A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements
By Barbara Baumeister, Matthew Dyer, Christian Stump, and Patrick Wegener
Abstract
In this note, we provide a short and self-contained proof that the braid group on $n$ strands acts transitively on the set of reduced factorizations of a Coxeter element in a Coxeter group of finite rank $n$ into products of reflections. We moreover use the same argument to also show that all factorizations of an element in a parabolic subgroup of $W$ also lie in this parabolic subgroup.
1. Introduction
Let $(W,T)$ be a dual Coxeter system of finite rank $m$ in the sense of Reference Bes03. This is to say that there is a subset $S \subseteq T$ with $|S| = m$ such that $(W,S)$ is a Coxeter system, and $T = \big \{ wsw^{-1} : w \in W, s \in S \big \}$ is the set of reflections for the Coxeter system $(W,S)$. We then call $(W,S)$ a simple system for $(W,T)$ and $S$ a set of simple reflections. Such simple systems for $(W,T)$ were studied by several authors, see e.g. Reference FHM06 and the references therein. In particular, if $S$ is a simple system for $(W,T)$ then so is $wSw^{-1}$ for any $w \in W$. It is moreover shown in Reference FHM06 that for important classes, all simple systems for $(W,T)$ are conjugate to one another in this sense.
A reflection subgroup$W'$ is a subgroup of $W$ generated by reflections. It is well known that $(W',W'\cap T)$ is again a dual Coxeter system, see e.g. Reference Dye90. For $w \in W$, a reduced $T$-factorization of $w$ is a shortest length factorization of $w$ into reflections, and we denote by $\operatorname {Red}_T(w)$ the set of all such reduced $T$-factorizations. Similarly for a given simple system $(W,S)$, a reduced $S$-factorization of $w$ is a shortest length factorization of $w$ into simple reflections. An element $c \in W$ is called a parabolic Coxeter element for $(W,T)$ if there is a simple system $S = \{ s_1,\ldots ,s_m \}$ such that $c = