with the structure of a monoid (i.e., a set with an associative and identity).\nIn fact, this operation is invertible; an example braid inverse is shown in Figure \\ref{fig-braid-inverse}.\nAs a result, the set of all braids on $n$ strands up to isotopy is in fact a group, the \\textit{braid group} $B_n$.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.55\\textwidth]{braid-inverse.png}\n    \\caption{A braid and its inverse}\n    \\label{fig-braid-inverse}\n\\end{figure}\n\nThe braid group $B_n$ is a well-studied mathematical object.\nEmil Artin first proposed it in 1925 \\cite{Artin1925} and further discussed it in a 1947 paper \\cite{10.2307\/1969218}.\nIt is well-known that $B_n$ has no torsion and that the full twist generates its center. Several efficient algorithms to solve $B_n$'s word and conjugacy problems are known \\cite{gonzalez2011basic}.\nIt is also well-known (as the Nielsen-Thurston classification) that all braids are either \\textit{periodic}, \\textit{reducible}, or \\textit{pseudo-Asonov} \\cite{gonzalez2011basic}.\nFurthermore, $B_n$ is linear, as there is a faithful (!) representation (the \\textit{Lawrence-Krammer representation}) $B_n\\to GL_{n(n-1)\/2}(\\mathbb{Z}[q^{\\pm 1}, t^{\\pm 1}])$ \\cite{krammer2002braid}.\n\nThe braid group $B_n$ has a canonical generating set of $n-1$ generators known as the \\textit{Artin generators} and conventionally denoted $\\sigma_1, \\dots, \\sigma_{n-1}$.\nThe Artin generator $\\sigma_i$ consists of the braid that twists the $i$th leftmost strand over and to the right of the $(i+1)$th strand.\nWith this generating set, the braid group has the following presentation:\n\\begin{align*}\n    B_n = \\langle \\sigma_1, \\dots, \\sigma_{n-1} | \\sigma_i\\sigma_{i+1}\\sigma_i = \\sigma_{i+1}\\sigma_i\\sigma_{i+1}\\forall i, \\sigma_i\\sigma_j = \\sigma_j\\sigma_i \\forall i, j: |i - j| > 1\\rangle.\n\\end{align*}\n\nThe \\textit{writhe} is a group homomorphism $B_n\\to\\mathbb{Z}$ defined by the relation $\\sigma_i\\mapsto 1$. The image of any braid $\\beta$ under writhe is called the \\textit{writhe} of the braid, denoted $|\\beta|$. The \\textit{permutation} is a group homomorphism $B_n\\to S_n$ defined by the relation $\\sigma_i\\mapsto(i, i + 1)$, where the right hand side is a transposition. The permutation of a braid is defined similarly and denoted $\\pi_{\\beta}$.\n\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.15\\textwidth]{Threedimensional_braid_remastered.png}\n    \\caption{The braid $(\\sigma_2^{-1}\\sigma_1^{-1})^2$, which has permutation (1,3,2) and writhe~$-4$.}\n\\end{figure}\n\nBoth the writhe and permutation are surjective homomorphisms but \\underline{not} injective.\nThe \\textit{writhe-permutation homomorphism} $WP: B_n\\to\\mathbb{Z}\\times S_n$ is defined by $\\beta\\mapsto (|\\beta|, \\pi_{\\beta})$. It is neither surjective nor injective, as we shall prove later.\n\n\\subsection{Artin groups of crystallographic type}\n\n\\hspace{\\parindent}An \\textit{Artin group} $\\mathcal{A}$ is a group presented by a finite set of generators and at most one braid relation (i.e., a relation of the form $a = b$, $ab = ba$, $aba = bab$, $abab = baba$, etc.) between any two generators.\nThe presentation of an Artin group can be depicted in a \\textit{Dynkin diagram}, as shown in Figure \\ref{fig-dynkin-diagrams}.\n\n\\begin{figure}[h]\n    \\vspace{10pt}\n    \\begin{center}\n    \\includegraphics[height=0.5\\textwidth]{349px-DynkinDiagrams.png}\n    \\caption{Dynkin diagrams of finite Coxeter groups\n    \\cite{dynkindiagrams2007commons}}\n    \\label{fig-dynkin-diagrams}\n    \\end{center}\n\\end{figure}\n\\begin{figure}\n    \\centering\n    \\includegraphics[height=100pt]{G2_simpleroots.png}\n    \\caption{The $G_2$ root system with two simple roots highlighted \\cite{nasmith2020medium}}\n    \\label{fig-g2-simpleroots}\n\\end{figure}\n\\vspace{-10pt}\nA \\textit{Coxeter group} $\\mathcal{C}$ is a group presented with the generators and relations of an Artin group and the relations that the square of any generator is the identity.\nA Coxeter group's presentation can also be depicted in a Dynkin diagram.\n\nSometimes, a Coxeter group is generated by the reflections upon a \\textit{root system} $\\Phi$, which is a special set of vectors.\n(In this case, they are known as \\textit{Weyl groups}.)\nThe associated Artin group $\\mathcal{A}$ is then called an \\textit{Artin group of crystallographic type}.\nFurthermore, in the cases where the Coxeter group is indeed associated with a root system, an isomorphism between the Coxeter group and the group of linear transformations sends the canonical Coxeter group generators to the reflections associated with some choice of \\textit{simple roots} $\\Delta\\subset\\Phi$.\n(We refer to the simple root associated to a canonical Artin generator $a$ of $\\mathcal{A}$ as $\\Delta_a$.)\nOne such set of simple roots, corresponding to the case commonly referred to as $G_2$, is shown in Figure \\ref{fig-g2-simpleroots}.\n\n\\subsection{Complex reflection groups}\n\n\\begin{wrapfigure}[22]{r}{0.3\\textwidth}\n    \\vspace{-40pt}\n        \\includegraphics[height=0.7\\textwidth]{complex-reflection-group-table.png}\n    \\caption{Partial table of complex reflection groups \\cite{Broue2010IntroductionTC}}\n\\end{wrapfigure}\n\\hspace{\\parindent}We adopt the terminology and notation of Brou\u00e9 \\cite{Broue2000}.\n\nLet $V$ be some finite-dimensional $\\mathbb{C}$-vector space.\nA \\textit{pseudo-reflection} is a linear transformation that acts trivially on some hyperplane (called its \\textit{reflecting hyperplane}).\n\nA \\textit{complex reflection group} $W$ is a finite subgroup of $GL(V)$ generated by pseudo-reflections.\n\nWe denote the set of pseudo-reflections in $W$ by $\\Psi$. (This notation is an exception; it comes from Dr. Trinh's notes \\cite{mqt200207notes}, not from Brou\u00e9.)\nThe hyperplane arrangement in $V$ consisting of the reflecting hyperplanes of the $\\Psi$ is denoted by $\\mathcal{A}$, and its hyperplane complement $V - \\cup_{H\\in\\mathcal{A}} H$ is denoted as $M$.\nWe define the analogous \\textit{braid group} $B := \\pi_1(M\/W)$; in other words, the fundamental group of $M$ up to rotation by elements of $W$.\n\nSimilar to the previous case, the braid groups $B$ have canonical Artin-like presentations, and the corresponding Coxeter groups $W$ have canonical Coxeter-like presentations \\cite{Broue2000}.\nMore precisely, there exists a subset $\\mathbf{S} = \\{\\mathbf{s}_1, \\dots, \\mathbf{s}_n\\}$ of $B$ consisting of distinguished braid reflections, and a set $R$ of relations of the form $w_1 = w_2$, where $w_1$ and $w_2$\nare positive words of equal length in the elements of $\\mathbf{S}$, such that $\\langle S|R\\rangle$ is\na presentation of $B$.\nMoreover, their images $s_1, \\dots, s_n$ in $W$ generate $W$, and the group $W$ is presented by\n$$\\langle S|R;(\\forall s\\in S)(s^{e_s} =1)\\rangle$$ where $e_s$ denotes the order of $s$ in $W$.\n\nThere is a quotient map $B\\to W$; we denote the image of an element $b\\in B$ under this quotient map simply as $W(b)$.\n\n\\subsection{Overview of results}\n\n\\hspace{\\parindent}We study a large family of mathematical objects that we call the \\textit{Gauss-Epple homomorphisms}.\nThis research helps us understand the structure of braid groups $B_n$, which describe the structure of \\textit{braids}, a kind of topological object.\n\nThe first such homomorphism was implicitly introduced by Epple \\cite{Epple98} as an action of $B_n$ based on a note by Gauss.\nAround the time of his note, Gauss was primarily interested in topology for its applications to electromagnetism and celestial dynamics. In this work, we generalize this concept to a broader family of homomorphisms from Artin groups of finite type, a large family of groups including the braid groups.\n\nFirstly, in Section \\ref{sec-theGEhomor}, we prove that there exists a well-defined and unique left group action of $B_n$ on $\\mathbb{Z}\\times[n]$, as implicit in \\cite{Epple98}.\nWe refer to this action as the \\textit{Gauss-Epple action}.\nIt is equivalent to a group homomorphism from $B_n$ to $\\Sym(\\mathbb{Z}\\times[n])$, which we call the \\textit{Gauss-Epple homomorphism} (denoted $GE$).\nIn Subsection \\ref{subsec-imWP}, we describe the \\textit{writhe-permutation} homomorphism $WP: B_n\\to\\mathbb{Z}\\times S_n$ (the homomorphism that maps a braid to its writhe and underlying permutation) in greater detail, as we treat this homomorphism as a toy model of $GE$. We show that the image of \\textit{writhe-permutation} homomorphism $WP: B_n\\to\\mathbb{Z}\\times S_n$ is a particular order 2 subgroup of $\\mathbb{Z}\\times S_n$.\nThen, in Subsection \\ref{subsec-imGE}, we show that the image of $GE$ is (isomorphic to) an order 2 subgroup of $\\mathbb{Z}^n\\rtimes S_n$.\nIn Subsection \\ref{subsec-kerGE}, we discuss the kernel of $GE$. We find that this group is strictly contained in the kernel of the \\textit{writhe-permutation homomorphism}.\n\nWe summarize these results with the following commutative diagram:\n\n\\[\\begin{tikzcd}\n\t{B_ n} & {\\mathbb{Z}^n\\rtimes S_n} & {\\Sym([n]\\times\\mathbb{Z})} \\\\\n\t& {\\mathbb{Z}\\times S_n}\n\t\\arrow[from=1-1, to=1-2]\n\t\\arrow[from=1-2, to=1-3]\n\t\\arrow[from=1-2, to=2-2]\n\t\\arrow[\"WP\", from=1-1, to=2-2]\n\t\\arrow[\"GE\", curve={height=-12pt}, from=1-1, to=1-3]\n\\end{tikzcd}\\]\n\nEach of the maps in this diagram is a group homomorphism, as explained below:\n\n\\begin{itemize}\n    \\item The map $GE$, from $B_n$ to $\\mathrm{Sym}(\\mathbb{Z}^2)$, is the Gauss-Epple action of $B_n$ on $\\mathbb{Z}^2$.\n    \\item The map from $B_n$ to $\\mathbb{Z}^n\\rtimes S_n$ maps a braid to the tuple of its vector and its permutation.\n    \\item The map from $\\mathbb{Z}^n\\rtimes S_n$ to $\\mathrm{Sym}(\\mathbb{Z}^2)$ is the map $(\\pi, \\ell)\\mapsto ((a, b) \\mapsto (\\pi(a), b + \\ell_a))$.\n    \\item The map $WP$, from $B_n$ to $\\mathbb{Z}\\times S_n$, maps a braid to its writhe and braid permutation.\n    \\item The map from $\\mathbb{Z}^n\\rtimes S_n$ to $\\mathbb{Z}\\times S_n$ maps $(\\ell, \\pi)$ to $(\\sum\\ell, \\pi)$.\n\\end{itemize}\n\nIn Section \\ref{sec-symGE}, we briefly discuss another action of $B_n$ (the \\textit{symmetric-Gauss-Epple action}) mentioned by Epple.\nIt turns out that this action has very similar properties to the Gauss-Epple action, including sharing the same kernel.\n\nIn Section \\ref{sec-superGE}, we discuss the \\textit{super-Gauss-Epple homomorphism} (denoted $SGE$), a homomorphism of $B_n$ that refines $GE$.\nTo describe the image of $SGE$, we introduce a 1-cocycle of the symmetric group $S_n$ on the set of $n\\times n$ antisymmetric matrices, which we prove has a remarkable nonnegativity property.\n\nFinally, in Section \\ref{sec-ATgroupsWeyl}, we generalize our results to the contexts of Artin groups of finite type, a broad family of groups that generalize the braid groups.\nMore specifically, we introduce a family of novel homomorphisms (with domain $\\mathcal{A}$ and range $\\mathbb{Z}^{\\Phi}\\rtimes\\mathcal{C}$) that are analogous to the super-Gauss-Epple homomorphism in the classical case of $B_n$.\nWe also note corresponding 1-cocycles, which also share an analogous nonnegativity property.\n\nWe close in Section \\ref{sec-complex} with remarks about complex reflection groups.\n\n\\subsubsection{Acknowledgements}\n\n\\hspace{\\parindent}We thank our advisor, Minh-T\u00e2m Trinh, for proposing this project and giving us helpful directions.\nWe also thank the MIT PRIMES program for supporting our research.\n\n\\section{The Gauss-Epple homomorphism}\\label{sec-theGEhomor}\n\n\\begin{figure}\n    \\begin{center}\\includegraphics[width=0.5\\textwidth]{Gauss_note.png}\\end{center}\n    \\caption{Page 283 of Gauss's Handbuch 7 \\cite{Epple98}}\n\\end{figure}\n\n\\hspace{\\parindent}We formally introduce the Gauss-Epple action. \nThis action was initially defined by Epple \\cite{Epple98} based on a note of Gauss.\nSince Gauss's notation involved complex numbers (more precisely, the Gaussian integers $\\mathbb{Z}[i]$), the Gauss-Epple action was originally defined in terms of complex numbers.\nFor notational and theoretical simplicity, however, we will define the Gauss-Epple action as an action of $B_n$ on $[n]\\times\\mathbb{Z}$ instead of on $\\mathbb{Z}[i]$.\nOur reasons for doing so should become clear shortly.\nWe also view the Gauss-Epple action as a group homomorphism from $B_n$ to $\\Sym([n]\\times\\mathbb{Z})$.\n(This is a specific instance of .the classical trick to convert between group actions $G\\times X \\to X$ and group homomorphisms $G \\to (X \\to X)$: by currying the input.)\n\n\nAs defined by Epple (based on Gauss's notation), the Gauss-Epple homomorphism is given as follows:\n\\begin{definition}[Gauss-Epple action]\\label{def-eppleGE}\nFor any $n\\in\\mathbb{N}$, the Gauss-Epple action $\\alpha: B_n\\times \\mathbb{Z}[i]\\to \\mathbb{Z}[i]$ is the unique left group action of $B_n$ (with canonical Artin generators $\\sigma_1, \\sigma_2, \\dots$) on the Gaussian integers $\\mathbb{Z}[i]$ defined by the following generating relation:\n\n$$\\alpha(\\sigma_k, z) := \\begin{cases}\nz &  \\Re(z)\\notin\\{k, k + 1\\}\\\\\nz + 1 &  \\Re(z) = k\\\\\nz - 1 + i &  \\Re(z) = k + 1\n\\end{cases}.$$\n\\end{definition}\n\nWe simplify this definition by replacing each complex number involved with the ordered pair of its real and complex parts, and then restricting the first component to the elements of $[n]$.\nThis notational change yields an action on the elements of $[n]\\times\\mathbb{Z}$.\nTherefore, as we define it, the Gauss-Epple action is given as follows:\n\n\\begin{definition}[Gauss-Epple action]\\label{def-GE}\nFor any $n\\in\\mathbb{N}$, the Gauss-Epple action $GE: B_n\\times ([n]\\times\\mathbb{Z})\\to [n]\\times\\mathbb{Z}$ is the unique left group action of $B_n$ (with canonical Artin generators $\\sigma_1, \\sigma_2, \\dots$) on $[n]\\times\\mathbb{Z}$ defined by the following generating relation:\n\n$$GE(\\sigma_k, (a, b)) := \\begin{cases}\n(a, b) & a\\notin\\{k, k + 1\\}\\\\\n(k + 1, b) & a = k\\\\\n(k, b + 1) & a = k + 1\n\\end{cases}.$$\n\\end{definition}\n\n\\begin{remark}\nNote that this definition implies that the inverses of the Artin generators act as follows:\n\n$$GE(\\sigma_k^{-1}, (a, b)) = \\begin{cases}\n(a, b) & a\\notin\\{k, k + 1\\}\\\\\n(k + 1, b - 1) & a = k\\\\\n(k, b) & a = k + 1\n\\end{cases}.$$\n\\end{remark}\n\\vspace{30pt}\n\nWe now prove that the Gauss-Epple action is well-defined, a fact stated without proof by Epple \\cite{Epple98}.\n\n\\begin{lemma}\\label{lem-GE-welldefined}\nThe Gauss-Epple action $GE$ is uniquely defined by Definition \\ref{def-GE} as a left group action of $B_n$ on $\\mathbb{Z}^2$.\n\\end{lemma}\n\\begin{proof}\nSince the Artin generators satisfy the generating relations $\\sigma_k \\sigma_l = \\sigma_l \\sigma_k$ for all $k, l$ such that $|k - l| \\geq 2$ and $\\sigma_k\\sigma_{k+1}\\sigma_k = \\sigma_{k+1}\\sigma_k\\sigma_{k+1}$ for all $k\\in[n - 1]$, it is enough to verify that that the analogous relations hold for the Gauss-Epple action to prove Lemma \\ref{lem-GE-welldefined}.\n\nSuppose that $k, l$ are two integers satisfying $|k - l|\\geq 2$. We observe that $\\{k, k + 1\\}\\cap\\{l, l + 1\\}=\\varnothing$, which will become important later. Let $a, b$ be arbitrary members of $\\mathbb{Z}$.\nWe prove that, for all integers $k, l$ satisfying $|k - l|\\geq 2$, we have $GE(\\sigma_k)GE(\\sigma_l) = GE(\\sigma_l)GE(\\sigma_k)$, as follows:\n\n\\begin{packed_item}\n\\item If $a\\notin\\{k, k + 1, l, l + 1\\}$, then $\\sigma_k\\sigma_l(a, b) = (a, b) = \\sigma_l\\sigma_k(a, b)$.\n\\item If $a\\in\\{k, k +1\\}$, then $\\sigma_k\\sigma_l(a, b) = \\sigma_k(a, b) = \\sigma_l\\sigma_k(a, b)$. \n(From the second to third step, we use the fact that the first component of $\\sigma_k(a, b)$ is in $\\{k, k + 1\\}$, and so outside of $\\{l, l + 1\\}$.)\n\\item If $a\\in\\{l, l + 1\\}$, then $\\sigma_k\\sigma_l(a, b) = \\sigma_l(a, b) = \\sigma_l\\sigma_k(a, b)$.\n\\end{packed_item}\n\nAgain, suppose that $k, a, b$ are arbitrary integers satisfying $k\\in\\{1, \\dots, n - 2\\}$.\nThen we prove that we have $GE(\\sigma_k)GE(\\sigma_{k+1})GE(\\sigma_k) = GE(\\sigma_{k+1})GE(\\sigma_k)GE(\\sigma_{k+1})$ as follows:\n\n\\begin{packed_item}\n\\item If $a\\notin\\{k, k + 1, k + 2\\}$, then $\\sigma_k\\sigma_{k+1}\\sigma_k(a, b) = (a, b) = \\sigma_{k+1}\\sigma_k\\sigma_{k+1}(a, b)$.\n\\item If $a = k$, then $\\sigma_k\\sigma_{k+1}\\sigma_k(a, b) = \\sigma_k\\sigma_{k+1}(k + 1, b) = \\sigma_k(k + 2, b) = (k + 2, b)$,\nand $\\sigma_{k+1}\\sigma_k\\sigma_{k+1}(a, b) = \\sigma_{k + 1}\\sigma_k(k, b) = \\sigma_{k+1}(k + 1, b) = (k + 2, b)$.\n\\item If $a = k + 1$, then $\\sigma_k\\sigma_{k+1}\\sigma_k(a, b) = \\sigma_k\\sigma_{k+1}(k, b + 1) = \\sigma_k(k, b + 1) = (k + 1, b + 1)$,\nand $\\sigma_{k+1}\\sigma_k\\sigma_{k+1}(a, b) = \\sigma_{k + 1}\\sigma_k(k + 2, b) = \\sigma_{k+1}(k + 2, b) = (k + 1, b + 1)$.\n\\item If $a = k + 2$, then $\\sigma_k\\sigma_{k+1}\\sigma_k(a, b) = \\sigma_k\\sigma_{k+1}(k + 2, b) = \\sigma_k(k + 1, b + 1) = (k, b + 2)$,\nand $\\sigma_{k+1}\\sigma_k\\sigma_{k+1}(a, b) = \\sigma_{k + 1}\\sigma_k(k + 1, b + 1) = \\sigma_{k+1}(k, b + 2) = (k, b + 2)$.\n\\end{packed_item}\n\\end{proof}\n\nFrom now on, we consider GE as a group homomorphism $GE: B_n\\to\\Sym([n]\\times\\mathbb{Z})$\n\n\\begin{remark}\nNote that, by letting $n$ vary, we can extend the Gauss-Epple homomorphism $GE: B_n \\to\\Sym([n]\\times\\mathbb{Z})$ to a homomorphism $GE: B_{\\infty}\\to\\Sym(\\{1, 2, \\dots\\}\\times\\mathbb{Z})$, where the group $B_{\\infty}$ is defined to be the direct limit of the groups $B_n$. \nHowever, we do not pursue this direction.\n\\end{remark}\n\n\\subsection{Image of WP}\\label{subsec-imWP}\n\\hspace{\\parindent}Firstly, note the following classical fact, which motivates the study of \n\\text{images} of group homomorphisms:\n\n\\begin{theorem}[First Isomorphism Theorem]\\label{thm-1st-isothm}\nLet $\\phi: G\\to H$ be a group homomorphism. Then $\\im(\\phi) \\cong G\/\\ker\\phi$.\n\\end{theorem}\n\nFor the group homomorphisms we study, it turns out that the images are much simpler than the kernels. Hence we will completely determine the images but only briefly discuss the kernels.\n\nSince the Gauss-Epple action permutes the $a$ part without respect to the $b$ part according to the permutation of the braid, it is evident that braids in the kernel of the Gauss-Epple action have the identity permutation (i.e., are pure).\nEarly on, however, we discovered that braids in the kernel of the Gauss-Epple homomorphism have zero writhe too (we will prove this in Lemma \\ref{lem-sum-link-writhe}).\nHence we decided to study the invariant $WP$ (which stands for ``writhe-permutation''), a weaker invariant than the Gauss-Epple homomorphism.\nThis invariant, too, turns out to be a group homomorphism: in this case, with a range of $\\mathbb{Z}\\times S_n$.\nIn fact, we find that the image of $WP$ is an order 2 subgroup of the range:\n\n\\begin{theorem}[Structure of imWP]\\label{thm-structure-imWP}\nThere is a braid on $n$ strands with writhe $w\\in\\mathbb{Z}$ and permutation $\\pi\\in S_n$ iff (if and only if) $\\pi$ and $w$ have the same parity.\n\\end{theorem}\n\n\\begin{proof}\n\\textbf{Necessity}: Suppose that there is a braid on $n$ strands with writhe $w\\in\\mathbb{Z}$ and permutation $\\pi\\in S_n$. \nIf the number of twists of the braid is even, both $\\pi$ and $w$ will also necessarily be even.\nSimilarly, if the number of twists of the braid is odd, both $\\pi$ and $w$ will necessarily be odd.\nThus the claim holds.\n\n\\textbf{Sufficiency}: Let $\\pi$ and $w$ be arbitrary permutations and integers with the same parity.\nSince $S_n$ is generated by $(1, 2), (2, 3), \\dots, (n - 1, n)$, we can write $\\pi = \\prod_{i = 1}^f t_{a_i}$, where $t_i = (i, i + 1)$ and $a_i$ is some arbitrary sequence of integers.\n\nConsider the braid $\\beta' = \\prod_{i = 1}^f \\sigma_{a_i}$, where $a_i$ is as before. \nTrivially, $\\beta'$ has permutation $\\pi$.\nBy the argument in the previous part of the proof, the writhe of $\\beta'$ has the same parity as $\\pi$, which has the same parity as $w$, so that $\\sigma_1^{w - |\\beta'|}$ is pure.\nWe conclude that the braid $\\beta = \\beta'\\sigma_1^{w - |\\beta'|}$ has writhe $w$ and permutation $\\pi$, as desired,  and we are done.\n\\end{proof}\nIt is clear that since the image of $WP$ is not $B_n$ (nor is it the image of $GE$), that $WP$ must have a nontrivial kernel, and moreover, an element lying outside of the kernel of $GE$.\nWe can make this explicit by giving the example $\\sigma_1^{-2}\\sigma_2^2$.\n\n\\subsection{Image of GE}\\label{subsec-imGE}\n\n\\hspace{\\parindent}The computation of the image of the Gauss-Epple homomorphism (henceforth abbreviated \\textit{imGE}) is similar, albeit much more tedious. Firstly, we prove that the Gauss-Epple action factors through $\\mathbb{Z}^n\\rtimes S_n$.\n(The group rule of $\\mathbb{Z}^n\\rtimes S_n$ is $(\\pi, \\ell)(\\pi', \\ell') = (\\pi\\pi', \\ell' + \\pi'\\ell)$.)\n\n\\begin{theorem}[Structure theorem for imGE]\\label{thm-factor-imGE}\nDefine the group inclusion $\\iota: \\mathbb{Z}^n\\rtimes S_n\\to\\Sym([n]\\times\\mathbb{Z})$ by the equation $(\\pi, \\ell)\\to ((a, b)\\to (\\pi(a), b + \\ell(a))$,\nand define the group surjection $\\varsigma: B_n\\to\\mathbb{Z}^n\\rtimes S_n$ by \n$$\\varsigma(\\sigma_k) := ((k, k + 1), \\vec{e_k}).$$\nThen $GE = \\varsigma\\circ\\iota$.\n\\end{theorem}\n\\begin{proof}\n\nIt is trivial to verify that $GE = \\varsigma\\circ\\iota$ for the Artin generators.\nBy induction, and by the fact that all three functions are group homomorphisms, it follows that $GE = \\varsigma\\circ\\iota$ for all braids.\n\\end{proof}\n\nAs an aside, we observe the Gauss-Epple action is transitive when restricted to $[n]\\times\\mathbb{Z}$.\nHowever, as we can deduce from Theorem \\ref{thm-factor-imGE}, it is NOT doubly transitive.\n\n\nWe also relate imWP to imGE with the following fact (which motivates the earlier study of imWP):\n\n\\begin{lemma}\\label{lem-sum-link-writhe}\nLet $\\beta$ be a braid, and suppose that $\\varsigma(\\beta) = (\\pi, \\ell)$ (where $\\varsigma$ is defined as in Theorem \\ref{thm-factor-imGE}). Then the sum of the components of $\\ell$ is the writhe of $\\beta$.\n\\end{lemma}\n\\begin{proof}\n\nNote that $\\varkappa: \\mathbb{Z}^n\\rtimes S_n\\to\\mathbb{Z}$ defined by $\\varkappa((\\pi, \\ell)) := \\sum_i \\ell_i$ is a group homomorphism.\nWe know that $\\varkappa$ is a group homomorphism since, for any $\\pi, \\ell, \\pi', \\ell'$, we have \\begin{align*}\n    \\varkappa((\\pi, \\ell)(\\pi', \\ell')) & = \\varkappa((\\pi\\pi', \\ell\\pi' + \\ell'))\\\\\n    &= \\sum_i (\\ell(\\pi'(i)) + \\ell'(i))\\\\\n    &= \\sum_i \\ell(\\pi'(i)) + \\sum_i \\ell'(i)\\\\\n    &= \\sum_i \\ell(i) + \\sum_i \\ell'(i)\\\\\n    &= \\varkappa((\\pi, \\ell)) + \\varkappa((\\pi', \\ell')),\n\\end{align*}\nas desired.\n\n\nHence $\\varkappa\\circ\\varsigma$ is a group homomorphism from $B_n$ to $\\mathbb{Z}$.\n\nWe also note that writhe is a group homomorphism from $B_n$ to $\\mathbb{Z}$.\nFurthermore, it is easy to verify that $\\varkappa\\circ\\varsigma(\\sigma_k) = |\\sigma_k| = 1$.\nSince $\\varkappa\\circ\\varsigma$ and writhe agree on the generators of $B_n$, they must agree everywhere by induction, and so be the same.\n\\end{proof}\nTo fully characterize imGE, we begin with the special case of pure braids.\n\n\\begin{lemma}\\label{lem-all-evensum-linkvecs-possible}\nFor any $\\ell\\in\\mathbb{Z}^n$, there is a pure braid $\\beta$ with $\\ell_{\\beta} = \\ell$ iff $\\ell$ has even sum.\n\\end{lemma}\n\\begin{proof}\n\\textbf{Necessity}: Suppose that there exists such a pure braid $\\beta$. Then $\\sum_a\\ell_a = |\\beta|$, and the latter is even since $\\beta$ is pure. Hence $\\sum_a\\ell_a$ must be pure, and we are done.\n\n\\textbf{Sufficiency}:\nLet $\\beta_1$ and $\\beta_2$ be arbitrary pure braids. Then $\\ell_{\\beta_1\\beta_2} = \\ell_{\\beta_1} + \\ell_{\\beta_2}$, and $\\ell_{\\beta_1^{-1}} = -\\ell_{\\beta_1}$.\nHence the set of all possible $\\ell_{\\beta}$ values for braids $\\beta$ forms a lattice in $\\mathbb{Z}^n$.\n\nWe compute that $\\ell_{\\sigma_k^2} = (0, \\dots, 0, 1, 1, 0, \\dots, 0)$, where the $1, 1$ is in the $k$th and $(k+1)$th places and where we have identified $\\ell$ with a vector in $\\mathbb{Z}^n$.\nSimilarly, we compute $\\ell_{(\\sigma_k\\sigma_{k + 1})^3} = (0, \\dots, 0, 2, 2, 2, 0, \\dots, 0)$.\n\nSince $(a, a + b - 2c, c) = (a - 2c)(1, 1, 0) + (b - 2c)(0, 1, 1) + c(2, 2, 2)$, we conclude that these vectors span the lattice of all elements of $\\mathbb{Z}^n$ with even sum, and we are done.\n\\end{proof}\n\nThen, we will combine this result with some further lemmas to characterize the image in Theorem \\ref{thm-imGE}.\nNow we can fully characterize imGE.\n\n\\begin{theorem}\\label{thm-imGE}\nLet $\\pi\\in S_n$ and $\\ell\\in\\mathbb{Z}^n$.\nThen there exists a braid $\\beta$ such that $\\pi_{\\beta} = \\pi, \\ell_{\\beta} = \\ell$ iff the sum of $\\ell$ has the same parity as $\\pi$.\n\\end{theorem}\n\\begin{proof}\n\\textbf{Necessity}: Suppose such a braid exists. By Lemma \\ref{lem-sum-link-writhe}, the sum of $\\ell = \\ell_{\\beta}$ must be the writhe of $\\beta$.\nThis must have the same parity as $\\pi_{\\beta} = \\pi$ by Theorem \\ref{thm-structure-imWP}, and we are done.\n\n\\textbf{Sufficiency}: Suppose that the sum of $\\ell$ has the same parity as $\\pi$.\nWe can construct a braid $\\beta_1$ with permutation $\\pi$. By Theorem \\ref{thm-structure-imWP} and Lemma \\ref{lem-sum-link-writhe}, $\\ell_{\\beta_1}$ must have a sum with the same parity as $\\pi$.\nBy Lemma \\ref{lem-all-evensum-linkvecs-possible}, there is a pure braid $\\beta_2$ such that $\\ell_{\\beta_2} = \\ell - \\ell_{\\beta_1}$ (since the right hand side has even sum).\nThen the braid $\\beta_3 := \\beta_1\\beta_2$ has permutation $\\pi$ and vector $\\ell$, as desired, and we are done.\n\\end{proof}\n\n\\subsection{Kernel of GE}\\label{subsec-kerGE}\n\n\\hspace{\\parindent}It is natural to consider the kernel of the Gauss-Epple homomorphism.\nWe find that imWP is a quotient group of imGE, as kerWP is a supergroup of kerGE.\n\nSince the image of GE is not $B_n$, the kernel of GE must necessarily be nontrivial.\nIn fact, by computational means, we found several explicit examples:\n\\begin{itemize}\n    \\item $\\sigma_1^2\\sigma_2^2\\sigma_1^{-2}\\sigma_2^{-2}$ (which corresponds to the Whitehead link \\cite{WMW-WhiteheadLink});\n    \\item  $\\sigma_1^{-1}\\sigma_3^{-1}\\sigma_2^2\\sigma_3^{-1}\\sigma_1^{-1}\\sigma_2^2$;\n    \\item $(\\sigma_1\\sigma_2^{-1})^3$;\n    \\item $(\\sigma_2\\sigma_1^{-1})^3$;\n    \\item $\\sigma_1\\sigma_2^{-1}\\sigma_1^2(\\sigma_1\\sigma_2^{-1})^2\\sigma_2^{-2}$;\n    \\item $(\\sigma_1\\sigma_2\\sigma_1^2\\sigma_2^{-1})^2\\sigma_1^{-2}$.\n\\end{itemize}\n\n\n\\hspace{\\parindent}We can make several observations about the structure of kerGE: \nWe know that the kernel of the Gauss-Epple action must be a subgroup of the group of pure braids, which in turn is a subgroup of the braid group $B_n$.\nSince $B_n$ has no torsion, the Gauss-Epple kernel must also have no torsion and be infinite.\n\n\\subsubsection{Random braids}\n\n\\hspace{\\parindent}We study the probability that a random braid of $n$ generators lies in the kernel of the Gauss-Epple action.\n\nLet $G$ be an arbitrary group and $H$ an arbitrary normal subgroup. \nThen, for all $g\\in G$, we know that $g\\in H$ iff $\\phi(g) = e$, where $\\phi$ is the quotient map $G\\to G\/H$.\n(This effectively reduces our problem to studying the quotient group $K := G\/H$.)\nNow, defining $V(N)$ to be the number of elements of $K$ that can be produced from words of length $N$, we have the following deep theorem:\n\\begin{theorem}\\label{thm:randwalkprob-vertcount-asy} \\cite{alexopoulos1997convolution, RWIGGreview, AMSNotice200109}\nFor any $d$, the probability of a random walk on $K$ returning to the identity on the $N$th step is on the order of $N^{-d\/2}$ iff $V(N)$ is comparable to $N^d$.\n\\end{theorem}\n\nTo apply this theorem, we take $G = B_n, H = \\ker GE, K = \\im GE$.\nSince $V(N)$ is comparable to $N^n$ thanks to Theorem \\ref{thm-imGE}, we conclude by Theorem \\ref{thm:randwalkprob-vertcount-asy} that the probability without filtering of an element being in the kernel of Gauss-Epple is asymptotically on the order of $N^{-n\/2}$, which is the result we seek.\n\n\\section{The symmetric Gauss-Epple homomorphism}\\label{sec-symGE}\n\n\\hspace{\\parindent}In his paper \\cite{Epple98}, Epple introduced yet another action of the braid group, which he called the \\textit{symmetric Gauss-Epple homomorphism}. This object is defined as follows:\n\n\\begin{definition}[Symmetric Gauss-Epple action]\nFor any $n\\in\\mathbb{N}$, the symmetric Gauss-Epple action $symGE: B_n\\times \\mathbb{Z}^2\\to \\mathbb{Z}^2$ is the unique left group action of $B_n$ (with canonical Artin generators $\\sigma_1, \\sigma_2, \\dots$) on $\\mathbb{Z}^2$ defined by the following relation:\n\n$$\\forall k\\in[n], a, b\\in\\mathbb{Z}:\n\\symGE(\\sigma_k, (a, b)) = \\begin{cases}\n(a, b) & a\\notin\\{k, k + 1\\}\\\\\n(k + 1, b + 1) & a = k\\\\\n(k, b + 1) & a = k + 1\n\\end{cases}.$$\n\\end{definition}\n\nFor similar reasons as in the proof of Theorem \\ref{thm-factor-imGE}, the symmetric Gauss-Epple action (which we abbreviate symGE) also factors through $\\mathbb{Z}^n\\rtimes S_n$ into $\\Sym(\\mathbb{Z}^2)$:\n\n\\begin{theorem}[Structure theorem for imGE]\\label{thm-factor-imsymGE}\nDefine the group inclusion $\\iota: \\mathbb{Z}^n\\rtimes S_n\\to\\Sym([n]\\times\\mathbb{Z})$ by the equation $(\\pi, \\ell)\\to ((a, b)\\to (\\pi(a), b + \\ell(a))$,\nand define the group surjection $\\varpi: B_n\\to\\mathbb{Z}^n\\rtimes S_n$ by \n$$\\varpi(\\sigma_k) := (\\vec{e_k} + \\vec{e_{k + 1}}, (k, k + 1)).$$\nThen $symGE = \\iota\\circ\\varpi$.\n\\end{theorem}\n\\begin{proof}\nTrivial.\n\\end{proof}\n\nWe then note that the symmetric Gauss-Epple action and the Gauss-Epple action have the same kernel:\n\n\\begin{theorem}\nThe kernel of the symmetric Gauss-Epple action is the same as the kernel of the ordinary Gauss-Epple action.\n\\end{theorem}\n\\begin{proof}\nSuppose $\\beta$ is an arbitrary pure braid. We show that $GE(\\beta) = (\\ell, \\text{id}) \\leftrightarrow \\symGE(\\beta) = (2\\ell, \\text{id})$.\nIt is clear that the ``permutation parts'' are all identity, so we only focus on the ``vector parts''.\nIt suffices to then show this claim for the the generators of the pure braid group $P_n$.\nNote that $P_n$ (the pure braid group on $n$ strands) is generated by $A_{i, j} := \\sigma_{j-1}\\dots\\sigma_{i+1}\\sigma_i^2\\sigma_{i+1}^{-1}\\dots\\sigma_{j-1}^{-1}$ \\cite{Suciu16}; for example, the generators for $P_3$ are $\\sigma_1^2, \\sigma_2^2, \\sigma_1\\sigma_2^2\\sigma_1^{-1}.$\n\nWe compute iteratively that $GE(A_{i, j}) = (\\vec{e_i} + \\vec{e_j}, \\text{id}), \\symGE(A_{i, j}) = (2\\vec{e_i} + 2\\vec{e_j}, \\text{id})$. Hence the claim holds, and we are done.\n\\end{proof}\n\n\\begin{remark}\nBy the First Isomorphism Theorem, the previous theorem implies that the images of the symmetric Gauss-Epple action and the images of the regular Gauss-Epple action must be isomorphic as groups.\n\\end{remark}\n\n\\section{The super-Gauss-Epple homomorphism}\\label{sec-superGE}\n\n\\hspace{\\parindent}We define another homomorphism, this time of type signature $B_n\\to \\mathbb{Z}^{n(n-1)}\\rtimes S_n$, which we name the \\textit{super-Gauss-Epple homomorphism} (abbreviated SGE).\nMore precisely, define $O_{i, j}$ to be the matrix with a 1 entry at the $(i, j)$th place and 0 entries everywhere else. \nThen we define SGE as follows:\n\\begin{definition}[super-Gauss-Epple homomorphism]\\label{def-SGE}\nThe super-Gauss-Epple homomorphism $SGE: B_n\\to \\mathbb{Z}^{n(n-1)}\\rtimes S_n$ is defined by the following equation:\n\n$$SGE(\\sigma_i) := (O_{i, i + 1}, (i, i + 1)).$$\n\\end{definition}\n\n(Here, by slight abuse of notation, we consider elements of $\\mathbb{Z}^{n(n-1)}$ to be $n\\times n$ matrices with all zeros along the diagonal.)\nAn example calculation of SGE is depicted in Figure \\ref{fig-calc-SGE}.\n\n\\begin{figure}[h]\n    \\begin{center}\\includegraphics[width=0.6\\textwidth]{SGE_calculation.png}\\end{center}\n    \\caption{An example calculation of SGE.}\n    \\label{fig-calc-SGE}\n\\end{figure}\n\nTo verify that this is indeed a homomorphism, we verify the braid relations (as is sufficient and necessary):\n\n\\begin{align*}\n    SGE(\\sigma_i\\sigma_{i+1}\\sigma_i) &= SGE(\\sigma_i)\\cdot SGE(\\sigma_{i+1})\\cdot SGE(\\sigma_i)\\\\\n    &= (O_{i, i + 1}, (i, i + 1))\\cdot(O_{i + 1, i + 2}, (i + 1, i + 2))\\cdot(O_{i, i + 1}, (i, i + 1))\\\\\n    &= (O_{i, i + 1} + (i, i+1)\\cdot O_{i + 1, i + 2} + (i, i+1)(i+1, i+2)\\cdot O_{i, i + 1}, \\\\\n    &\\hspace{15pt}(i, i+1)(i+1, i+2)(i, i+1))\\\\\n    &= (O_{i, i + 1} + O_{i, i + 2} + O_{i + 1, i + 2},(i, i+1)(i+1, i+2)(i, i+1))\\\\\n    &= (O_{i, i + 1} + O_{i, i + 2} + O_{i + 1, i + 2}, (i, i+2, i+1)),\n\\end{align*}\nand\n\\begin{align*}\n    SGE(\\sigma_{i+1}\\sigma_i\\sigma_{i+1}) &= SGE(\\sigma_{i+1})SGE(\\sigma_i)SGE(\\sigma_{i+1})\\\\\n    &= (O_{i + 1, i + 2}, (i + 1, i + 2))\\cdot(O_{i, i + 1}, (i, i + 1))\\cdot(O_{i + 1, i + 2}, (i + 1, i + 2))\\\\\n    &= (O_{i + 1, i + 2} + (i+1, i+2)\\cdot O_{i, i + 1} + (i+1, i+2)(i, i+1)\\cdot O_{i + 1, i + 2},\\\\\n    &\\hspace{15pt}(i+1, i+2)(i, i+1)(i+1, i+2))\\\\\n    &= (O_{i, i + 1} + O_{i, i + 2} + O_{i + 1, i + 2},\\\\\n    &\\hspace{15pt}(i, i+1)(i+1, i+2)(i, i+1))\\\\\n    &= (O_{i, i + 1} + O_{i, i + 2} + O_{i + 1, i + 2}, (i, i+2, i+1)),\n\\end{align*}\nas desired.\n\nA braid's image under the super-Gauss-Epple homomorphism stores at least as much information about it as the ordinary Gauss-Epple homomorphism. Note that this is strictly more information, as we can give the explicit example $\\sigma_2^2\\sigma_3^{-1}\\sigma_2\\sigma_1^2\\sigma_2^{-1}\\sigma_3^{-1}\\sigma_1^{-2}$.\n\n(By the Third Isomorphism Theorem, this also follows iff the image of the super-Gauss-Epple homomorphism is not isomorphic to that of the ordinary Gauss-Epple homomorphism.)\n\nIt is to be noted that the kernel of SGE is still nontrivial.\nFor example, it contains the braid $(\\sigma_1\\sigma_2^{-1})^3$, which was mentioned before as being a nontrivial example of the kernel of regular GE.\n\nHence, we have the following commutative diagram:\n\n\\[\\begin{tikzcd}\n\t&& {B_n\/[P_n, P_n]} \\\\\n\t{B_n} && {P_n^{ab}\\rtimes S_n} & {\\Sym(\\mathbb{Z}\\times\\binom{n}{2})} \\\\\n\t&& {\\mathbb{Z}^n\\rtimes S_n} & {\\Sym(\\mathbb{Z}\\times[n])} \\\\\n\t&& {\\mathbb{Z}\\times S_n}\n\t\\arrow[from=1-3, to=2-3]\n\t\\arrow[from=2-3, to=3-3]\n\t\\arrow[from=3-3, to=4-3]\n\t\\arrow[\"UGE\", from=2-1, to=1-3]\n\t\\arrow[\"SGE\", from=2-1, to=2-3]\n\t\\arrow[\"GE\", from=2-1, to=3-3]\n\t\\arrow[\"WP\", from=2-1, to=4-3]\n\t\\arrow[from=2-3, to=2-4]\n\t\\arrow[from=3-3, to=3-4]\n\\end{tikzcd}\\]\n\n\\subsection{A 1-cocycle}\\label{section-braids-1cocy}\n\n\\hspace{\\parindent}Firstly, note that the matrix of (the super-Gauss-Epple homomorphism of) a pure braid is always symmetric:\n\n\\begin{lemma}\\label{lem-SGE-pure-symmetric}\nLet $\\beta\\in P_n$, and suppose that $SGE(\\beta) = (M, \\text{id})$. Then $M$ is a symmetric matrix.\n\\end{lemma}\n\\begin{proof}\nRecall that $P_n$ (the pure braid group on $n$ strands) is generated by \n$$A_{i, j} := \\sigma_{j-1}\\dots\\sigma_{i+1}\\sigma_i^2\\sigma_{i+1}^{-1}\\dots\\sigma_{j-1}^{-1}$$ \\cite{Suciu16}; for example, the generators for $P_3$ are $\\sigma_1^2, \\sigma_2^2, \\sigma_1\\sigma_2^2\\sigma_1^{-1}.$\n\nNote that $SGE(A_{i, j}) = (O_{i, j} + O_{j, i}, id)$, and the right component is clearly symmetric.\nSince SGE matrices add for pure braids, the conclusion holds by induction.\n\\end{proof}\n\nHence, the difference between the upper half and the (transposed) lower half is always only determined by the braid permutation.\n\nNow, let $F: B_n\\to M_{n, n}(\\mathbb{Z})$ be defined so that $SGE(\\beta) = (M, s_{\\beta}) \\implies F(\\beta) = M - M^T$.\nThen $F(\\beta) = 0\\forall\\beta\\in P_n$.\nFurthermore, $F$ satisfies the equality $F(\\beta\\gamma) = F(\\beta) + M_{s_{\\beta}}F(\\gamma)M_{s_{\\beta}}^{-1}$.\nHence $F(\\pi\\beta) = F(\\beta) + id\\cdot F(\\beta) = F(\\beta)~\\forall\\beta\\in B_n, \\pi\\in P_n$, so that the value of $F(\\beta)$ only depends on the permutation of the braid $\\beta$, and that in general $F$ projects down to a function $\\overline{F}: S_n\\to M_{n, n}(\\mathbb{Z})$.\nFurthermore, we have $\\overline{F}(\\beta\\gamma) = F(\\beta) + M_{s_{\\beta}}F(\\gamma)M_{s_{\\beta}}^{-1}$ (which we shall abbreviate as $\\overline{F}(\\beta\\gamma) = \\overline{F}(\\beta) + \\beta\\cdot\\overline{F}(\\gamma)$. \nTherefore, $\\overline{F}$ is a 1-cocycle of the action of $S_n$ on $M_{n, n}(\\mathbb{Z})$, and in fact of the action on the set of antisymmetric $n\\times n$ matrices over the integers.\n\nWhile computing values of $\\overline{F}$ for some random braids, we noticed that the entries in the upper half were always in $\\{0, 1\\}$. We prove this inductively as follows:\n\n\\begin{theorem}\\label{thm-overlineF-upperhalf-zeroone}\nLet $\\varrho$ be an arbitrary permutation, and $i, j$ be integers with $i > j$. Then $\\overline{F}(\\varrho)[i, j]\\in\\{0, 1\\}$.\n\\end{theorem}\n\\begin{proof}\nWe assume this for some $\\varrho$ and prove it for $\\tau\\varrho$, where $\\tau := (\\iota, \\iota + 1)$ is a transposition.\n\nSince $\\overline{F}(\\tau\\varrho) = \\overline{F}(\\tau) + \\tau\\cdot\\overline{F}(\\varrho)t$, the conjecture is instantly verified for all $i, j$ except for $\\iota, \\iota + 1$ (since the first term is trivially zero and the second term is in $\\{0, 1\\}$ by the inductive hypothesis).\nIn the special case $i = \\iota, j = \\iota + 1$, we compute \\begin{align*}\n\\overline{F}(\\tau\\varrho)[\\iota, \\iota + 1] &= \\overline{F}(\\tau)[\\iota, \\iota + 1] + (\\tau\\cdot\\overline{F}(\\varrho)[\\iota, \\iota]\\\\\n&= 1 - \\overline{F}(\\varrho)[\\iota, \\iota + 1]\\\\\n&\\in\\{0, 1\\}.\n\\end{align*}\n(Here, we use the inductive hypothesis $\\overline{F}(\\varrho)[\\iota, \\iota + 1]\\in\\{0, 1\\}$.)\nHence the claim holds.\n\nBy induction on the minimum number of simple transpositions required to be multiplied together to represent an arbitrary permutation, we are then done.\n\\end{proof}\n\nThis means that the image of the super-Gauss-Epple homomorphism is a group with $\\mathbb{Z}^{\\frac{n(n-1)}{2}}$ as a normal subgroup and $S_n$ as the quotient group.\nNote that it is \\textit{not} a semidirect product, since no braid with permutation $(1,2)$ has a square in the kernel of the super-Gauss-Epple homomorphism.\n(Indeed, suppose such a braid $\\beta = \\sigma_1\\varpi$ existed, and let $SGE(\\beta) = (M, (1, 2))$ for some matrix $M$. Then the matrix of $SGE(\\beta^2)$ is $M + (1, 2)\\cdot M$. But since $M[1, 2]$ is one more than $((1, 2)\\cdot M)[1, 2] = M[2, 1]$ and both are integers, the two cannot sum to zero, which is a contradiction.)\n\n\\section{Artin groups of crystallographic type}\\label{sec-ATgroupsWeyl}\n\n\\hspace{\\parindent}In this section, we explore generalizations of the Gauss-Epple homomorphism to the setting of Artin groups, which naturally project to corresponding Coxeter groups (generalizing how the braid group projects to the symmetric group).\n\n\\subsection{Special cases}\n\n\\subsubsection{The $I_2(4), I_2(6)$ cases}\n\n\\hspace{\\parindent}This are special cases of the \\textit{dihedral Artin groups}, which are the Artin groups of type $I_2(2n)$ for integers $n$.\nThe abelianization is $\\mathbb{Z}^2$ and the Coxeter group is $D_{2n}$. Hence, the generators map to $D_{2n}\\times\\mathbb{Z}^2$ as follows:\n\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline\ngenerator & $D_{2n}\\times\\mathbb{Z}^2$\\\\\n\\hline\n$a$ & $(s, (1, 0))$\\\\\n$b$ & $(sr, (0, 1))$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\nFor GE-like purposes, we may consider $D_{2n}$ as a subset of $S_n$.\n\nFor the case $n = 4$, with $a\\to (13), b\\to (01)(23)$ in permutation notation, this gives rise to the following system of equations:\n\n$$\\begin{cases}\n\\ell_{a, 1} + \\ell_{b, 2} + \\ell_{a, 2} + \\ell_{b, 3}  &= \\ell_{b, 1} + \\ell_{a, 1} + \\ell_{b, 4} + \\ell_{a, 4} \\\\\n\\ell_{a, 2} + \\ell_{b, 1} + \\ell_{a, 3} + \\ell_{b, 2}  &= \\ell_{b, 2} + \\ell_{a, 4} + \\ell_{b, 1} + \\ell_{a, 3} \\\\\n\\ell_{a, 3} + \\ell_{b, 4} + \\ell_{a, 4} + \\ell_{b, 1}  &= \\ell_{b, 3} + \\ell_{a, 3} + \\ell_{b, 2} + \\ell_{a, 2} \\\\\n\\ell_{a, 4} + \\ell_{b, 3} + \\ell_{a, 1} + \\ell_{b, 4}  &= \\ell_{b, 4} + \\ell_{a, 2} + \\ell_{b, 3} + \\ell_{a, 1} \\\\\n\\end{cases}$$\n(We used a computer program to generate this system of equations.)\n\nSolving this system manually reveals a six-dimensional space: \n$$\\ell_a = (\\ell_{a, 1}, \\ell_{a, 2}, \\ell_{a, 3}, \\ell_{a, 2}), \\ell_b = (\\ell_{b, 1}, \\ell_{b, 2}, \\ell_{b, 3}, \\ell_{b, 2} + \\ell_{b, 3} - \\ell_{b, 1}).$$\n\nSimilarly, for the case $n = 6$ (also denoted $G_2$), we make the ansatz $a\\to(26)(35), b\\to(12)(36)(45)$, based on an action of the Coxeter group on the hexagon.\nWith the map $a\\to\\ell_a:= (\\ell_{a, 1}, \\ell_{a, 1}, \\dots, \\ell_{a, 6}), b\\to\\ell_b := (\\ell_{b, 1}, \\dots, \\ell_{b, 6})$, this produces the following system of linear equations:\n\n$$\\begin{cases}\n\\ell_{a, 1} + \\ell_{b, 1} + \\ell_{a, 2} + \\ell_{b, 6} + \\ell_{a, 3} + \\ell_{b, 5} &= \\ell_{b, 1} + \\ell_{a, 2} + \\ell_{b, 6} + \\ell_{a, 3} + \\ell_{b, 5} + \\ell_{a, 4}\\\\\n\\ell_{a, 2} + \\ell_{b, 6} + \\ell_{a, 3} + \\ell_{b, 5} + \\ell_{a, 4} + \\ell_{b, 4} &= \\ell_{b, 2} + \\ell_{a, 1} + \\ell_{b, 1} + \\ell_{a, 2} + \\ell_{b, 6} + \\ell_{a, 3}\\\\\n\\ell_{a, 3} + \\ell_{b, 5} + \\ell_{a, 4} + \\ell_{b, 4} + \\ell_{a, 5} + \\ell_{b, 3} &= \\ell_{b, 3} + \\ell_{a, 6} + \\ell_{b, 2} + \\ell_{a, 1} + \\ell_{b, 1} + \\ell_{a, 2}\\\\\n\\ell_{a, 4} + \\ell_{b, 4} + \\ell_{a, 5} + \\ell_{b, 3} + \\ell_{a, 6} + \\ell_{b, 2} &= \\ell_{b, 4} + \\ell_{a, 5} + \\ell_{b, 3} + \\ell_{a, 6} + \\ell_{b, 2} + \\ell_{a, 1}\\\\\n\\ell_{a, 5} + \\ell_{b, 3} + \\ell_{a, 6} + \\ell_{b, 2} + \\ell_{a, 1} + \\ell_{b, 1} &= \\ell_{b, 5} + \\ell_{a, 4} + \\ell_{b, 4} + \\ell_{a, 5} + \\ell_{b, 3} + \\ell_{a, 6}\\\\\n\\ell_{a, 6} + \\ell_{b, 2} + \\ell_{a, 1} + \\ell_{b, 1} + \\ell_{a, 2} + \\ell_{b, 6} &= \\ell_{b, 6} + \\ell_{a, 3} + \\ell_{b, 5} + \\ell_{a, 4} + \\ell_{b, 4} + \\ell_{a, 5}\\\\\n\\end{cases}$$\n\nThis system is highly symmetric and many of the variables cancel. \nTherefore, we can solve it exactly, producing the following linear parametrization of the entire solution space over the integers, which is nine-dimensional: \n$$\\ell_a = (a_1, a_2, a_3, a_1, a_2 + x_2, a_3 + x_2), \\ell_b = (b_1, b_2, b_3, b_1 + y_1, b_2 - y_1, b_3 + y_3),$$\nwhere $a_1, a_2, a_3, b_1, b_2, b_3, x_2, y_1, y_3$ range independently over $\\mathbb{Z}$.\n\n\\subsubsection{The $B_n$ case}\n\n\\hspace{\\parindent}The isomorphism between the Coxeter group of this case and $C_2\\wr S_n = C_2^n\\rtimes S_n$ can be tabulated in generators (where we list the generators of the Coxeter graph as $a_1, a_2, \\dots, a_n$) as follows:\n\n\\begin{center}\n    \\begin{tabular}{|c|c|c|}\n    \\hline\n    Coxeter & $S_n$ & $C_2^n$\\\\\n    \\hline\n    $a_1$ & id & $(1, 0, \\dots, 0)$\\\\\n    $a_2$ & (1, 2) & $(0, 0, \\dots, 0)$\\\\\n    $a_3$ & (2, 3) & $(0, 0, \\dots, 0)$\\\\\n    \\vdots & \\vdots & \\vdots\\\\\n    $a_n$ & ($n - 1$, $n$) & $(0, 0, \\dots, 0)$\\\\\n    \\hline\n    \\end{tabular}\n\\end{center}\n\nIn the classical case of the braid group $B_3$, the space of possible ``linking vector'' analogues is four-dimensional, and spanned by ``symmetric GE'' ($\\sigma_1 \\mapsto (1, 1, 0), \\sigma_2 \\mapsto (0, 1, 1)$), ``row GE'' ($\\sigma_1 \\mapsto (0, 0, 1), \\sigma_2 \\mapsto (1, 0, 0)$), ``zero-ish GE 1'' ($\\sigma_1 \\mapsto (1, -1, 0), \\sigma_2 \\mapsto 0$), and ``zero-ish GE 2'' ($\\sigma_1 \\mapsto 0, \\sigma_2 \\mapsto (0, 1, -1)$). We can derive this by noting that if we have $\\sigma_1 \\mapsto (a_1, b_1, c_1), \\sigma_2 \\mapsto (a_2, b_2, c_2)$, then the single braiding relation $\\sigma_1\\sigma_2\\sigma_1 = \\sigma_2\\sigma_1\\sigma_2$ gives us $c_1 = a_2$ and $a_1 + b_1 = b_2 + c_2$ as constraints after simplification; some elementary algebra then gives us the four basis vectors. \n\nAs for the images of the corresponding homomorphisms of $B_3$ over the pure braids only, the image of symmetric GE is the set of vectors of 3 even integers which sum to 0 modulo 4, the image of row GE is just the set of vectors of 3 even integers, and the images of the two ``zero-ish'' analogs are just all zero.\n\nAs for higher braid groups $B_n$, the ``far commutativity relations'' $\\sigma_i\\sigma_j = \\sigma_j\\sigma_i$ for $|i - j|\\geq 2$ basically constrain all the entries of the ``linking vector'' corresponding to $\\sigma_i$ (which I'll call $\\ell_i$ by analogy) right of the $(i+1)^{\\text{th}}$ to be the same within, and the entries of $\\ell_i$ left of the $i^{\\text{th}}$ to also be the same within. Combined with the braid relations $\\sigma_i\\sigma_{i+1}\\sigma_i = \\sigma_{i+1}\\sigma_i\\sigma_{i+1}$, we get $\\ell_{i, i+2} = \\ell_{i + 1, i}$ as well as $\\ell_{i, i} + \\ell_{i, i+1} = \\ell_{i+1, i+1} + \\ell_{i+1, i+2}$. Hence the set of possible $\\ell_i$ combinations can be spanned by the following:\n\n\\begin{itemize}\n\\item Classical symmetric GE ($\\ell_i = e_i + e_{i + 1}$)\n\\item Row GE for row $i_0$ ($\\ell_{i_0} = \\sum_{i: i > i_0 + 1} e_i, \\ell_{i_0 + 1} = \\sum_{i: i < i_0} e_i, \\ell_i = 0$ elsewise)\n\\item Zero-ish GE for row $i_0$ ($\\ell_{i_0} = e_{i_0} - e_{i_0 + 1}, \\ell_i = 0$ elsewise)\n\\end{itemize}\n\nThere are also corresponding 1-cocycles. We can take linear combinations thereof to get even more 1-cocycles since $\\mathbb{Z}^3$ is abelian.\n\n\\subsection{Super-Gauss-Epple analogues}\n\n\\hspace{\\parindent}Since $\\mathcal{C}$ acts on $\\Phi$, this induces an action of $\\mathcal{C}$ on $\\mathbb{Z}^\\Phi$.\nHence, there is a homomorphism $\\mathcal{A}\\to\\mathbb{Z}^{\\Phi}\\rtimes\\mathcal{C}$, defined by $a\\mapsto(\\alpha_a, \\mathcal{C}(a))$, where $\\mathcal{C}(a)$ is the image of $a$ under the quotient map $\\mathcal{A}\\to\\mathcal{C}$; by analogy, we shall also denote it $SGE$, or $SGE_{\\mathcal{A}}$ when necessary.\n\nThis is a generalization of the super-Gauss-Epple homomorphism, when $\\mathcal{A} = B_n, \\mathcal{C} = S_n$.\nHere, the root system $\\Phi$ is $\\{\\vec{e_i} - \\vec{e_j}|1\\leq i\\neq j\\leq n\\}$, which has size $n(n-1)$, and we identify the $\\vec{e_i} - \\vec{e_j}$ component of $\\mathbb{Z}^{\\Phi}$ with the $(i, j)$th component of a matrix whose diagonal elements are all zero. \n\nWe can show that this is indeed a homomorphism with geometric arguments.\n\n\\begin{theorem}\\label{thm-SGE-welldef-Artingroups}\nThere is a unique homomorphism $\\mathcal{A}\\to\\mathbb{Z}^{\\Phi}\\rtimes\\mathcal{C}$, defined by $a\\mapsto(\\alpha_a, \\mathcal{C}(a))$, where $\\mathcal{C}(a)$ is the image of $a$ under the quotient map $\\mathcal{A}\\to\\mathcal{C}$.\n\\end{theorem}\n\\begin{proof}\nWe show that the braid relations of $\\mathcal{A}$ are satisfied.\nTo do this, we show that the $\\mathbb{Z}^{\\Phi}$ sides are identical.\nAs these values are formal linear combinations of roots (elements of $\\Phi$), we also consider them as multisets, and prove them equal accordingly.\n(We do not consider the $\\mathcal{C}$ sides because these are trivially equal due to the fact that there is a bona-fide homomorphism $\\mathcal{A} \\to \\mathcal{C}$.)\n\nLet $a_1$ and $a_2$ be two generators of $\\mathcal{A}$ for which there is a braid relation.\nFor notational simplicity, we will refer to the corresponding vectors as $\\vec{v}_1 := \\mathcal{C}(a_1), \\vec{v}_2 := \\mathcal{C}(a_2)$.\nWe shall also introduce the notation $*$.\nFor two arbitrary vectors $\\vec{v}$ and $\\vec{w}$, we use $\\vec{v} * \\vec{w}$ to denote the vector that results from reflecting $\\vec{w}$ over the orthogonal hyperplane of $\\vec{v}$.\nWe will compose it right-associatively, so that $\\vec{v} * \\vec{w} * \\vec{x}$ shall refer to $\\vec{v} * (\\vec{w} * \\vec{x})$.\nWe now perform casework depending on the length of this braid relation:\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[height=0.2\\textwidth]{ArtinGenRelLen2.png}\n    \\caption{Two vectors for Coxeter group generators with a braid relation of length 2.}\n    \\label{fig:artingenrellen2}\n\\end{figure}\n\\textbf{Length 2}: We have $a_1 a_2 = a_2 a_1$, so $\\vec{v}_1$ and $\\vec{v}_2$ are orthogonal, as shown in Figure \\ref{fig:artingenrellen2}.\nWe find that $\\vec{v}_1 * \\vec{v}_2 = \\vec{v}_2$ and that $\\vec{v}_2 * \\vec{v}_1 = \\vec{v}_1$.\nTherefore, we have $\\{\\vec{v}_1, \\vec{v}_1 * \\vec{v}_2\\} = \\{\\vec{v}_2, \\vec{v}_2 * \\vec{v}_1\\}$, as desired.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[height=0.19\\textwidth]{ArtinGenRelLen3.png}\n    \\caption{Two vectors for Coxeter group generators with a braid relation of length 3.}\n    \\label{fig:artingenrellen3}\n\\end{figure}\n\\textbf{Length 3}: We have $a_1 a_2 a_1 = a_2 a_1 a_2$, so $\\vec{v}_1$ and $\\vec{v}_2$ have an angle of $120^\\circ$ between them, as shown in Figure \\ref{fig:artingenrellen3}.\nWe find that $\\vec{v}_1 * \\vec{v}_2 = \\vec{v}_2 * \\vec{v}_1$ (both being the vector that bisects the angle between $\\vec{v}_1$ and $\\vec{v}_2$).\nFurthermore, we find that $\\vec{v}_1 * (\\vec{v}_2 * \\vec{v}_1) = \\vec{v}_2$ and $\\vec{v}_2 * (\\vec{v}_1 * \\vec{v}_2) = \\vec{v}_1$.\nTherefore, we have $\\{\\vec{v}_1, \\vec{v}_1 * \\vec{v}_2, \\vec{v}_1 * (\\vec{v}_2 * \\vec{v}_1)\\} = \\{\\vec{v}_2, \\vec{v}_2 * \\vec{v}_1, \\vec{v}_2 * (\\vec{v}_1 * \\vec{v}_2)\\}$, as desired.\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[height=0.22\\textwidth]{ArtinGenRelLen4.png}\n    \\caption{Two vectors for Coxeter group generators with a braid relation of length 4.}\n    \\label{fig:artingenrellen4}\n\\end{figure}\n\\textbf{Length 4}: We have $(a_1 a_2)^2 = (a_2 a_1)^2$, so $\\vec{v}_1$ and $\\vec{v}_2$ have an angle of $135^\\circ$ between them, as shown in Figure \\ref{fig:artingenrellen4}.\nWe find that $\\vec{v}_1 * \\vec{v}_2$ has an angle of $45^\\circ$ from $\\vec{v}_1$ towards $\\vec{v}_2$, and is so orthogonal to $\\vec{v}_2$.\nSimilarly, $\\vec{v}_2 * \\vec{v}_1$ has an angle of $45^\\circ$ from $\\vec{v}_2$ towards $\\vec{v}_1$, and is so orthogonal to $\\vec{v}_1$.\nTherefore, we have $\\vec{v}_1 * (\\vec{v}_2 * \\vec{v}_1) = \\vec{v}_2 * \\vec{v}_1$ and $\\vec{v}_2 * (\\vec{v}_1 * \\vec{v}_2) = \\vec{v}_1 * \\vec{v}_2$.\nFurthermore, we have $\\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2 = \\vec{v}_2$ and $\\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1 = \\vec{v}_1$.\nTherefore, we conclude that $\\{\\vec{v}_1, \\vec{v}_1 * \\vec{v}_2, \\vec{v}_1 * (\\vec{v}_2 * \\vec{v}_1), \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2\\} = \\{\\vec{v}_2, \\vec{v}_2 * \\vec{v}_1, \\vec{v}_2 * (\\vec{v}_1 * \\vec{v}_2, \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1\\}$, as desired.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[height=0.22\\textwidth]{ArtinGenRelLen6.png}\n    \\caption{Two vectors for Coxeter group generators with a braid relation of length 6.}\n    \\label{fig:artingenrellen6}\n\\end{figure}\n\\textbf{Length 6}: We have $(a_1 a_2)^3 = (a_2 a_1)^3$, so $\\vec{v}_1$ and $\\vec{v}_2$ have an angle of $150^\\circ$ between them, as shown in Figure \\ref{fig:artingenrellen6}.\nDivide this angle by four evenly spaced vectors, named $\\vec{w}_1, \\vec{w}_2, \\vec{w}_3, \\vec{w}_4$, so that the order of the vectors is $\\vec{v}_1, \\vec{w}_1, \\vec{w}_2, \\vec{w}_3, \\vec{w}_4, \\vec{v}_2$ (with $30$-degree gaps between each consecutive pair).\nAs before, we can calculate that $\\vec{v}_1, \\vec{v}_1 * \\vec{v}_2, \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1, \\dots, \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2$ is precisely $\\vec{v}_1, \\vec{w}_1, \\vec{w}_2, \\vec{w}_3, \\vec{w}_4, \\vec{v}_2$, in that order.\nSimilarly, we can compute that $\\vec{v}_2, \\vec{v}_2 * \\vec{v}_1, \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2, \\dots, \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1$ is precisely $\\vec{v}_2, \\vec{w}_4, \\vec{w}_3, \\vec{w}_2, \\vec{w}_1, \\vec{v}_1$, in that order.\nHence, we have $\\{\\vec{v}_1, \\vec{v}_1 * \\vec{v}_2, \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1, \\dots, \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2\\} = \\{\\vec{v}_2, \\vec{v}_2 * \\vec{v}_1, \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2, \\dots, \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1\\}$ as desired, and we are done.\n\\end{proof}\n\n\n\\subsection{Another commutative diagram}\n\n\\hspace{\\parindent}We can summarize this in the following commutative diagram:\n\n\\[\\begin{tikzcd}\n\t{\\mathcal{P}} & {\\mathcal{A}} & {\\mathcal{C}} \\\\\n\t{\\mathcal{P}\/[\\mathcal{P}, \\mathcal{P}]} & {\\mathcal{A}\/[\\mathcal{P}, \\mathcal{P}]}\n\t\\arrow[two heads, from=1-1, to=2-1]\n\t\\arrow[hook', from=2-1, to=2-2]\n\t\\arrow[hook, from=1-1, to=1-2]\n\t\\arrow[\"SGE\", two heads, from=1-2, to=2-2]\n\t\\arrow[two heads, from=2-2, to=1-3]\n\t\\arrow[two heads, from=1-2, to=1-3]\n\\end{tikzcd}\\]\n\nHere, every arrow is either an inclusion map (indicated as being injective) or a quotient map (indicated as being surjective).\n\n\\subsection{More 1-cocycles}\n\n\\hspace{\\parindent}Again, suppose that $\\mathcal{A}$ is an Artin group of finite type, $\\mathcal{C}$ is its associated Coxeter group, $\\Phi$ is an associated root system, and $\\Delta\\subset\\Phi$ is an associated subset of simple roots (and let $\\Delta_a$ be the associated simple root for any Artin generator $a$ of $\\mathcal{A}$).\nDefine the function $F: \\mathcal{A}\\to\\mathbb{Z}^{\\Phi}$ by \n$$\\forall a\\in\\mathcal{A}: SGE(a) = (\\ell, \\mathcal{C}(a)) \\implies F(a) = \\ell - \\overline{\\ell},$$\nwhere $\\ell$ is a vector of $\\mathbb{Z}^{\\Phi}$ and $\\overline{\\ell}$ is the image of $\\ell$ under the rotation\/reflection of $\\mathbb{Z}^{\\Phi}$ given by $e_{\\alpha}\\to e_{-\\alpha}\\forall\\alpha\\in\\Phi$.\nIt is clear that $F$ satisfies the relation $F(ab) = F(a) + \\mathcal{C}(a)\\cdot F(b)$.\n\nThen, we have the following lemmas (generalizing those of Section \\ref{section-braids-1cocy}):\n\n\\begin{lemma}\\label{lem-Fbar-exists-artingroups}\nLet $F: \\mathcal{A}\\to\\mathbb{Z}^{\\Phi}$ be as defined for Artin groups of finite type, and let $p\\in\\mathcal{P}$ (where $\\mathcal{P}$ is the kernel of the quotient map $\\mathcal{A}\\to\\mathcal{C}$). Then $F(p) = \\vec{0}$, and hence the map $F$ descends to a map $\\overline{F}: \\mathcal{C}\\to\\mathbb{Z}^{\\Phi}$\n\\end{lemma}\n\\begin{proof}\nLet $a$ be an Artin generator.\nThen $F(a^2) = F(a) + \\mathcal{C}(a)\\cdot F(a) = (\\phi_{\\Delta_a} - \\phi_{-\\Delta_a}) + (\\phi_{-\\Delta_a} - \\phi_{\\Delta_a}) = \\vec{0}$.\n\nAgain, let $p\\in\\mathcal{P}$, $\\upsilon$ be an Artin generator of $\\mathcal{A}$, and suppose that $F(p) = 0$.\nThen $F(\\upsilon p\\upsilon^{-1}) = F(\\upsilon) + \\mathcal{C}(\\upsilon)\\cdot F(p) + \\mathcal{C}(\\upsilon p)\\cdot F(\\upsilon^{-1}) = (\\phi_{\\Delta_{\\upsilon}} - \\phi_{-\\Delta_{\\upsilon}}) + \\mathcal{C}(\\upsilon)\\cdot\\vec{0} + (\\phi_{-\\Delta_{\\upsilon}} - \\phi_{\\Delta_{\\upsilon}}) = \\vec{0}$.\n\nBy induction, we thus show that $p\\in\\mathcal{P}$, $F(p) = 0$.\nHence, $F$ maps cosets of $\\mathcal{P}$ in $\\mathcal{A}$ into single elements of $\\mathbb{Z}^{\\phi}$, so we conclude $\\overline{F}$ exists and we are done.\n\\end{proof}\n\n\\begin{lemma}\\label{lem-Fbar-zeroone-artingroups}\nLet $\\overline{F}: \\mathcal{C}\\to\\mathbb{Z}^{\\Phi}$ be as defined in Lemma \\ref{lem-Fbar-exists-artingroups}, and let $\\Phi^+$ be the set of positive roots associated to the choice $\\Delta$ of simple roots.\nThen for all $\\phi\\in\\Phi^+, c\\in\\mathcal{C}$, we have $\\overline{F}(c)[\\phi]\\in\\{0, 1\\}$ (where $\\overline{F}(c)[\\phi]$ is the $\\phi$ component of $\\overline{F}(c)$).\n\\end{lemma}\n\\begin{proof}\nIf $c = e$, then the statement is trivial.\nHence, suppose it holds for $c$; we shall prove it for $\\vartheta c$, where $\\vartheta$ is a Coxeter generator of $\\mathcal{C}$ (that is, the image of an Artin generator of $\\mathcal{A}$).\nTo this end, we find, for any $\\phi\\in\\Phi$, that $\\overline{F}(\\vartheta c)[\\phi] = \\overline{F}(\\vartheta)[\\phi] + (\\vartheta\\cdot\\overline{F}(c))[\\phi] = \\overline{F}(\\vartheta)[\\phi] + \\overline{F}(c)[\\vartheta\\star\\phi]$, where $\\vartheta\\star\\phi$ is the image of $\\phi$ under the reflection across the hyperplane orthogonal to $\\alpha_{\\vartheta}$.\nIf $\\alpha_{\\vartheta}\\neq\\phi$, then the first term vanishes, and since $\\vartheta\\star\\phi$ also belongs to $\\Phi^+$, the conclusion follows.\nIf $\\alpha_{\\vartheta} = \\phi$, then we have $\\overline{F}(\\vartheta c)[\\phi] = 1 + \\overline{F}(c)[-\\phi] = 1 - \\overline{F}(c)[\\phi]\\in\\{0, 1\\}$, and the conclusion again follows.\nBy induction on the length of $c$, we are done.\n\\end{proof}\n\nSince this cocycle exists, the image of the super-Gauss-Epple homomorphism is not a semidirect product (as there is no way to embed the Coxeter group as a subgroup).\nHowever, we can show that the image of $\\mathcal{P}$ under the super-Gauss-Epple homomorphism is abelian.\nNote that this is \\textit{not} a semidirect product, for reasons analogous to those given at the end of Section \\ref{section-braids-1cocy} (namely, that no element of $\\mathcal{A}$ whose image in $\\mathcal{C}$ is a canonical generator can have a square whose image under super-Gauss-Epple is that of the identity).\n\nWe can also characterize the image of the super-Gauss-Epple homomorphism the following way:\nNote that, if $c\\in\\mathcal{A}$ and $a$ is a canonical generator of $\\mathcal{A}$, then $SGE(ca^2c^{-1}) = (v_c, \\mathcal{C}(c))(\\Phi_{\\Delta_a} + \\Phi_{-\\Delta_a}, e)(-\\mathcal{C}(c)^{-1}v_c, W(c)) = (\\Phi_{\\mathcal{C}(c)\\cdot\\Delta_a} + \\Phi_{\\mathcal{C}(c)\\cdot\\Delta_a}, e)$.\nHence, the set of all pairs $(v, c)\\in\\mathbb{Z}^{\\Phi}\\rtimes C$ that belong to the image of SGE are the pairs such that $v - \\overline{v} = \\overline{F}(c)$.\n\n\n\\section{Complex reflection groups}\\label{sec-complex}\n\nWe considered a candidate analogue of a ``symmetric super-Gauss-Epple homomorphism'', which would have type signature $B \\to \\mathbb{Z}^{\\mathcal{A}}\\rtimes W$: namely, a homomorphism given by generating relations $s\\to (e_{H_s}, W(s))$, where $H_s$ is a distinguished generator of $B$ whose corresponding element in $W$ is a member of $\\Psi$.\nFor example, in the case of $\\mathcal{C} = G(n, 1, 1) = \\mathbb{Z}\/n\\mathbb{Z}$, we have $\\mathcal{A} = \\mathbb{Z}$, and $\\mathcal{HA}$ consists of a single element $\\mathcal{H}$.\nTherefore, the associated homomorphism is an injective homomorphism of type $\\mathbb{Z}\\to\\mathbb{Z}\/n\\mathbb{Z}\\times\\mathbb{Z}$ given by the generating relation $1\\to (1, 1)$.\nIt is trivial to verify that this is indeed a homomorphism.\n\nHowever, using a geometric argument, we show that no such homomorphism exists, unless $B$ is an Artin group of finite type:\n\n\\begin{theorem}[No SGE for complex reflection groups]\nLet $B$ be some braid group associated to a complex reflection group, $W$ be the associated complex reflection group, and $\\mathcal{HA}$ the associated hyperplane arrangement.\nSuppose that, in the canonical Artin-like presentation of $B$,\nthere is a relation of the form $aba = bab$, where $a$ and $b$ are two distinct generators.\nThen there is no homomorphism $B \\to \\mathbb{Z}^{\\mathcal{A}}\\rtimes W$ given by generating relations $s\\to (e_{H_s}, W(s))$, where $H_s$ is a distinguished generator of $B$ whose corresponding element in $W$ is a member of $\\Psi$.\n\\end{theorem}\n\\begin{proof}\nWe prove $(e_{H_a}, W(a))(e_{H_b}, W(b))(e_{H_a}, W(a)) \\neq (e_{H_b}, W(b))(e_{H_a}, W(a))(e_{H_b}, W(b))$.\n\nWe can easily compute that the former part of the left hand side  will be $e_{H_a} + e_{W(a)H_b} + e_{W(a)W(b)e_{H_a}}$,\nand the former part of the right hand side would be $e_{H_b} + e_{W(b)H_a} + e_{W(b)W(a)e_{H_b}}$.\nWe claim that $H_a = W(b)W(a)H_b, H_b = W(a)W(b)H_a$, but that $W(a)H_b \\neq W(b)H_a$.\n\nFirstly, however, we shall need to use the determinant trick.\nWe have $W(a)W(b)W(a) = W(b)W(a)W(b)$.\nSince both sides are linear transformations, we can take determinants: $\\det(W(a))\\det(W(b))\\det(W(a)) = \\det(W(b))\\det(W(a))\\det(W(b))$.\nSince these determinants are nonzero real numbers, we can cancel to obtain $\\det(W(a)) = \\det(W(b))$.\nThis equality is important because letting $\\zeta_{W(a)}$ be the multiplier of $W(a)$ and similarly for $\\zeta_{W(b)}$, we can conclude that $\\zeta_{W(a)} = \\zeta_{W(b)}$, and will refer to both constants as merely $\\zeta$.\n\nNow, to show $H_a = W(b)W(a)H_b$, let $s_{\\zeta, H}$ be the pseudo-reflection that multiplies by $\\zeta$ and preserves $H$. \nWe compute\n\\begin{align*}\n    s_{W(b)W(a)H_b, \\zeta} &= W(b)W(a)s_{H_b, \\zeta}(W(b)W(a))^{-1}\\\\\n    &= W(b)W(a)W(b)(W(b)W(a))^{-1}\\\\\n    &= W(a)W(b)W(a)(W(b)W(a))^{-1}\\\\\n    &= W(a)\\\\\n    &= s_{H_a, \\zeta},\n\\end{align*}\nwhich gives the claim.\nBy symmetry, we can also obtain $H_b = W(a)W(b)H_a$.\n\nNow to show that $W(a)H_b \\neq W(b)H_a$, we compute\n\n\\begin{align*}\n    s_{W(a)H_b, \\zeta} &= W(a)s_{H_b, \\zeta}W(a)^{-1}\\\\\n    &= W(a)W(b)W(a)^{-1}\\\\\n    &= (W(a)W(b)W(a))W(a)^{-2},\n\\end{align*}\nand similarly $s_{W(b)H_a, \\zeta} = (W(b)W(a)W(b))W(b)^{-2} = (W(a)W(b)W(a))W(b)^{-2}$.\nObserve now that $W(a)^2 = s_{H_a, \\zeta^2}$ and $W(b)^2 = s_{H_b, \\zeta^2}$. \nSince $a$ and $b$ are distinct, this means that $H_a$ and $H_b$ are distinct.\nThis in turn implies that $s_{H_a, \\zeta^2}$ and $s_{H_b, \\zeta^2}$ are distinct unless $\\zeta^2 = 1$.\nHowever, if $\\zeta^2 = 1$, then $\\zeta = -1$, so that $a$ and $b$ are merely real reflections, as desired.\n\\end{proof}\n\n\n\n\\bibliographystyle{plain}\n\n\\section{Introduction}\n\n\\subsection{Braid groups}\n\n\\begin{wrapfigure}[6]{r}{0.2\\textwidth}\n    \\vspace{-20pt}\n    \\includegraphics[height=0.18\\textwidth]{Threedimensional_braid_remastered.png}\n    \\caption{An example braid}\n    \\label{fig-example-braid}\n\\end{wrapfigure}\n\n\\hspace{\\parindent}Fix, once and for all, an arbitrary integer $n\\geq 2$; we shall use the notation $[n] := \\{1, \\dots, n\\}$.\nA \\textit{braid} on $n$ strands is a topological object consisting of $n$ strands in 3-dimensional space whose endpoints are fixed to two distinguished parallel planes, such as in Figure \\ref{fig-example-braid}.\n(We assume, as is common, that the endpoints on each plane are collinear.\nWe also choose one plane to be the ``top'' and number the strands from $1$ to $n$ in their order on this plane; we choose the other to be the ``bottom''.)\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{braid-composition.png}\n    \\caption{Braid multiplication}\n    \\label{fig-braid-multiplication}\n\\end{figure}\n\nBy ``gluing'' two braids on $n$ strands together (i.e., identifying the ``bottom'' plane of the first with the ``top'' plane of the second, as depicted in Figure \\ref{fig-braid-multiplication}), we can create a third braid on $n$ strands.\nThis operation, which we view as composition, endows the set of all braids on $n$ strands up to isotopy (i.e., topological deformation) with the structure of a monoid (i.e., a set with an associative and identity).\nIn fact, this operation is invertible; an example braid inverse is shown in Figure \\ref{fig-braid-inverse}.\nAs a result, the set of all braids on $n$ strands up to isotopy is in fact a group, the \\textit{braid group} $B_n$.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.55\\textwidth]{braid-inverse.png}\n    \\caption{A braid and its inverse}\n    \\label{fig-braid-inverse}\n\\end{figure}\n\nThe braid group $B_n$ is a well-studied mathematical object.\nEmil Artin first proposed it in 1925 \\cite{Artin1925} and further discussed it in a 1947 paper \\cite{10.2307\/1969218}.\nIt is well-known that $B_n$ has no torsion and that the full twist generates its center. Several efficient algorithms to solve $B_n$'s word and conjugacy problems are known \\cite{gonzalez2011basic}.\nIt is also well-known (as the Nielsen-Thurston classification) that all braids are either \\textit{periodic}, \\textit{reducible}, or \\textit{pseudo-Asonov} \\cite{gonzalez2011basic}.\nFurthermore, $B_n$ is linear, as there is a faithful (!) representation (the \\textit{Lawrence-Krammer representation}) $B_n\\to GL_{n(n-1)\/2}(\\mathbb{Z}[q^{\\pm 1}, t^{\\pm 1}])$ \\cite{krammer2002braid}.\n\nThe braid group $B_n$ has a canonical generating set of $n-1$ generators known as the \\textit{Artin generators} and conventionally denoted $\\sigma_1, \\dots, \\sigma_{n-1}$.\nThe Artin generator $\\sigma_i$ consists of the braid that twists the $i$th leftmost strand over and to the right of the $(i+1)$th strand.\nWith this generating set, the braid group has the following presentation:\n\\begin{align*}\n    B_n = \\langle \\sigma_1, \\dots, \\sigma_{n-1} | \\sigma_i\\sigma_{i+1}\\sigma_i = \\sigma_{i+1}\\sigma_i\\sigma_{i+1}\\forall i, \\sigma_i\\sigma_j = \\sigma_j\\sigma_i \\forall i, j: |i - j| > 1\\rangle.\n\\end{align*}\n\nThe \\textit{writhe} is a group homomorphism $B_n\\to\\mathbb{Z}$ defined by the relation $\\sigma_i\\mapsto 1$. The image of any braid $\\beta$ under writhe is called the \\textit{writhe} of the braid, denoted $|\\beta|$. The \\textit{permutation} is a group homomorphism $B_n\\to S_n$ defined by the relation $\\sigma_i\\mapsto(i, i + 1)$, where the right hand side is a transposition. The permutation of a braid is defined similarly and denoted $\\pi_{\\beta}$.\n\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.15\\textwidth]{Threedimensional_braid_remastered.png}\n    \\caption{The braid $(\\sigma_2^{-1}\\sigma_1^{-1})^2$, which has permutation (1,3,2) and writhe~$-4$.}\n\\end{figure}\n\nBoth the writhe and permutation are surjective homomorphisms but \\underline{not} injective.\nThe \\textit{writhe-permutation homomorphism} $WP: B_n\\to\\mathbb{Z}\\times S_n$ is defined by $\\beta\\mapsto (|\\beta|, \\pi_{\\beta})$. It is neither surjective nor injective, as we shall prove later.\n\n\\subsection{Artin groups of crystallographic type}\n\n\\hspace{\\parindent}An \\textit{Artin group} $\\mathcal{A}$ is a group presented by a finite set of generators and at most one braid relation (i.e., a relation of the form $a = b$, $ab = ba$, $aba = bab$, $abab = baba$, etc.) between any two generators.\nThe presentation of an Artin group can be depicted in a \\textit{Dynkin diagram}, as shown in Figure \\ref{fig-dynkin-diagrams}.\n\n\\begin{figure}[h]\n    \\vspace{10pt}\n    \\begin{center}\n    \\includegraphics[height=0.5\\textwidth]{349px-DynkinDiagrams.png}\n    \\caption{Dynkin diagrams of finite Coxeter groups\n    \\cite{dynkindiagrams2007commons}}\n    \\label{fig-dynkin-diagrams}\n    \\end{center}\n\\end{figure}\n\\begin{figure}\n    \\centering\n    \\includegraphics[height=100pt]{G2_simpleroots.png}\n    \\caption{The $G_2$ root system with two simple roots highlighted \\cite{nasmith2020medium}}\n    \\label{fig-g2-simpleroots}\n\\end{figure}\n\\vspace{-10pt}\nA \\textit{Coxeter group} $\\mathcal{C}$ is a group presented with the generators and relations of an Artin group and the relations that the square of any generator is the identity.\nA Coxeter group's presentation can also be depicted in a Dynkin diagram.\n\nSometimes, a Coxeter group is generated by the reflections upon a \\textit{root system} $\\Phi$, which is a special set of vectors.\n(In this case, they are known as \\textit{Weyl groups}.)\nThe associated Artin group $\\mathcal{A}$ is then called an \\textit{Artin group of crystallographic type}.\nFurthermore, in the cases where the Coxeter group is indeed associated with a root system, an isomorphism between the Coxeter group and the group of linear transformations sends the canonical Coxeter group generators to the reflections associated with some choice of \\textit{simple roots} $\\Delta\\subset\\Phi$.\n(We refer to the simple root associated to a canonical Artin generator $a$ of $\\mathcal{A}$ as $\\Delta_a$.)\nOne such set of simple roots, corresponding to the case commonly referred to as $G_2$, is shown in Figure \\ref{fig-g2-simpleroots}.\n\n\\subsection{Complex reflection groups}\n\n\\begin{wrapfigure}[22]{r}{0.3\\textwidth}\n    \\vspace{-40pt}\n        \\includegraphics[height=0.7\\textwidth]{complex-reflection-group-table.png}\n    \\caption{Partial table of complex reflection groups \\cite{Broue2010IntroductionTC}}\n\\end{wrapfigure}\n\\hspace{\\parindent}We adopt the terminology and notation of Brou\u00e9 \\cite{Broue2000}.\n\nLet $V$ be some finite-dimensional $\\mathbb{C}$-vector space.\nA \\textit{pseudo-reflection} is a linear transformation that acts trivially on some hyperplane (called its \\textit{reflecting hyperplane}).\n\nA \\textit{complex reflection group} $W$ is a finite subgroup of $GL(V)$ generated by pseudo-reflections.\n\nWe denote the set of pseudo-reflections in $W$ by $\\Psi$. (This notation is an exception; it comes from Dr. Trinh's notes \\cite{mqt200207notes}, not from Brou\u00e9.)\nThe hyperplane arrangement in $V$ consisting of the reflecting hyperplanes of the $\\Psi$ is denoted by $\\mathcal{A}$, and its hyperplane complement $V - \\cup_{H\\in\\mathcal{A}} H$ is denoted as $M$.\nWe define the analogous \\textit{braid group} $B := \\pi_1(M\/W)$; in other words, the fundamental group of $M$ up to rotation by elements of $W$.\n\nSimilar to the previous case, the braid groups $B$ have canonical Artin-like presentations, and the corresponding Coxeter groups $W$ have canonical Coxeter-like presentations \\cite{Broue2000}.\nMore precisely, there exists a subset $\\mathbf{S} = \\{\\mathbf{s}_1, \\dots, \\mathbf{s}_n\\}$ of $B$ consisting of distinguished braid reflections, and a set $R$ of relations of the form $w_1 = w_2$, where $w_1$ and $w_2$\nare positive words of equal length in the elements of $\\mathbf{S}$, such that $\\langle S|R\\rangle$ is\na presentation of $B$.\nMoreover, their images $s_1, \\dots, s_n$ in $W$ generate $W$, and the group $W$ is presented by\n$$\\langle S|R;(\\forall s\\in S)(s^{e_s} =1)\\rangle$$ where $e_s$ denotes the order of $s$ in $W$.\n\nThere is a quotient map $B\\to W$; we denote the image of an element $b\\in B$ under this quotient map simply as $W(b)$.\n\n\\subsection{Overview of results}\n\n\\hspace{\\parindent}We study a large family of mathematical objects that we call the \\textit{Gauss-Epple homomorphisms}.\nThis research helps us understand the structure of braid groups $B_n$, which describe the structure of \\textit{braids}, a kind of topological object.\n\nThe first such homomorphism was implicitly introduced by Epple \\cite{Epple98} as an action of $B_n$ based on a note by Gauss.\nAround the time of his note, Gauss was primarily interested in topology for its applications to electromagnetism and celestial dynamics. In this work, we generalize this concept to a broader family of homomorphisms from Artin groups of finite type, a large family of groups including the braid groups.\n\nFirstly, in Section \\ref{sec-theGEhomor}, we prove that there exists a well-defined and unique left group action of $B_n$ on $\\mathbb{Z}\\times[n]$, as implicit in \\cite{Epple98}.\nWe refer to this action as the \\textit{Gauss-Epple action}.\nIt is equivalent to a group homomorphism from $B_n$ to $\\Sym(\\mathbb{Z}\\times[n])$, which we call the \\textit{Gauss-Epple homomorphism} (denoted $GE$).\nIn Subsection \\ref{subsec-imWP}, we describe the \\textit{writhe-permutation} homomorphism $WP: B_n\\to\\mathbb{Z}\\times S_n$ (the homomorphism that maps a braid to its writhe and underlying permutation) in greater detail, as we treat this homomorphism as a toy model of $GE$. We show that the image of \\textit{writhe-permutation} homomorphism $WP: B_n\\to\\mathbb{Z}\\times S_n$ is a particular order 2 subgroup of $\\mathbb{Z}\\times S_n$.\nThen, in Subsection \\ref{subsec-imGE}, we show that the image of $GE$ is (isomorphic to) an order 2 subgroup of $\\mathbb{Z}^n\\rtimes S_n$.\nIn Subsection \\ref{subsec-kerGE}, we discuss the kernel of $GE$. We find that this group is strictly contained in the kernel of the \\textit{writhe-permutation homomorphism}.\n\nWe summarize these results with the following commutative diagram:\n\n\\[\\begin{tikzcd}\n\t{B_ n} & {\\mathbb{Z}^n\\rtimes S_n} & {\\Sym([n]\\times\\mathbb{Z})} \\\\\n\t& {\\mathbb{Z}\\times S_n}\n\t\\arrow[from=1-1, to=1-2]\n\t\\arrow[from=1-2, to=1-3]\n\t\\arrow[from=1-2, to=2-2]\n\t\\arrow[\"WP\", from=1-1, to=2-2]\n\t\\arrow[\"GE\", curve={height=-12pt}, from=1-1, to=1-3]\n\\end{tikzcd}\\]\n\nEach of the maps in this diagram is a group homomorphism, as explained below:\n\n\\begin{itemize}\n    \\item The map $GE$, from $B_n$ to $\\mathrm{Sym}(\\mathbb{Z}^2)$, is the Gauss-Epple action of $B_n$ on $\\mathbb{Z}^2$.\n    \\item The map from $B_n$ to $\\mathbb{Z}^n\\rtimes S_n$ maps a braid to the tuple of its vector and its permutation.\n    \\item The map from $\\mathbb{Z}^n\\rtimes S_n$ to $\\mathrm{Sym}(\\mathbb{Z}^2)$ is the map $(\\pi, \\ell)\\mapsto ((a, b) \\mapsto (\\pi(a), b + \\ell_a))$.\n    \\item The map $WP$, from $B_n$ to $\\mathbb{Z}\\times S_n$, maps a braid to its writhe and braid permutation.\n    \\item The map from $\\mathbb{Z}^n\\rtimes S_n$ to $\\mathbb{Z}\\times S_n$ maps $(\\ell, \\pi)$ to $(\\sum\\ell, \\pi)$.\n\\end{itemize}\n\nIn Section \\ref{sec-symGE}, we briefly discuss another action of $B_n$ (the \\textit{symmetric-Gauss-Epple action}) mentioned by Epple.\nIt turns out that this action has very similar properties to the Gauss-Epple action, including sharing the same kernel.\n\nIn Section \\ref{sec-superGE}, we discuss the \\textit{super-Gauss-Epple homomorphism} (denoted $SGE$), a homomorphism of $B_n$ that refines $GE$.\nTo describe the image of $SGE$, we introduce a 1-cocycle of the symmetric group $S_n$ on the set of $n\\times n$ antisymmetric matrices, which we prove has a remarkable nonnegativity property.\n\nFinally, in Section \\ref{sec-ATgroupsWeyl}, we generalize our results to the contexts of Artin groups of finite type, a broad family of groups that generalize the braid groups.\nMore specifically, we introduce a family of novel homomorphisms (with domain $\\mathcal{A}$ and range $\\mathbb{Z}^{\\Phi}\\rtimes\\mathcal{C}$) that are analogous to the super-Gauss-Epple homomorphism in the classical case of $B_n$.\nWe also note corresponding 1-cocycles, which also share an analogous nonnegativity property.\n\nWe close in Section \\ref{sec-complex} with remarks about complex reflection groups.\n\n\\subsubsection{Acknowledgements}\n\n\\hspace{\\parindent}We thank our advisor, Minh-T\u00e2m Trinh, for proposing this project and giving us helpful directions.\nWe also thank the MIT PRIMES program for supporting our research.\n\n\\section{The Gauss-Epple homomorphism}\\label{sec-theGEhomor}\n\n\\begin{figure}\n    \\begin{center}\\includegraphics[width=0.5\\textwidth]{Gauss_note.png}\\end{center}\n    \\caption{Page 283 of Gauss's Handbuch 7 \\cite{Epple98}}\n\\end{figure}\n\n\\hspace{\\parindent}We formally introduce the Gauss-Epple action. \nThis action was initially defined by Epple \\cite{Epple98} based on a note of Gauss.\nSince Gauss's notation involved complex numbers (more precisely, the Gaussian integers $\\mathbb{Z}[i]$), the Gauss-Epple action was originally defined in terms of complex numbers.\nFor notational and theoretical simplicity, however, we will define the Gauss-Epple action as an action of $B_n$ on $[n]\\times\\mathbb{Z}$ instead of on $\\mathbb{Z}[i]$.\nOur reasons for doing so should become clear shortly.\nWe also view the Gauss-Epple action as a group homomorphism from $B_n$ to $\\Sym([n]\\times\\mathbb{Z})$.\n(This is a specific instance of .the classical trick to convert between group actions $G\\times X \\to X$ and group homomorphisms $G \\to (X \\to X)$: by currying the input.)\n\n\nAs defined by Epple (based on Gauss's notation), the Gauss-Epple homomorphism is given as follows:\n\\begin{definition}[Gauss-Epple action]\\label{def-eppleGE}\nFor any $n\\in\\mathbb{N}$, the Gauss-Epple action $\\alpha: B_n\\times \\mathbb{Z}[i]\\to \\mathbb{Z}[i]$ is the unique left group action of $B_n$ (with canonical Artin generators $\\sigma_1, \\sigma_2, \\dots$) on the Gaussian integers $\\mathbb{Z}[i]$ defined by the following generating relation:\n\n$$\\alpha(\\sigma_k, z) := \\begin{cases}\nz &  \\Re(z)\\notin\\{k, k + 1\\}\\\\\nz + 1 &  \\Re(z) = k\\\\\nz - 1 + i &  \\Re(z) = k + 1\n\\end{cases}.$$\n\\end{definition}\n\nWe simplify this definition by replacing each complex number involved with the ordered pair of its real and complex parts, and then restricting the first component to the elements of $[n]$.\nThis notational change yields an action on the elements of $[n]\\times\\mathbb{Z}$.\nTherefore, as we define it, the Gauss-Epple action is given as follows:\n\n\\begin{definition}[Gauss-Epple action]\\label{def-GE}\nFor any $n\\in\\mathbb{N}$, the Gauss-Epple action $GE: B_n\\times ([n]\\times\\mathbb{Z})\\to [n]\\times\\mathbb{Z}$ is the unique left group action of $B_n$ (with canonical Artin generators $\\sigma_1, \\sigma_2, \\dots$) on $[n]\\times\\mathbb{Z}$ defined by the following generating relation:\n\n$$GE(\\sigma_k, (a, b)) := \\begin{cases}\n(a, b) & a\\notin\\{k, k + 1\\}\\\\\n(k + 1, b) & a = k\\\\\n(k, b + 1) & a = k + 1\n\\end{cases}.$$\n\\end{definition}\n\n\\begin{remark}\nNote that this definition implies that the inverses of the Artin generators act as follows:\n\n$$GE(\\sigma_k^{-1}, (a, b)) = \\begin{cases}\n(a, b) & a\\notin\\{k, k + 1\\}\\\\\n(k + 1, b - 1) & a = k\\\\\n(k, b) & a = k + 1\n\\end{cases}.$$\n\\end{remark}\n\\vspace{30pt}\n\nWe now prove that the Gauss-Epple action is well-defined, a fact stated without proof by Epple \\cite{Epple98}.\n\n\\begin{lemma}\\label{lem-GE-welldefined}\nThe Gauss-Epple action $GE$ is uniquely defined by Definition \\ref{def-GE} as a left group action of $B_n$ on $\\mathbb{Z}^2$.\n\\end{lemma}\n\\begin{proof}\nSince the Artin generators satisfy the generating relations $\\sigma_k \\sigma_l = \\sigma_l \\sigma_k$ for all $k, l$ such that $|k - l| \\geq 2$ and $\\sigma_k\\sigma_{k+1}\\sigma_k = \\sigma_{k+1}\\sigma_k\\sigma_{k+1}$ for all $k\\in[n - 1]$, it is enough to verify that that the analogous relations hold for the Gauss-Epple action to prove Lemma \\ref{lem-GE-welldefined}.\n\nSuppose that $k, l$ are two integers satisfying $|k - l|\\geq 2$. We observe that $\\{k, k + 1\\}\\cap\\{l, l + 1\\}=\\varnothing$, which will become important later. Let $a, b$ be arbitrary members of $\\mathbb{Z}$.\nWe prove that, for all integers $k, l$ satisfying $|k - l|\\geq 2$, we have $GE(\\sigma_k)GE(\\sigma_l) = GE(\\sigma_l)GE(\\sigma_k)$, as follows:\n\n\\begin{packed_item}\n\\item If $a\\notin\\{k, k + 1, l, l + 1\\}$, then $\\sigma_k\\sigma_l(a, b) = (a, b) = \\sigma_l\\sigma_k(a, b)$.\n\\item If $a\\in\\{k, k +1\\}$, then $\\sigma_k\\sigma_l(a, b) = \\sigma_k(a, b) = \\sigma_l\\sigma_k(a, b)$. \n(From the second to third step, we use the fact that the first component of $\\sigma_k(a, b)$ is in $\\{k, k + 1\\}$, and so outside of $\\{l, l + 1\\}$.)\n\\item If $a\\in\\{l, l + 1\\}$, then $\\sigma_k\\sigma_l(a, b) = \\sigma_l(a, b) = \\sigma_l\\sigma_k(a, b)$.\n\\end{packed_item}\n\nAgain, suppose that $k, a, b$ are arbitrary integers satisfying $k\\in\\{1, \\dots, n - 2\\}$.\nThen we prove that we have $GE(\\sigma_k)GE(\\sigma_{k+1})GE(\\sigma_k) = GE(\\sigma_{k+1})GE(\\sigma_k)GE(\\sigma_{k+1})$ as follows:\n\n\\begin{packed_item}\n\\item If $a\\notin\\{k, k + 1, k + 2\\}$, then $\\sigma_k\\sigma_{k+1}\\sigma_k(a, b) = (a, b) = \\sigma_{k+1}\\sigma_k\\sigma_{k+1}(a, b)$.\n\\item If $a = k$, then $\\sigma_k\\sigma_{k+1}\\sigma_k(a, b) = \\sigma_k\\sigma_{k+1}(k + 1, b) = \\sigma_k(k + 2, b) = (k + 2, b)$,\nand $\\sigma_{k+1}\\sigma_k\\sigma_{k+1}(a, b) = \\sigma_{k + 1}\\sigma_k(k, b) = \\sigma_{k+1}(k + 1, b) = (k + 2, b)$.\n\\item If $a = k + 1$, then $\\sigma_k\\sigma_{k+1}\\sigma_k(a, b) = \\sigma_k\\sigma_{k+1}(k, b + 1) = \\sigma_k(k, b + 1) = (k + 1, b + 1)$,\nand $\\sigma_{k+1}\\sigma_k\\sigma_{k+1}(a, b) = \\sigma_{k + 1}\\sigma_k(k + 2, b) = \\sigma_{k+1}(k + 2, b) = (k + 1, b + 1)$.\n\\item If $a = k + 2$, then $\\sigma_k\\sigma_{k+1}\\sigma_k(a, b) = \\sigma_k\\sigma_{k+1}(k + 2, b) = \\sigma_k(k + 1, b + 1) = (k, b + 2)$,\nand $\\sigma_{k+1}\\sigma_k\\sigma_{k+1}(a, b) = \\sigma_{k + 1}\\sigma_k(k + 1, b + 1) = \\sigma_{k+1}(k, b + 2) = (k, b + 2)$.\n\\end{packed_item}\n\\end{proof}\n\nFrom now on, we consider GE as a group homomorphism $GE: B_n\\to\\Sym([n]\\times\\mathbb{Z})$\n\n\\begin{remark}\nNote that, by letting $n$ vary, we can extend the Gauss-Epple homomorphism $GE: B_n \\to\\Sym([n]\\times\\mathbb{Z})$ to a homomorphism $GE: B_{\\infty}\\to\\Sym(\\{1, 2, \\dots\\}\\times\\mathbb{Z})$, where the group $B_{\\infty}$ is defined to be the direct limit of the groups $B_n$. \nHowever, we do not pursue this direction.\n\\end{remark}\n\n\\subsection{Image of WP}\\label{subsec-imWP}\n\\hspace{\\parindent}Firstly, note the following classical fact, which motivates the study of \n\\text{images} of group homomorphisms:\n\n\\begin{theorem}[First Isomorphism Theorem]\\label{thm-1st-isothm}\nLet $\\phi: G\\to H$ be a group homomorphism. Then $\\im(\\phi) \\cong G\/\\ker\\phi$.\n\\end{theorem}\n\nFor the group homomorphisms we study, it turns out that the images are much simpler than the kernels. Hence we will completely determine the images but only briefly discuss the kernels.\n\nSince the Gauss-Epple action permutes the $a$ part without respect to the $b$ part according to the permutation of the braid, it is evident that braids in the kernel of the Gauss-Epple action have the identity permutation (i.e., are pure).\nEarly on, however, we discovered that braids in the kernel of the Gauss-Epple homomorphism have zero writhe too (we will prove this in Lemma \\ref{lem-sum-link-writhe}).\nHence we decided to study the invariant $WP$ (which stands for ``writhe-permutation''), a weaker invariant than the Gauss-Epple homomorphism.\nThis invariant, too, turns out to be a group homomorphism: in this case, with a range of $\\mathbb{Z}\\times S_n$.\nIn fact, we find that the image of $WP$ is an order 2 subgroup of the range:\n\n\\begin{theorem}[Structure of imWP]\\label{thm-structure-imWP}\nThere is a braid on $n$ strands with writhe $w\\in\\mathbb{Z}$ and permutation $\\pi\\in S_n$ iff (if and only if) $\\pi$ and $w$ have the same parity.\n\\end{theorem}\n\n\\begin{proof}\n\\textbf{Necessity}: Suppose that there is a braid on $n$ strands with writhe $w\\in\\mathbb{Z}$ and permutation $\\pi\\in S_n$. \nIf the number of twists of the braid is even, both $\\pi$ and $w$ will also necessarily be even.\nSimilarly, if the number of twists of the braid is odd, both $\\pi$ and $w$ will necessarily be odd.\nThus the claim holds.\n\n\\textbf{Sufficiency}: Let $\\pi$ and $w$ be arbitrary permutations and integers with the same parity.\nSince $S_n$ is generated by $(1, 2), (2, 3), \\dots, (n - 1, n)$, we can write $\\pi = \\prod_{i = 1}^f t_{a_i}$, where $t_i = (i, i + 1)$ and $a_i$ is some arbitrary sequence of integers.\n\nConsider the braid $\\beta' = \\prod_{i = 1}^f \\sigma_{a_i}$, where $a_i$ is as before. \nTrivially, $\\beta'$ has permutation $\\pi$.\nBy the argument in the previous part of the proof, the writhe of $\\beta'$ has the same parity as $\\pi$, which has the same parity as $w$, so that $\\sigma_1^{w - |\\beta'|}$ is pure.\nWe conclude that the braid $\\beta = \\beta'\\sigma_1^{w - |\\beta'|}$ has writhe $w$ and permutation $\\pi$, as desired,  and we are done.\n\\end{proof}\nIt is clear that since the image of $WP$ is not $B_n$ (nor is it the image of $GE$), that $WP$ must have a nontrivial kernel, and moreover, an element lying outside of the kernel of $GE$.\nWe can make this explicit by giving the example $\\sigma_1^{-2}\\sigma_2^2$.\n\n\\subsection{Image of GE}\\label{subsec-imGE}\n\n\\hspace{\\parindent}The computation of the image of the Gauss-Epple homomorphism (henceforth abbreviated \\textit{imGE}) is similar, albeit much more tedious. Firstly, we prove that the Gauss-Epple action factors through $\\mathbb{Z}^n\\rtimes S_n$.\n(The group rule of $\\mathbb{Z}^n\\rtimes S_n$ is $(\\pi, \\ell)(\\pi', \\ell') = (\\pi\\pi', \\ell' + \\pi'\\ell)$.)\n\n\\begin{theorem}[Structure theorem for imGE]\\label{thm-factor-imGE}\nDefine the group inclusion $\\iota: \\mathbb{Z}^n\\rtimes S_n\\to\\Sym([n]\\times\\mathbb{Z})$ by the equation $(\\pi, \\ell)\\to ((a, b)\\to (\\pi(a), b + \\ell(a))$,\nand define the group surjection $\\varsigma: B_n\\to\\mathbb{Z}^n\\rtimes S_n$ by \n$$\\varsigma(\\sigma_k) := ((k, k + 1), \\vec{e_k}).$$\nThen $GE = \\varsigma\\circ\\iota$.\n\\end{theorem}\n\\begin{proof}\n\nIt is trivial to verify that $GE = \\varsigma\\circ\\iota$ for the Artin generators.\nBy induction, and by the fact that all three functions are group homomorphisms, it follows that $GE = \\varsigma\\circ\\iota$ for all braids.\n\\end{proof}\n\nAs an aside, we observe the Gauss-Epple action is transitive when restricted to $[n]\\times\\mathbb{Z}$.\nHowever, as we can deduce from Theorem \\ref{thm-factor-imGE}, it is NOT doubly transitive.\n\n\nWe also relate imWP to imGE with the following fact (which motivates the earlier study of imWP):\n\n\\begin{lemma}\\label{lem-sum-link-writhe}\nLet $\\beta$ be a braid, and suppose that $\\varsigma(\\beta) = (\\pi, \\ell)$ (where $\\varsigma$ is defined as in Theorem \\ref{thm-factor-imGE}). Then the sum of the components of $\\ell$ is the writhe of $\\beta$.\n\\end{lemma}\n\\begin{proof}\n\nNote that $\\varkappa: \\mathbb{Z}^n\\rtimes S_n\\to\\mathbb{Z}$ defined by $\\varkappa((\\pi, \\ell)) := \\sum_i \\ell_i$ is a group homomorphism.\nWe know that $\\varkappa$ is a group homomorphism since, for any $\\pi, \\ell, \\pi', \\ell'$, we have \\begin{align*}\n    \\varkappa((\\pi, \\ell)(\\pi', \\ell')) & = \\varkappa((\\pi\\pi', \\ell\\pi' + \\ell'))\\\\\n    &= \\sum_i (\\ell(\\pi'(i)) + \\ell'(i))\\\\\n    &= \\sum_i \\ell(\\pi'(i)) + \\sum_i \\ell'(i)\\\\\n    &= \\sum_i \\ell(i) + \\sum_i \\ell'(i)\\\\\n    &= \\varkappa((\\pi, \\ell)) + \\varkappa((\\pi', \\ell')),\n\\end{align*}\nas desired.\n\n\nHence $\\varkappa\\circ\\varsigma$ is a group homomorphism from $B_n$ to $\\mathbb{Z}$.\n\nWe also note that writhe is a group homomorphism from $B_n$ to $\\mathbb{Z}$.\nFurthermore, it is easy to verify that $\\varkappa\\circ\\varsigma(\\sigma_k) = |\\sigma_k| = 1$.\nSince $\\varkappa\\circ\\varsigma$ and writhe agree on the generators of $B_n$, they must agree everywhere by induction, and so be the same.\n\\end{proof}\nTo fully characterize imGE, we begin with the special case of pure braids.\n\n\\begin{lemma}\\label{lem-all-evensum-linkvecs-possible}\nFor any $\\ell\\in\\mathbb{Z}^n$, there is a pure braid $\\beta$ with $\\ell_{\\beta} = \\ell$ iff $\\ell$ has even sum.\n\\end{lemma}\n\\begin{proof}\n\\textbf{Necessity}: Suppose that there exists such a pure braid $\\beta$. Then $\\sum_a\\ell_a = |\\beta|$, and the latter is even since $\\beta$ is pure. Hence $\\sum_a\\ell_a$ must be pure, and we are done.\n\n\\textbf{Sufficiency}:\nLet $\\beta_1$ and $\\beta_2$ be arbitrary pure braids. Then $\\ell_{\\beta_1\\beta_2} = \\ell_{\\beta_1} + \\ell_{\\beta_2}$, and $\\ell_{\\beta_1^{-1}} = -\\ell_{\\beta_1}$.\nHence the set of all possible $\\ell_{\\beta}$ values for braids $\\beta$ forms a lattice in $\\mathbb{Z}^n$.\n\nWe compute that $\\ell_{\\sigma_k^2} = (0, \\dots, 0, 1, 1, 0, \\dots, 0)$, where the $1, 1$ is in the $k$th and $(k+1)$th places and where we have identified $\\ell$ with a vector in $\\mathbb{Z}^n$.\nSimilarly, we compute $\\ell_{(\\sigma_k\\sigma_{k + 1})^3} = (0, \\dots, 0, 2, 2, 2, 0, \\dots, 0)$.\n\nSince $(a, a + b - 2c, c) = (a - 2c)(1, 1, 0) + (b - 2c)(0, 1, 1) + c(2, 2, 2)$, we conclude that these vectors span the lattice of all elements of $\\mathbb{Z}^n$ with even sum, and we are done.\n\\end{proof}\n\nThen, we will combine this result with some further lemmas to characterize the image in Theorem \\ref{thm-imGE}.\nNow we can fully characterize imGE.\n\n\\begin{theorem}\\label{thm-imGE}\nLet $\\pi\\in S_n$ and $\\ell\\in\\mathbb{Z}^n$.\nThen there exists a braid $\\beta$ such that $\\pi_{\\beta} = \\pi, \\ell_{\\beta} = \\ell$ iff the sum of $\\ell$ has the same parity as $\\pi$.\n\\end{theorem}\n\\begin{proof}\n\\textbf{Necessity}: Suppose such a braid exists. By Lemma \\ref{lem-sum-link-writhe}, the sum of $\\ell = \\ell_{\\beta}$ must be the writhe of $\\beta$.\nThis must have the same parity as $\\pi_{\\beta} = \\pi$ by Theorem \\ref{thm-structure-imWP}, and we are done.\n\n\\textbf{Sufficiency}: Suppose that the sum of $\\ell$ has the same parity as $\\pi$.\nWe can construct a braid $\\beta_1$ with permutation $\\pi$. By Theorem \\ref{thm-structure-imWP} and Lemma \\ref{lem-sum-link-writhe}, $\\ell_{\\beta_1}$ must have a sum with the same parity as $\\pi$.\nBy Lemma \\ref{lem-all-evensum-linkvecs-possible}, there is a pure braid $\\beta_2$ such that $\\ell_{\\beta_2} = \\ell - \\ell_{\\beta_1}$ (since the right hand side has even sum).\nThen the braid $\\beta_3 := \\beta_1\\beta_2$ has permutation $\\pi$ and vector $\\ell$, as desired, and we are done.\n\\end{proof}\n\n\\subsection{Kernel of GE}\\label{subsec-kerGE}\n\n\\hspace{\\parindent}It is natural to consider the kernel of the Gauss-Epple homomorphism.\nWe find that imWP is a quotient group of imGE, as kerWP is a supergroup of kerGE.\n\nSince the image of GE is not $B_n$, the kernel of GE must necessarily be nontrivial.\nIn fact, by computational means, we found several explicit examples:\n\\begin{itemize}\n    \\item $\\sigma_1^2\\sigma_2^2\\sigma_1^{-2}\\sigma_2^{-2}$ (which corresponds to the Whitehead link \\cite{WMW-WhiteheadLink});\n    \\item  $\\sigma_1^{-1}\\sigma_3^{-1}\\sigma_2^2\\sigma_3^{-1}\\sigma_1^{-1}\\sigma_2^2$;\n    \\item $(\\sigma_1\\sigma_2^{-1})^3$;\n    \\item $(\\sigma_2\\sigma_1^{-1})^3$;\n    \\item $\\sigma_1\\sigma_2^{-1}\\sigma_1^2(\\sigma_1\\sigma_2^{-1})^2\\sigma_2^{-2}$;\n    \\item $(\\sigma_1\\sigma_2\\sigma_1^2\\sigma_2^{-1})^2\\sigma_1^{-2}$.\n\\end{itemize}\n\n\n\\hspace{\\parindent}We can make several observations about the structure of kerGE: \nWe know that the kernel of the Gauss-Epple action must be a subgroup of the group of pure braids, which in turn is a subgroup of the braid group $B_n$.\nSince $B_n$ has no torsion, the Gauss-Epple kernel must also have no torsion and be infinite.\n\n\\subsubsection{Random braids}\n\n\\hspace{\\parindent}We study the probability that a random braid of $n$ generators lies in the kernel of the Gauss-Epple action.\n\nLet $G$ be an arbitrary group and $H$ an arbitrary normal subgroup. \nThen, for all $g\\in G$, we know that $g\\in H$ iff $\\phi(g) = e$, where $\\phi$ is the quotient map $G\\to G\/H$.\n(This effectively reduces our problem to studying the quotient group $K := G\/H$.)\nNow, defining $V(N)$ to be the number of elements of $K$ that can be produced from words of length $N$, we have the following deep theorem:\n\\begin{theorem}\\label{thm:randwalkprob-vertcount-asy} \\cite{alexopoulos1997convolution, RWIGGreview, AMSNotice200109}\nFor any $d$, the probability of a random walk on $K$ returning to the identity on the $N$th step is on the order of $N^{-d\/2}$ iff $V(N)$ is comparable to $N^d$.\n\\end{theorem}\n\nTo apply this theorem, we take $G = B_n, H = \\ker GE, K = \\im GE$.\nSince $V(N)$ is comparable to $N^n$ thanks to Theorem \\ref{thm-imGE}, we conclude by Theorem \\ref{thm:randwalkprob-vertcount-asy} that the probability without filtering of an element being in the kernel of Gauss-Epple is asymptotically on the order of $N^{-n\/2}$, which is the result we seek.\n\n\\section{The symmetric Gauss-Epple homomorphism}\\label{sec-symGE}\n\n\\hspace{\\parindent}In his paper \\cite{Epple98}, Epple introduced yet another action of the braid group, which he called the \\textit{symmetric Gauss-Epple homomorphism}. This object is defined as follows:\n\n\\begin{definition}[Symmetric Gauss-Epple action]\nFor any $n\\in\\mathbb{N}$, the symmetric Gauss-Epple action $symGE: B_n\\times \\mathbb{Z}^2\\to \\mathbb{Z}^2$ is the unique left group action of $B_n$ (with canonical Artin generators $\\sigma_1, \\sigma_2, \\dots$) on $\\mathbb{Z}^2$ defined by the following relation:\n\n$$\\forall k\\in[n], a, b\\in\\mathbb{Z}:\n\\symGE(\\sigma_k, (a, b)) = \\begin{cases}\n(a, b) & a\\notin\\{k, k + 1\\}\\\\\n(k + 1, b + 1) & a = k\\\\\n(k, b + 1) & a = k + 1\n\\end{cases}.$$\n\\end{definition}\n\nFor similar reasons as in the proof of Theorem \\ref{thm-factor-imGE}, the symmetric Gauss-Epple action (which we abbreviate symGE) also factors through $\\mathbb{Z}^n\\rtimes S_n$ into $\\Sym(\\mathbb{Z}^2)$:\n\n\\begin{theorem}[Structure theorem for imGE]\\label{thm-factor-imsymGE}\nDefine the group inclusion $\\iota: \\mathbb{Z}^n\\rtimes S_n\\to\\Sym([n]\\times\\mathbb{Z})$ by the equation $(\\pi, \\ell)\\to ((a, b)\\to (\\pi(a), b + \\ell(a))$,\nand define the group surjection $\\varpi: B_n\\to\\mathbb{Z}^n\\rtimes S_n$ by \n$$\\varpi(\\sigma_k) := (\\vec{e_k} + \\vec{e_{k + 1}}, (k, k + 1)).$$\nThen $symGE = \\iota\\circ\\varpi$.\n\\end{theorem}\n\\begin{proof}\nTrivial.\n\\end{proof}\n\nWe then note that the symmetric Gauss-Epple action and the Gauss-Epple action have the same kernel:\n\n\\begin{theorem}\nThe kernel of the symmetric Gauss-Epple action is the same as the kernel of the ordinary Gauss-Epple action.\n\\end{theorem}\n\\begin{proof}\nSuppose $\\beta$ is an arbitrary pure braid. We show that $GE(\\beta) = (\\ell, \\text{id}) \\leftrightarrow \\symGE(\\beta) = (2\\ell, \\text{id})$.\nIt is clear that the ``permutation parts'' are all identity, so we only focus on the ``vector parts''.\nIt suffices to then show this claim for the the generators of the pure braid group $P_n$.\nNote that $P_n$ (the pure braid group on $n$ strands) is generated by $A_{i, j} := \\sigma_{j-1}\\dots\\sigma_{i+1}\\sigma_i^2\\sigma_{i+1}^{-1}\\dots\\sigma_{j-1}^{-1}$ \\cite{Suciu16}; for example, the generators for $P_3$ are $\\sigma_1^2, \\sigma_2^2, \\sigma_1\\sigma_2^2\\sigma_1^{-1}.$\n\nWe compute iteratively that $GE(A_{i, j}) = (\\vec{e_i} + \\vec{e_j}, \\text{id}), \\symGE(A_{i, j}) = (2\\vec{e_i} + 2\\vec{e_j}, \\text{id})$. Hence the claim holds, and we are done.\n\\end{proof}\n\n\\begin{remark}\nBy the First Isomorphism Theorem, the previous theorem implies that the images of the symmetric Gauss-Epple action and the images of the regular Gauss-Epple action must be isomorphic as groups.\n\\end{remark}\n\n\\section{The super-Gauss-Epple homomorphism}\\label{sec-superGE}\n\n\\hspace{\\parindent}We define another homomorphism, this time of type signature $B_n\\to \\mathbb{Z}^{n(n-1)}\\rtimes S_n$, which we name the \\textit{super-Gauss-Epple homomorphism} (abbreviated SGE).\nMore precisely, define $O_{i, j}$ to be the matrix with a 1 entry at the $(i, j)$th place and 0 entries everywhere else. \nThen we define SGE as follows:\n\\begin{definition}[super-Gauss-Epple homomorphism]\\label{def-SGE}\nThe super-Gauss-Epple homomorphism $SGE: B_n\\to \\mathbb{Z}^{n(n-1)}\\rtimes S_n$ is defined by the following equation:\n\n$$SGE(\\sigma_i) := (O_{i, i + 1}, (i, i + 1)).$$\n\\end{definition}\n\n(Here, by slight abuse of notation, we consider elements of $\\mathbb{Z}^{n(n-1)}$ to be $n\\times n$ matrices with all zeros along the diagonal.)\nAn example calculation of SGE is depicted in Figure \\ref{fig-calc-SGE}.\n\n\\begin{figure}[h]\n    \\begin{center}\\includegraphics[width=0.6\\textwidth]{SGE_calculation.png}\\end{center}\n    \\caption{An example calculation of SGE.}\n    \\label{fig-calc-SGE}\n\\end{figure}\n\nTo verify that this is indeed a homomorphism, we verify the braid relations (as is sufficient and necessary):\n\n\\begin{align*}\n    SGE(\\sigma_i\\sigma_{i+1}\\sigma_i) &= SGE(\\sigma_i)\\cdot SGE(\\sigma_{i+1})\\cdot SGE(\\sigma_i)\\\\\n    &= (O_{i, i + 1}, (i, i + 1))\\cdot(O_{i + 1, i + 2}, (i + 1, i + 2))\\cdot(O_{i, i + 1}, (i, i + 1))\\\\\n    &= (O_{i, i + 1} + (i, i+1)\\cdot O_{i + 1, i + 2} + (i, i+1)(i+1, i+2)\\cdot O_{i, i + 1}, \\\\\n    &\\hspace{15pt}(i, i+1)(i+1, i+2)(i, i+1))\\\\\n    &= (O_{i, i + 1} + O_{i, i + 2} + O_{i + 1, i + 2},(i, i+1)(i+1, i+2)(i, i+1))\\\\\n    &= (O_{i, i + 1} + O_{i, i + 2} + O_{i + 1, i + 2}, (i, i+2, i+1)),\n\\end{align*}\nand\n\\begin{align*}\n    SGE(\\sigma_{i+1}\\sigma_i\\sigma_{i+1}) &= SGE(\\sigma_{i+1})SGE(\\sigma_i)SGE(\\sigma_{i+1})\\\\\n    &= (O_{i + 1, i + 2}, (i + 1, i + 2))\\cdot(O_{i, i + 1}, (i, i + 1))\\cdot(O_{i + 1, i + 2}, (i + 1, i + 2))\\\\\n    &= (O_{i + 1, i + 2} + (i+1, i+2)\\cdot O_{i, i + 1} + (i+1, i+2)(i, i+1)\\cdot O_{i + 1, i + 2},\\\\\n    &\\hspace{15pt}(i+1, i+2)(i, i+1)(i+1, i+2))\\\\\n    &= (O_{i, i + 1} + O_{i, i + 2} + O_{i + 1, i + 2},\\\\\n    &\\hspace{15pt}(i, i+1)(i+1, i+2)(i, i+1))\\\\\n    &= (O_{i, i + 1} + O_{i, i + 2} + O_{i + 1, i + 2}, (i, i+2, i+1)),\n\\end{align*}\nas desired.\n\nA braid's image under the super-Gauss-Epple homomorphism stores at least as much information about it as the ordinary Gauss-Epple homomorphism. Note that this is strictly more information, as we can give the explicit example $\\sigma_2^2\\sigma_3^{-1}\\sigma_2\\sigma_1^2\\sigma_2^{-1}\\sigma_3^{-1}\\sigma_1^{-2}$.\n\n(By the Third Isomorphism Theorem, this also follows iff the image of the super-Gauss-Epple homomorphism is not isomorphic to that of the ordinary Gauss-Epple homomorphism.)\n\nIt is to be noted that the kernel of SGE is still nontrivial.\nFor example, it contains the braid $(\\sigma_1\\sigma_2^{-1})^3$, which was mentioned before as being a nontrivial example of the kernel of regular GE.\n\nHence, we have the following commutative diagram:\n\n\\[\\begin{tikzcd}\n\t&& {B_n\/[P_n, P_n]} \\\\\n\t{B_n} && {P_n^{ab}\\rtimes S_n} & {\\Sym(\\mathbb{Z}\\times\\binom{n}{2})} \\\\\n\t&& {\\mathbb{Z}^n\\rtimes S_n} & {\\Sym(\\mathbb{Z}\\times[n])} \\\\\n\t&& {\\mathbb{Z}\\times S_n}\n\t\\arrow[from=1-3, to=2-3]\n\t\\arrow[from=2-3, to=3-3]\n\t\\arrow[from=3-3, to=4-3]\n\t\\arrow[\"UGE\", from=2-1, to=1-3]\n\t\\arrow[\"SGE\", from=2-1, to=2-3]\n\t\\arrow[\"GE\", from=2-1, to=3-3]\n\t\\arrow[\"WP\", from=2-1, to=4-3]\n\t\\arrow[from=2-3, to=2-4]\n\t\\arrow[from=3-3, to=3-4]\n\\end{tikzcd}\\]\n\n\\subsection{A 1-cocycle}\\label{section-braids-1cocy}\n\n\\hspace{\\parindent}Firstly, note that the matrix of (the super-Gauss-Epple homomorphism of) a pure braid is always symmetric:\n\n\\begin{lemma}\\label{lem-SGE-pure-symmetric}\nLet $\\beta\\in P_n$, and suppose that $SGE(\\beta) = (M, \\text{id})$. Then $M$ is a symmetric matrix.\n\\end{lemma}\n\\begin{proof}\nRecall that $P_n$ (the pure braid group on $n$ strands) is generated by \n$$A_{i, j} := \\sigma_{j-1}\\dots\\sigma_{i+1}\\sigma_i^2\\sigma_{i+1}^{-1}\\dots\\sigma_{j-1}^{-1}$$ \\cite{Suciu16}; for example, the generators for $P_3$ are $\\sigma_1^2, \\sigma_2^2, \\sigma_1\\sigma_2^2\\sigma_1^{-1}.$\n\nNote that $SGE(A_{i, j}) = (O_{i, j} + O_{j, i}, id)$, and the right component is clearly symmetric.\nSince SGE matrices add for pure braids, the conclusion holds by induction.\n\\end{proof}\n\nHence, the difference between the upper half and the (transposed) lower half is always only determined by the braid permutation.\n\nNow, let $F: B_n\\to M_{n, n}(\\mathbb{Z})$ be defined so that $SGE(\\beta) = (M, s_{\\beta}) \\implies F(\\beta) = M - M^T$.\nThen $F(\\beta) = 0\\forall\\beta\\in P_n$.\nFurthermore, $F$ satisfies the equality $F(\\beta\\gamma) = F(\\beta) + M_{s_{\\beta}}F(\\gamma)M_{s_{\\beta}}^{-1}$.\nHence $F(\\pi\\beta) = F(\\beta) + id\\cdot F(\\beta) = F(\\beta)~\\forall\\beta\\in B_n, \\pi\\in P_n$, so that the value of $F(\\beta)$ only depends on the permutation of the braid $\\beta$, and that in general $F$ projects down to a function $\\overline{F}: S_n\\to M_{n, n}(\\mathbb{Z})$.\nFurthermore, we have $\\overline{F}(\\beta\\gamma) = F(\\beta) + M_{s_{\\beta}}F(\\gamma)M_{s_{\\beta}}^{-1}$ (which we shall abbreviate as $\\overline{F}(\\beta\\gamma) = \\overline{F}(\\beta) + \\beta\\cdot\\overline{F}(\\gamma)$. \nTherefore, $\\overline{F}$ is a 1-cocycle of the action of $S_n$ on $M_{n, n}(\\mathbb{Z})$, and in fact of the action on the set of antisymmetric $n\\times n$ matrices over the integers.\n\nWhile computing values of $\\overline{F}$ for some random braids, we noticed that the entries in the upper half were always in $\\{0, 1\\}$. We prove this inductively as follows:\n\n\\begin{theorem}\\label{thm-overlineF-upperhalf-zeroone}\nLet $\\varrho$ be an arbitrary permutation, and $i, j$ be integers with $i > j$. Then $\\overline{F}(\\varrho)[i, j]\\in\\{0, 1\\}$.\n\\end{theorem}\n\\begin{proof}\nWe assume this for some $\\varrho$ and prove it for $\\tau\\varrho$, where $\\tau := (\\iota, \\iota + 1)$ is a transposition.\n\nSince $\\overline{F}(\\tau\\varrho) = \\overline{F}(\\tau) + \\tau\\cdot\\overline{F}(\\varrho)t$, the conjecture is instantly verified for all $i, j$ except for $\\iota, \\iota + 1$ (since the first term is trivially zero and the second term is in $\\{0, 1\\}$ by the inductive hypothesis).\nIn the special case $i = \\iota, j = \\iota + 1$, we compute \\begin{align*}\n\\overline{F}(\\tau\\varrho)[\\iota, \\iota + 1] &= \\overline{F}(\\tau)[\\iota, \\iota + 1] + (\\tau\\cdot\\overline{F}(\\varrho)[\\iota, \\iota]\\\\\n&= 1 - \\overline{F}(\\varrho)[\\iota, \\iota + 1]\\\\\n&\\in\\{0, 1\\}.\n\\end{align*}\n(Here, we use the inductive hypothesis $\\overline{F}(\\varrho)[\\iota, \\iota + 1]\\in\\{0, 1\\}$.)\nHence the claim holds.\n\nBy induction on the minimum number of simple transpositions required to be multiplied together to represent an arbitrary permutation, we are then done.\n\\end{proof}\n\nThis means that the image of the super-Gauss-Epple homomorphism is a group with $\\mathbb{Z}^{\\frac{n(n-1)}{2}}$ as a normal subgroup and $S_n$ as the quotient group.\nNote that it is \\textit{not} a semidirect product, since no braid with permutation $(1,2)$ has a square in the kernel of the super-Gauss-Epple homomorphism.\n(Indeed, suppose such a braid $\\beta = \\sigma_1\\varpi$ existed, and let $SGE(\\beta) = (M, (1, 2))$ for some matrix $M$. Then the matrix of $SGE(\\beta^2)$ is $M + (1, 2)\\cdot M$. But since $M[1, 2]$ is one more than $((1, 2)\\cdot M)[1, 2] = M[2, 1]$ and both are integers, the two cannot sum to zero, which is a contradiction.)\n\n\\section{Artin groups of crystallographic type}\\label{sec-ATgroupsWeyl}\n\n\\hspace{\\parindent}In this section, we explore generalizations of the Gauss-Epple homomorphism to the setting of Artin groups, which naturally project to corresponding Coxeter groups (generalizing how the braid group projects to the symmetric group).\n\n\\subsection{Special cases}\n\n\\subsubsection{The $I_2(4), I_2(6)$ cases}\n\n\\hspace{\\parindent}This are special cases of the \\textit{dihedral Artin groups}, which are the Artin groups of type $I_2(2n)$ for integers $n$.\nThe abelianization is $\\mathbb{Z}^2$ and the Coxeter group is $D_{2n}$. Hence, the generators map to $D_{2n}\\times\\mathbb{Z}^2$ as follows:\n\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline\ngenerator & $D_{2n}\\times\\mathbb{Z}^2$\\\\\n\\hline\n$a$ & $(s, (1, 0))$\\\\\n$b$ & $(sr, (0, 1))$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\nFor GE-like purposes, we may consider $D_{2n}$ as a subset of $S_n$.\n\nFor the case $n = 4$, with $a\\to (13), b\\to (01)(23)$ in permutation notation, this gives rise to the following system of equations:\n\n$$\\begin{cases}\n\\ell_{a, 1} + \\ell_{b, 2} + \\ell_{a, 2} + \\ell_{b, 3}  &= \\ell_{b, 1} + \\ell_{a, 1} + \\ell_{b, 4} + \\ell_{a, 4} \\\\\n\\ell_{a, 2} + \\ell_{b, 1} + \\ell_{a, 3} + \\ell_{b, 2}  &= \\ell_{b, 2} + \\ell_{a, 4} + \\ell_{b, 1} + \\ell_{a, 3} \\\\\n\\ell_{a, 3} + \\ell_{b, 4} + \\ell_{a, 4} + \\ell_{b, 1}  &= \\ell_{b, 3} + \\ell_{a, 3} + \\ell_{b, 2} + \\ell_{a, 2} \\\\\n\\ell_{a, 4} + \\ell_{b, 3} + \\ell_{a, 1} + \\ell_{b, 4}  &= \\ell_{b, 4} + \\ell_{a, 2} + \\ell_{b, 3} + \\ell_{a, 1} \\\\\n\\end{cases}$$\n(We used a computer program to generate this system of equations.)\n\nSolving this system manually reveals a six-dimensional space: \n$$\\ell_a = (\\ell_{a, 1}, \\ell_{a, 2}, \\ell_{a, 3}, \\ell_{a, 2}), \\ell_b = (\\ell_{b, 1}, \\ell_{b, 2}, \\ell_{b, 3}, \\ell_{b, 2} + \\ell_{b, 3} - \\ell_{b, 1}).$$\n\nSimilarly, for the case $n = 6$ (also denoted $G_2$), we make the ansatz $a\\to(26)(35), b\\to(12)(36)(45)$, based on an action of the Coxeter group on the hexagon.\nWith the map $a\\to\\ell_a:= (\\ell_{a, 1}, \\ell_{a, 1}, \\dots, \\ell_{a, 6}), b\\to\\ell_b := (\\ell_{b, 1}, \\dots, \\ell_{b, 6})$, this produces the following system of linear equations:\n\n$$\\begin{cases}\n\\ell_{a, 1} + \\ell_{b, 1} + \\ell_{a, 2} + \\ell_{b, 6} + \\ell_{a, 3} + \\ell_{b, 5} &= \\ell_{b, 1} + \\ell_{a, 2} + \\ell_{b, 6} + \\ell_{a, 3} + \\ell_{b, 5} + \\ell_{a, 4}\\\\\n\\ell_{a, 2} + \\ell_{b, 6} + \\ell_{a, 3} + \\ell_{b, 5} + \\ell_{a, 4} + \\ell_{b, 4} &= \\ell_{b, 2} + \\ell_{a, 1} + \\ell_{b, 1} + \\ell_{a, 2} + \\ell_{b, 6} + \\ell_{a, 3}\\\\\n\\ell_{a, 3} + \\ell_{b, 5} + \\ell_{a, 4} + \\ell_{b, 4} + \\ell_{a, 5} + \\ell_{b, 3} &= \\ell_{b, 3} + \\ell_{a, 6} + \\ell_{b, 2} + \\ell_{a, 1} + \\ell_{b, 1} + \\ell_{a, 2}\\\\\n\\ell_{a, 4} + \\ell_{b, 4} + \\ell_{a, 5} + \\ell_{b, 3} + \\ell_{a, 6} + \\ell_{b, 2} &= \\ell_{b, 4} + \\ell_{a, 5} + \\ell_{b, 3} + \\ell_{a, 6} + \\ell_{b, 2} + \\ell_{a, 1}\\\\\n\\ell_{a, 5} + \\ell_{b, 3} + \\ell_{a, 6} + \\ell_{b, 2} + \\ell_{a, 1} + \\ell_{b, 1} &= \\ell_{b, 5} + \\ell_{a, 4} + \\ell_{b, 4} + \\ell_{a, 5} + \\ell_{b, 3} + \\ell_{a, 6}\\\\\n\\ell_{a, 6} + \\ell_{b, 2} + \\ell_{a, 1} + \\ell_{b, 1} + \\ell_{a, 2} + \\ell_{b, 6} &= \\ell_{b, 6} + \\ell_{a, 3} + \\ell_{b, 5} + \\ell_{a, 4} + \\ell_{b, 4} + \\ell_{a, 5}\\\\\n\\end{cases}$$\n\nThis system is highly symmetric and many of the variables cancel. \nTherefore, we can solve it exactly, producing the following linear parametrization of the entire solution space over the integers, which is nine-dimensional: \n$$\\ell_a = (a_1, a_2, a_3, a_1, a_2 + x_2, a_3 + x_2), \\ell_b = (b_1, b_2, b_3, b_1 + y_1, b_2 - y_1, b_3 + y_3),$$\nwhere $a_1, a_2, a_3, b_1, b_2, b_3, x_2, y_1, y_3$ range independently over $\\mathbb{Z}$.\n\n\\subsubsection{The $B_n$ case}\n\n\\hspace{\\parindent}The isomorphism between the Coxeter group of this case and $C_2\\wr S_n = C_2^n\\rtimes S_n$ can be tabulated in generators (where we list the generators of the Coxeter graph as $a_1, a_2, \\dots, a_n$) as follows:\n\n\\begin{center}\n    \\begin{tabular}{|c|c|c|}\n    \\hline\n    Coxeter & $S_n$ & $C_2^n$\\\\\n    \\hline\n    $a_1$ & id & $(1, 0, \\dots, 0)$\\\\\n    $a_2$ & (1, 2) & $(0, 0, \\dots, 0)$\\\\\n    $a_3$ & (2, 3) & $(0, 0, \\dots, 0)$\\\\\n    \\vdots & \\vdots & \\vdots\\\\\n    $a_n$ & ($n - 1$, $n$) & $(0, 0, \\dots, 0)$\\\\\n    \\hline\n    \\end{tabular}\n\\end{center}\n\nIn the classical case of the braid group $B_3$, the space of possible ``linking vector'' analogues is four-dimensional, and spanned by ``symmetric GE'' ($\\sigma_1 \\mapsto (1, 1, 0), \\sigma_2 \\mapsto (0, 1, 1)$), ``row GE'' ($\\sigma_1 \\mapsto (0, 0, 1), \\sigma_2 \\mapsto (1, 0, 0)$), ``zero-ish GE 1'' ($\\sigma_1 \\mapsto (1, -1, 0), \\sigma_2 \\mapsto 0$), and ``zero-ish GE 2'' ($\\sigma_1 \\mapsto 0, \\sigma_2 \\mapsto (0, 1, -1)$). We can derive this by noting that if we have $\\sigma_1 \\mapsto (a_1, b_1, c_1), \\sigma_2 \\mapsto (a_2, b_2, c_2)$, then the single braiding relation $\\sigma_1\\sigma_2\\sigma_1 = \\sigma_2\\sigma_1\\sigma_2$ gives us $c_1 = a_2$ and $a_1 + b_1 = b_2 + c_2$ as constraints after simplification; some elementary algebra then gives us the four basis vectors. \n\nAs for the images of the corresponding homomorphisms of $B_3$ over the pure braids only, the image of symmetric GE is the set of vectors of 3 even integers which sum to 0 modulo 4, the image of row GE is just the set of vectors of 3 even integers, and the images of the two ``zero-ish'' analogs are just all zero.\n\nAs for higher braid groups $B_n$, the ``far commutativity relations'' $\\sigma_i\\sigma_j = \\sigma_j\\sigma_i$ for $|i - j|\\geq 2$ basically constrain all the entries of the ``linking vector'' corresponding to $\\sigma_i$ (which I'll call $\\ell_i$ by analogy) right of the $(i+1)^{\\text{th}}$ to be the same within, and the entries of $\\ell_i$ left of the $i^{\\text{th}}$ to also be the same within. Combined with the braid relations $\\sigma_i\\sigma_{i+1}\\sigma_i = \\sigma_{i+1}\\sigma_i\\sigma_{i+1}$, we get $\\ell_{i, i+2} = \\ell_{i + 1, i}$ as well as $\\ell_{i, i} + \\ell_{i, i+1} = \\ell_{i+1, i+1} + \\ell_{i+1, i+2}$. Hence the set of possible $\\ell_i$ combinations can be spanned by the following:\n\n\\begin{itemize}\n\\item Classical symmetric GE ($\\ell_i = e_i + e_{i + 1}$)\n\\item Row GE for row $i_0$ ($\\ell_{i_0} = \\sum_{i: i > i_0 + 1} e_i, \\ell_{i_0 + 1} = \\sum_{i: i < i_0} e_i, \\ell_i = 0$ elsewise)\n\\item Zero-ish GE for row $i_0$ ($\\ell_{i_0} = e_{i_0} - e_{i_0 + 1}, \\ell_i = 0$ elsewise)\n\\end{itemize}\n\nThere are also corresponding 1-cocycles. We can take linear combinations thereof to get even more 1-cocycles since $\\mathbb{Z}^3$ is abelian.\n\n\\subsection{Super-Gauss-Epple analogues}\n\n\\hspace{\\parindent}Since $\\mathcal{C}$ acts on $\\Phi$, this induces an action of $\\mathcal{C}$ on $\\mathbb{Z}^\\Phi$.\nHence, there is a homomorphism $\\mathcal{A}\\to\\mathbb{Z}^{\\Phi}\\rtimes\\mathcal{C}$, defined by $a\\mapsto(\\alpha_a, \\mathcal{C}(a))$, where $\\mathcal{C}(a)$ is the image of $a$ under the quotient map $\\mathcal{A}\\to\\mathcal{C}$; by analogy, we shall also denote it $SGE$, or $SGE_{\\mathcal{A}}$ when necessary.\n\nThis is a generalization of the super-Gauss-Epple homomorphism, when $\\mathcal{A} = B_n, \\mathcal{C} = S_n$.\nHere, the root system $\\Phi$ is $\\{\\vec{e_i} - \\vec{e_j}|1\\leq i\\neq j\\leq n\\}$, which has size $n(n-1)$, and we identify the $\\vec{e_i} - \\vec{e_j}$ component of $\\mathbb{Z}^{\\Phi}$ with the $(i, j)$th component of a matrix whose diagonal elements are all zero. \n\nWe can show that this is indeed a homomorphism with geometric arguments.\n\n\\begin{theorem}\\label{thm-SGE-welldef-Artingroups}\nThere is a unique homomorphism $\\mathcal{A}\\to\\mathbb{Z}^{\\Phi}\\rtimes\\mathcal{C}$, defined by $a\\mapsto(\\alpha_a, \\mathcal{C}(a))$, where $\\mathcal{C}(a)$ is the image of $a$ under the quotient map $\\mathcal{A}\\to\\mathcal{C}$.\n\\end{theorem}\n\\begin{proof}\nWe show that the braid relations of $\\mathcal{A}$ are satisfied.\nTo do this, we show that the $\\mathbb{Z}^{\\Phi}$ sides are identical.\nAs these values are formal linear combinations of roots (elements of $\\Phi$), we also consider them as multisets, and prove them equal accordingly.\n(We do not consider the $\\mathcal{C}$ sides because these are trivially equal due to the fact that there is a bona-fide homomorphism $\\mathcal{A} \\to \\mathcal{C}$.)\n\nLet $a_1$ and $a_2$ be two generators of $\\mathcal{A}$ for which there is a braid relation.\nFor notational simplicity, we will refer to the corresponding vectors as $\\vec{v}_1 := \\mathcal{C}(a_1), \\vec{v}_2 := \\mathcal{C}(a_2)$.\nWe shall also introduce the notation $*$.\nFor two arbitrary vectors $\\vec{v}$ and $\\vec{w}$, we use $\\vec{v} * \\vec{w}$ to denote the vector that results from reflecting $\\vec{w}$ over the orthogonal hyperplane of $\\vec{v}$.\nWe will compose it right-associatively, so that $\\vec{v} * \\vec{w} * \\vec{x}$ shall refer to $\\vec{v} * (\\vec{w} * \\vec{x})$.\nWe now perform casework depending on the length of this braid relation:\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[height=0.2\\textwidth]{ArtinGenRelLen2.png}\n    \\caption{Two vectors for Coxeter group generators with a braid relation of length 2.}\n    \\label{fig:artingenrellen2}\n\\end{figure}\n\\textbf{Length 2}: We have $a_1 a_2 = a_2 a_1$, so $\\vec{v}_1$ and $\\vec{v}_2$ are orthogonal, as shown in Figure \\ref{fig:artingenrellen2}.\nWe find that $\\vec{v}_1 * \\vec{v}_2 = \\vec{v}_2$ and that $\\vec{v}_2 * \\vec{v}_1 = \\vec{v}_1$.\nTherefore, we have $\\{\\vec{v}_1, \\vec{v}_1 * \\vec{v}_2\\} = \\{\\vec{v}_2, \\vec{v}_2 * \\vec{v}_1\\}$, as desired.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[height=0.19\\textwidth]{ArtinGenRelLen3.png}\n    \\caption{Two vectors for Coxeter group generators with a braid relation of length 3.}\n    \\label{fig:artingenrellen3}\n\\end{figure}\n\\textbf{Length 3}: We have $a_1 a_2 a_1 = a_2 a_1 a_2$, so $\\vec{v}_1$ and $\\vec{v}_2$ have an angle of $120^\\circ$ between them, as shown in Figure \\ref{fig:artingenrellen3}.\nWe find that $\\vec{v}_1 * \\vec{v}_2 = \\vec{v}_2 * \\vec{v}_1$ (both being the vector that bisects the angle between $\\vec{v}_1$ and $\\vec{v}_2$).\nFurthermore, we find that $\\vec{v}_1 * (\\vec{v}_2 * \\vec{v}_1) = \\vec{v}_2$ and $\\vec{v}_2 * (\\vec{v}_1 * \\vec{v}_2) = \\vec{v}_1$.\nTherefore, we have $\\{\\vec{v}_1, \\vec{v}_1 * \\vec{v}_2, \\vec{v}_1 * (\\vec{v}_2 * \\vec{v}_1)\\} = \\{\\vec{v}_2, \\vec{v}_2 * \\vec{v}_1, \\vec{v}_2 * (\\vec{v}_1 * \\vec{v}_2)\\}$, as desired.\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[height=0.22\\textwidth]{ArtinGenRelLen4.png}\n    \\caption{Two vectors for Coxeter group generators with a braid relation of length 4.}\n    \\label{fig:artingenrellen4}\n\\end{figure}\n\\textbf{Length 4}: We have $(a_1 a_2)^2 = (a_2 a_1)^2$, so $\\vec{v}_1$ and $\\vec{v}_2$ have an angle of $135^\\circ$ between them, as shown in Figure \\ref{fig:artingenrellen4}.\nWe find that $\\vec{v}_1 * \\vec{v}_2$ has an angle of $45^\\circ$ from $\\vec{v}_1$ towards $\\vec{v}_2$, and is so orthogonal to $\\vec{v}_2$.\nSimilarly, $\\vec{v}_2 * \\vec{v}_1$ has an angle of $45^\\circ$ from $\\vec{v}_2$ towards $\\vec{v}_1$, and is so orthogonal to $\\vec{v}_1$.\nTherefore, we have $\\vec{v}_1 * (\\vec{v}_2 * \\vec{v}_1) = \\vec{v}_2 * \\vec{v}_1$ and $\\vec{v}_2 * (\\vec{v}_1 * \\vec{v}_2) = \\vec{v}_1 * \\vec{v}_2$.\nFurthermore, we have $\\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2 = \\vec{v}_2$ and $\\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1 = \\vec{v}_1$.\nTherefore, we conclude that $\\{\\vec{v}_1, \\vec{v}_1 * \\vec{v}_2, \\vec{v}_1 * (\\vec{v}_2 * \\vec{v}_1), \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2\\} = \\{\\vec{v}_2, \\vec{v}_2 * \\vec{v}_1, \\vec{v}_2 * (\\vec{v}_1 * \\vec{v}_2, \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1\\}$, as desired.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[height=0.22\\textwidth]{ArtinGenRelLen6.png}\n    \\caption{Two vectors for Coxeter group generators with a braid relation of length 6.}\n    \\label{fig:artingenrellen6}\n\\end{figure}\n\\textbf{Length 6}: We have $(a_1 a_2)^3 = (a_2 a_1)^3$, so $\\vec{v}_1$ and $\\vec{v}_2$ have an angle of $150^\\circ$ between them, as shown in Figure \\ref{fig:artingenrellen6}.\nDivide this angle by four evenly spaced vectors, named $\\vec{w}_1, \\vec{w}_2, \\vec{w}_3, \\vec{w}_4$, so that the order of the vectors is $\\vec{v}_1, \\vec{w}_1, \\vec{w}_2, \\vec{w}_3, \\vec{w}_4, \\vec{v}_2$ (with $30$-degree gaps between each consecutive pair).\nAs before, we can calculate that $\\vec{v}_1, \\vec{v}_1 * \\vec{v}_2, \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1, \\dots, \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2$ is precisely $\\vec{v}_1, \\vec{w}_1, \\vec{w}_2, \\vec{w}_3, \\vec{w}_4, \\vec{v}_2$, in that order.\nSimilarly, we can compute that $\\vec{v}_2, \\vec{v}_2 * \\vec{v}_1, \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2, \\dots, \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1$ is precisely $\\vec{v}_2, \\vec{w}_4, \\vec{w}_3, \\vec{w}_2, \\vec{w}_1, \\vec{v}_1$, in that order.\nHence, we have $\\{\\vec{v}_1, \\vec{v}_1 * \\vec{v}_2, \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1, \\dots, \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2\\} = \\{\\vec{v}_2, \\vec{v}_2 * \\vec{v}_1, \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2, \\dots, \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1 * \\vec{v}_2 * \\vec{v}_1\\}$ as desired, and we are done.\n\\end{proof}\n\n\n\\subsection{Another commutative diagram}\n\n\\hspace{\\parindent}We can summarize this in the following commutative diagram:\n\n\\[\\begin{tikzcd}\n\t{\\mathcal{P}} & {\\mathcal{A}} & {\\mathcal{C}} \\\\\n\t{\\mathcal{P}\/[\\mathcal{P}, \\mathcal{P}]} & {\\mathcal{A}\/[\\mathcal{P}, \\mathcal{P}]}\n\t\\arrow[two heads, from=1-1, to=2-1]\n\t\\arrow[hook', from=2-1, to=2-2]\n\t\\arrow[hook, from=1-1, to=1-2]\n\t\\arrow[\"SGE\", two heads, from=1-2, to=2-2]\n\t\\arrow[two heads, from=2-2, to=1-3]\n\t\\arrow[two heads, from=1-2, to=1-3]\n\\end{tikzcd}\\]\n\nHere, every arrow is either an inclusion map (indicated as being injective) or a quotient map (indicated as being surjective).\n\n\\subsection{More 1-cocycles}\n\n\\hspace{\\parindent}Again, suppose that $\\mathcal{A}$ is an Artin group of finite type, $\\mathcal{C}$ is its associated Coxeter group, $\\Phi$ is an associated root system, and $\\Delta\\subset\\Phi$ is an associated subset of simple roots (and let $\\Delta_a$ be the associated simple root for any Artin generator $a$ of $\\mathcal{A}$).\nDefine the function $F: \\mathcal{A}\\to\\mathbb{Z}^{\\Phi}$ by \n$$\\forall a\\in\\mathcal{A}: SGE(a) = (\\ell, \\mathcal{C}(a)) \\implies F(a) = \\ell - \\overline{\\ell},$$\nwhere $\\ell$ is a vector of $\\mathbb{Z}^{\\Phi}$ and $\\overline{\\ell}$ is the image of $\\ell$ under the rotation\/reflection of $\\mathbb{Z}^{\\Phi}$ given by $e_{\\alpha}\\to e_{-\\alpha}\\forall\\alpha\\in\\Phi$.\nIt is clear that $F$ satisfies the relation $F(ab) = F(a) + \\mathcal{C}(a)\\cdot F(b)$.\n\nThen, we have the following lemmas (generalizing those of Section \\ref{section-braids-1cocy}):\n\n\\begin{lemma}\\label{lem-Fbar-exists-artingroups}\nLet $F: \\mathcal{A}\\to\\mathbb{Z}^{\\Phi}$ be as defined for Artin groups of finite type, and let $p\\in\\mathcal{P}$ (where $\\mathcal{P}$ is the kernel of the quotient map $\\mathcal{A}\\to\\mathcal{C}$). Then $F(p) = \\vec{0}$, and hence the map $F$ descends to a map $\\overline{F}: \\mathcal{C}\\to\\mathbb{Z}^{\\Phi}$\n\\end{lemma}\n\\begin{proof}\nLet $a$ be an Artin generator.\nThen $F(a^2) = F(a) + \\mathcal{C}(a)\\cdot F(a) = (\\phi_{\\Delta_a} - \\phi_{-\\Delta_a}) + (\\phi_{-\\Delta_a} - \\phi_{\\Delta_a}) = \\vec{0}$.\n\nAgain, let $p\\in\\mathcal{P}$, $\\upsilon$ be an Artin generator of $\\mathcal{A}$, and suppose that $F(p) = 0$.\nThen $F(\\upsilon p\\upsilon^{-1}) = F(\\upsilon) + \\mathcal{C}(\\upsilon)\\cdot F(p) + \\mathcal{C}(\\upsilon p)\\cdot F(\\upsilon^{-1}) = (\\phi_{\\Delta_{\\upsilon}} - \\phi_{-\\Delta_{\\upsilon}}) + \\mathcal{C}(\\upsilon)\\cdot\\vec{0} + (\\phi_{-\\Delta_{\\upsilon}} - \\phi_{\\Delta_{\\upsilon}}) = \\vec{0}$.\n\nBy induction, we thus show that $p\\in\\mathcal{P}$, $F(p) = 0$.\nHence, $F$ maps cosets of $\\mathcal{P}$ in $\\mathcal{A}$ into single elements of $\\mathbb{Z}^{\\phi}$, so we conclude $\\overline{F}$ exists and we are done.\n\\end{proof}\n\n\\begin{lemma}\\label{lem-Fbar-zeroone-artingroups}\nLet $\\overline{F}: \\mathcal{C}\\to\\mathbb{Z}^{\\Phi}$ be as defined in Lemma \\ref{lem-Fbar-exists-artingroups}, and let $\\Phi^+$ be the set of positive roots associated to the choice $\\Delta$ of simple roots.\nThen for all $\\phi\\in\\Phi^+, c\\in\\mathcal{C}$, we have $\\overline{F}(c)[\\phi]\\in\\{0, 1\\}$ (where $\\overline{F}(c)[\\phi]$ is the $\\phi$ component of $\\overline{F}(c)$).\n\\end{lemma}\n\\begin{proof}\nIf $c = e$, then the statement is trivial.\nHence, suppose it holds for $c$; we shall prove it for $\\vartheta c$, where $\\vartheta$ is a Coxeter generator of $\\mathcal{C}$ (that is, the image of an Artin generator of $\\mathcal{A}$).\nTo this end, we find, for any $\\phi\\in\\Phi$, that $\\overline{F}(\\vartheta c)[\\phi] = \\overline{F}(\\vartheta)[\\phi] + (\\vartheta\\cdot\\overline{F}(c))[\\phi] = \\overline{F}(\\vartheta)[\\phi] + \\overline{F}(c)[\\vartheta\\star\\phi]$, where $\\vartheta\\star\\phi$ is the image of $\\phi$ under the reflection across the hyperplane orthogonal to $\\alpha_{\\vartheta}$.\nIf $\\alpha_{\\vartheta}\\neq\\phi$, then the first term vanishes, and since $\\vartheta\\star\\phi$ also belongs to $\\Phi^+$, the conclusion follows.\nIf $\\alpha_{\\vartheta} = \\phi$, then we have $\\overline{F}(\\vartheta c)[\\phi] = 1 + \\overline{F}(c)[-\\phi] = 1 - \\overline{F}(c)[\\phi]\\in\\{0, 1\\}$, and the conclusion again follows.\nBy induction on the length of $c$, we are done.\n\\end{proof}\n\nSince this cocycle exists, the image of the super-Gauss-Epple homomorphism is not a semidirect product (as there is no way to embed the Coxeter group as a subgroup).\nHowever, we can show that the image of $\\mathcal{P}$ under the super-Gauss-Epple homomorphism is abelian.\nNote that this is \\textit{not} a semidirect product, for reasons analogous to those given at the end of Section \\ref{section-braids-1cocy} (namely, that no element of $\\mathcal{A}$ whose image in $\\mathcal{C}$ is a canonical generator can have a square whose image under super-Gauss-Epple is that of the identity).\n\nWe can also characterize the image of the super-Gauss-Epple homomorphism the following way:\nNote that, if $c\\in\\mathcal{A}$ and $a$ is a canonical generator of $\\mathcal{A}$, then $SGE(ca^2c^{-1}) = (v_c, \\mathcal{C}(c))(\\Phi_{\\Delta_a} + \\Phi_{-\\Delta_a}, e)(-\\mathcal{C}(c)^{-1}v_c, W(c)) = (\\Phi_{\\mathcal{C}(c)\\cdot\\Delta_a} + \\Phi_{\\mathcal{C}(c)\\cdot\\Delta_a}, e)$.\nHence, the set of all pairs $(v, c)\\in\\mathbb{Z}^{\\Phi}\\rtimes C$ that belong to the image of SGE are the pairs such that $v - \\overline{v} = \\overline{F}(c)$.\n\n\n\\section{Complex reflection groups}\\label{sec-complex}\n\nWe considered a candidate analogue of a ``symmetric super-Gauss-Epple homomorphism'', which would have type signature $B \\to \\mathbb{Z}^{\\mathcal{A}}\\rtimes W$: namely, a homomorphism given by generating relations $s\\to (e_{H_s}, W(s))$, where $H_s$ is a distinguished generator of $B$ whose corresponding element in $W$ is a member of $\\Psi$.\nFor example, in the case of $\\mathcal{C} = G(n, 1, 1) = \\mathbb{Z}\/n\\mathbb{Z}$, we have $\\mathcal{A} = \\mathbb{Z}$, and $\\mathcal{HA}$ consists of a single element $\\mathcal{H}$.\nTherefore, the associated homomorphism is an injective homomorphism of type $\\mathbb{Z}\\to\\mathbb{Z}\/n\\mathbb{Z}\\times\\mathbb{Z}$ given by the generating relation $1\\to (1, 1)$.\nIt is trivial to verify that this is indeed a homomorphism.\n\nHowever, using a geometric argument, we show that no such homomorphism exists, unless $B$ is an Artin group of finite type:\n\n\\begin{theorem}[No SGE for complex reflection groups]\nLet $B$ be some braid group associated to a complex reflection group, $W$ be the associated complex reflection group, and $\\mathcal{HA}$ the associated hyperplane arrangement.\nSuppose that, in the canonical Artin-like presentation of $B$,\nthere is a relation of the form $aba = bab$, where $a$ and $b$ are two distinct generators.\nThen there is no homomorphism $B \\to \\mathbb{Z}^{\\mathcal{A}}\\rtimes W$ given by generating relations $s\\to (e_{H_s}, W(s))$, where $H_s$ is a distinguished generator of $B$ whose corresponding element in $W$ is a member of $\\Psi$.\n\\end{theorem}\n\\begin{proof}\nWe prove $(e_{H_a}, W(a))(e_{H_b}, W(b))(e_{H_a}, W(a)) \\neq (e_{H_b}, W(b))(e_{H_a}, W(a))(e_{H_b}, W(b))$.\n\nWe can easily compute that the former part of the left hand side  will be $e_{H_a} + e_{W(a)H_b} + e_{W(a)W(b)e_{H_a}}$,\nand the former part of the right hand side would be $e_{H_b} + e_{W(b)H_a} + e_{W(b)W(a)e_{H_b}}$.\nWe claim that $H_a = W(b)W(a)H_b, H_b = W(a)W(b)H_a$, but that $W(a)H_b \\neq W(b)H_a$.\n\nFirstly, however, we shall need to use the determinant trick.\nWe have $W(a)W(b)W(a) = W(b)W(a)W(b)$.\nSince both sides are linear transformations, we can take determinants: $\\det(W(a))\\det(W(b))\\det(W(a)) = \\det(W(b))\\det(W(a))\\det(W(b))$.\nSince these determinants are nonzero real numbers, we can cancel to obtain $\\det(W(a)) = \\det(W(b))$.\nThis equality is important because letting $\\zeta_{W(a)}$ be the multiplier of $W(a)$ and similarly for $\\zeta_{W(b)}$, we can conclude that $\\zeta_{W(a)} = \\zeta_{W(b)}$, and will refer to both constants as merely $\\zeta$.\n\nNow, to show $H_a = W(b)W(a)H_b$, let $s_{\\zeta, H}$ be the pseudo-reflection that multiplies by $\\zeta$ and preserves $H$. \nWe compute\n\\begin{align*}\n    s_{W(b)W(a)H_b, \\zeta} &= W(b)W(a)s_{H_b, \\zeta}(W(b)W(a))^{-1}\\\\\n    &= W(b)W(a)W(b)(W(b)W(a))^{-1}\\\\\n    &= W(a)W(b)W(a)(W(b)W(a))^{-1}\\\\\n    &= W(a)\\\\\n    &= s_{H_a, \\zeta},\n\\end{align*}\nwhich gives the claim.\nBy symmetry, we can also obtain $H_b = W(a)W(b)H_a$.\n\nNow to show that $W(a)H_b \\neq W(b)H_a$, we compute\n\n\\begin{align*}\n    s_{W(a)H_b, \\zeta} &= W(a)s_{H_b, \\zeta}W(a)^{-1}\\\\\n    &= W(a)W(b)W(a)^{-1}\\\\\n    &= (W(a)W(b)W(a))W(a)^{-2},\n\\end{align*}\nand similarly $s_{W(b)H_a, \\zeta} = (W(b)W(a)W(b))W(b)^{-2} = (W(a)W(b)W(a))W(b)^{-2}$.\nObserve now that $W(a)^2 = s_{H_a, \\zeta^2}$ and $W(b)^2 = s_{H_b, \\zeta^2}$. \nSince $a$ and $b$ are distinct, this means that $H_a$ and $H_b$ are distinct.\nThis in turn implies that $s_{H_a, \\zeta^2}$ and $s_{H_b, \\zeta^2}$ are distinct unless $\\zeta^2 = 1$.\nHowever, if $\\zeta^2 = 1$, then $\\zeta = -1$, so that $a$ and $b$ are merely real reflections, as desired.\n\\end{proof}\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nAll groups considered in this article are finite. \nThe following sufficient condition for nilpotency of a group $G$ was discovered by B. Baumslag and J. Wiegold \\cite{baubau}.\n\\medskip\n\n{\\it Let $G$ be a group in which $|ab|=|a||b|$ whenever the elements $a,b$ have coprime orders. Then $G$ is nilpotent.}\n\\medskip\n\nHere the symbol $|x|$ stands for the order of the element $x$ in a group $G$. In \\cite{BS} a similar sufficient condition for nilpotency of the commutator subgroup $G'$ was established. \n\\medskip\n\n{\\it Let $G$ be a group in which $|ab|=|a||b|$ whenever the elements $a,b$ are commutators of coprime orders. Then $G'$ is nilpotent.}\n\\medskip\n\nOf course, the conditions in both above results are also necessary for the nilpotency of $G$ and $G'$, respectively. In the present article we extend the above results as follows.\n\nGiven an integer $k\\geq1$, the word $\\gamma_{k}=\\gamma_k(x_1,\\dots,x_k)$ is defined inductively by the formulae\n\\[\n\\gamma_1=x_1,\n\\qquad \\text{and} \\qquad\n\\gamma_k=[\\gamma_{k-1},x_k]=[x_1,\\ldots,x_k]\n\\quad\n\\text{for $k\\ge 2$.}\n\\]\nThe subgroup of a group $G$ generated by all values of the word $\\gamma_k$ is denoted by $\\gamma_k(G)$. Of course, this is the familiar $k$th term of the lower central series of $G$. If $k=2$ we have $\\gamma_k(G)=G'$. In the sequel the values of the word $\\gamma_k$ in $G$ will be called $\\gamma_k$-commutators.\n\n\\begin{theorem}\\label{main} The $k$th term of the lower central series of a group $G$ is nilpotent if and only if $|ab|=|a||b|$ for any $\\gamma_k$-commutators $a,b\\in G$ of coprime orders. \n\\end{theorem}\n\nRecall that a group $G$ is called metanilpotent if there is a normal nilpotent subgroup $N$ such that $G\/N$ is nilpotent. The following corollary is immediate.\n\n\\begin{corollary}\\label{ain} A group $G$ is metanilpotent if and only if there exists a positive integer $k$ such that $|ab|=|a||b|$ for any $\\gamma_k$-commutators $a,b\\in G$ of coprime orders. \n\\end{corollary}\n\nWe suspect that a similar criterion of nilpotency of the $k$th term of the derived series of $G$ can be established. On the other hand, Kassabov and Nikolov showed in \\cite{kani} that for any $n\\geq7$ the alternating group $A_n$ admits a commutator word all of whose nontrivial values have order 3. Thus, the verbal subgroup $w(G)$ need not be nilpotent even if all $w$-values have order dividing 3.\n\n\n\\section{Proofs}\nAs usual, if $\\pi$ is a set of primes, we denote by $\\pi'$ the set of all primes that do not belong to $\\pi$. For a group $G$ we denote by $\\pi(G)$ the set of primes dividing the order of $G$. The maximal normal $\\pi$-subgroup of $G$ is denoted by $O_{\\pi}(G)$. The Fitting subgroup of $G$ is denoted by $F(G)$. The Fitting height of $G$ is denoted by $h(G)$. Throughout the article we use without special references the well-known properties of coprime actions:  if $\\alpha$ is an automorphism of a finite group $G$ of coprime order, $(|\\alpha|,|G|)=1$, then $C_{G\/N}(\\alpha)=C_G(\\alpha)N\/N$ for any $\\alpha$-invariant normal subgroup $N$, the equality $[G,\\alpha]=[[G,\\alpha],\\alpha]$ holds, and if $G$ is in addition abelian, then $G=[G,\\alpha]\\times C_G(\\alpha)$. Here $[G,\\alpha]$ is the subgroup of $G$ generated by the elements of the form $g^{-1}g^\\alpha$, where $g\\in G$.\n\nFor elements $x,y$ of a group $G$ write $[x,{}_0y]=x$ and $[x,{}_{i+1}y]=[[x,{}_{i}y],y]$ for $i\\geq0$. An element $y\\in G$ is called Engel if for any $x\\in G$ there is a positive integer $n=n(x)$ such that $[x,{}_{n}y]=1$.\n\nThe following lemma is well-known. \n\\begin{lemma} \\label{meta} Let $p$ be a prime and $G$ a metanilpotent group.  Suppose that $x$ is a $p$-element in $G$ such that $[O_{p'}(F(G)),x]=1$. Then $x\\in F(G)$. \n\\end{lemma}\n\n\\begin{proof} Since all Engel elements of a finite group lie in the Fitting subgroup \\cite[12.3.7]{Rob}, it is sufficient to show that $x$ is an Engel element. Let $F=F(G)$ and $P$ be the Sylow $p$-subgroup of $F$. We have $F= P\\times O_{p'}(F)$. By hypothesis, $G\/F$ is nilpotent of class $n$ for some positive integer $n$. We deduce that $[G,{}_{n}x]\\leq F$ and so $[G,_{n+1}x] \\leq P$. Therefore $\\langle[G,_{n+1}x],x\\rangle$ is contained in a Sylow $p$-subgroup of $G$. Hence, $x$ is an Engel element in $G$ and so the lemma follows. \n\\end{proof}\n\n\\begin{lemma}\\label{gamma} Let $k$ be a positive integer and $G$ a group such that $G=G'$. Let $q\\in\\pi(G)$. Then $G$ is generated by $\\gamma_k$-commutators of $p$-power order for primes $p\\neq q$.  \n\\end{lemma}\n\\begin{proof} For each prime $p\\in\\pi(G)\\setminus\\{q\\}$ let $N_p$ denote the subgroup generated by all $\\gamma_k$-commutators of $p$-power order. Let us show first that for each $p$ the Sylow $p$-subgroups of $G$ are contained in $N_p$. Suppose that this is false and choose $p$ such that a Sylow $p$-subgroup of $G$ is not contained in $N_p$. We can pass to the quotient $G\/N_p$ and assume that $N_p=1$. Since $G=G'$, it is clear that $G$ does not possess a normal $p$-complement. Therefore the Frobenius Theorem \\cite[Theorem 7.4.5]{go} shows that $G$ has a $p$-subgroup $H$ and a $p'$-element $a\\in N_G(H)$ such that $[H,a]\\neq1$. We have $$1\\neq[H,a]=[H,\\underbrace{a,\\ldots,a}_{(k-1)\\ times}]\\leq N_p,$$ a contradiction. Therefore indeed $N_p$ contains the Sylow $p$-subgroups of $G$. Let $T$ be the product of all $N_p$ for $p\\neq q$. We see that $G\/T$ is a $q$-group. Since $G=G'$, we conclude that $G=T$. The proof is complete.\n\\end{proof}\n\nLet us call a subgroup $H$ of $G$ a tower of height $h$ if $H$ can be written as a product $H=P_1\\cdots P_h$, where\n\n(1) $P_i$ is a $p_i$-group ($p_i$ a prime) for $i=1,\\dots,h$.\n\n(2) $P_i$ normalizes $P_j$ for $i<j$.\n\n(3) $[P_i,P_{i-1}]=P_i$ for $i=2,\\dots,h$. \\\\ \n\nIt follows from (3) that $p_i\\neq p_{i+1}$ for $i=1,\\dots,h-1$. A finite soluble group $G$ has Fitting height at least $h$ if and only if $G$ possesses a tower of height $h$ (see for example \\cite{turull}). \\\\\n\nThroughout the remaining part of the article $G$ denotes a finite group for which there exists $k\\geq1$ such that $|ab|=|a||b|$ for any $\\gamma_k$-commutators $a,b\\in G$ of coprime orders. We denote by $X$ the set of all $\\gamma_k$-commutators in $G$.\n\n\n\\begin{lemma}\\label{bbb} Let $x\\in X$ and $N$ be a subgroup normalized by $x$. If $(|N|,|x|)=1$, then $[N,x]=1$.\n\\end{lemma}\n\\begin{proof} Choose $y\\in N$. The order of the $\\gamma_k$-commutator $[x,y]$ is prime to that of $x$. Therefore we must have $|x[x,y]|=|x||[x,y]|$. However $x[x,y]=y^{-1}xy$. This is a conjugate of $x$ and so $|x[x,y]|=|x|$. Therefore $[x,y]=1$.\n\\end{proof}\n\n\\begin{lemma}\\label{solu} If $G$ is soluble, then the subgroup $\\gamma_k(G)$ is nilpotent.\n\\end{lemma}\n\\begin{proof} Set $h=h(G)$ and $F = F(G)$. If $G$ is nilpotent, the result is immediate so we assume that $h\\geq 2$.\n\nWe first examine the case $h=2$. If $G\/F$ has nilpotency class at most $k-1$, then $\\gamma_k(G) \\leq F$ is nilpotent. So we will assume that $G\/F$ is nilpotent of class at least $k$. Hence, the image of some Sylow $p$-subgroup $P$ of $G$ in $G\/F$ has nilpotency class at least $k$. Therefore, there exists a $\\gamma_k$-commutator $x$ in elements of $P$ which does not belong to $F$. By Lemma \\ref{bbb} $[O_{p'}(F),x]=1$, whence by Lemma \\ref{meta}, $x\\in F$. This is a contradiction.\n\nNow assume that $h\\geq 3$. By \\cite[Lemma 1.9]{turull}, there exists a tower $P_1P_2P_3\\ldots P_h$ of height $h$ in $G$. Since $P_2=[P_2,P_1]$, it follows that $$P_2=[P_2,\\underbrace{P_1, \\ldots,P_1}_{(k-1)\\ times}].$$ \nCombining Lemma \\ref{bbb} with the fact that $P_2$ is generated by $\\gamma_k$-commutators of $G$ of $p_2$-orders, we deduce that $P_3$ commutes with $P_2$. On the other hand, $[P_3,P_2]=P_3$, because $P_1P_2P_3\\ldots P_h$ is a tower. This is a contradiction.  The proof is complete. \n\\end{proof}\n\n\n\n\nWe are now in a position to complete the proof of Theorem \\ref{main}.  \n\n\n\\begin{proof}[Proof of Theorem \\ref{main}] It is clear that if $\\gamma_k(G)$ is nilpotent, then $|ab|=|a||b|$ for any $\\gamma_k$-commutators $a,b\\in G$ of coprime orders. So we only need to prove the converse. Since the case where $k\\leq2$ was considered in \\cite{BS} and \\cite{baubau}, we will assume that $k\\geq3$.\n\nSuppose that the theorem is false and let $G$ be a counterexample of minimal order. In view of Lemma \\ref{solu} $G$ is not soluble while all proper subgroups of $G$ are. Therefore $G=G'$. Let $R$ be the soluble radical of $G$. It follows that $G\/R$ is nonabelian simple. By Lemma \\ref{solu} $\\gamma_k(R)$ is nilpotent. Suppose that $R\\neq1$. \n\nChoose $q\\in\\pi(F(G))$. According to Lemma \\ref{gamma} $G$ is generated by the $\\gamma_k$-commutators of $p$-power order for primes $p\\neq q$. Let $Q$ be the Sylow $q$-subgroup of $F(G)$. By Lemma \\ref{bbb}, $[Q,x]=1$, for every $\\gamma_k$-commutator $x$ of $q'$-order. Therefore $Q\\leq Z(G)$. This happens for each choice of $q\\in\\pi(F(G))$ so we conclude that $F(G)\\leq Z(G)$.  \n\nFor each $x\\in R$ and $y\\in G$ we have $$[y,\\underbrace{x,\\ldots,x}_{k\\ times}]\\in\\gamma_k(R)\\leq F(G)=Z(G).$$ Consequently, every element $x\\in R$ is Engel. Therefore $R\\leq F(G)$ (\\cite[12.7.4]{Rob}). Hence, $R=Z(G)$ and $G$ is quasisimple.   \n\nSince $G$ does not possess a normal 2-complement, it follows from the Frobenius Theorem \\cite[Theorem 7.4.5]{go} that $G$ contains a 2-subgroup $H$ and an element of odd order $b\\in N_G(H)$ such that $[H,b]\\neq1$. In view of Thompson's Theorem \\cite[Theorem 5.3.11]{go} we can assume that $H$ is of nilpotency class at most two and $H\/Z(H)$ is elementary abelian. We claim that $G$ contains an element $a\\in X$ such that $a$ is a 2-element and $a$ has order 2 modulo $Z(G)$. \n\nRecall that $k\\geq3$. Let $Y$ be the set of all elements $[h,{}_{(k-2)}b]$, where $h\\in H$. Since $H$ is nilpotent of class at most $2$, it follows that for every $y\\in Y$ all elements in $[H,y]$ are $\\gamma_k$-commutators. If for some $y\\in Y$ the subgroup $[H,y]$ does not lie in $Z(G)$, every element of $[H,y]$ having order $2$ modulo $Z(G)$ enjoys the required properties. Therefore we assume that $[H,y]\\leq Z(G)$ for every $y\\in Y$. Let $K$ be the subgroup generated by $Y$. Obviously, $K=[K,b]$ and $K'\\leq Z(G)$. It follows that $C_K(b)\\leq Z(G)$ and all elements in $K$ are $\\gamma_k$-commutators modulo $Z(G)$. If $d\\in K$ such that $d\\not\\in Z(G)$ and $d^2\\in Z(G)$, then the commutator $[d,b]$ is as required.\n\nNow we fix an element $a$ with the above properties. Since $G\/Z(G)$ is nonabelian simple, it follows from the Baer-Suzuki Theorem \\cite[Theorem 3.8.2]{go} that there exists an element $t\\in G$ such that the order of $[a,t]$ is odd. On the one hand, it is clear that $a$ inverts $[a,t]$. On the other hand, by Lemma \\ref{bbb}, $a$ centralizes $[a,t]$. This is a contradiction.\n\\end{proof}\n\\section{Acknowledgment}\nThe work of the first and the third authors was supported by CNPq-Brazil and FAPDF. The second author was partially supported by the National Group for Algebraic and Geometric Structures, and their Applications  (GNSAGA -- INdAM). This article was written during the second author's visit to the University of Brasilia. He wishes to thank the Department of Mathematics for excellent hospitality.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}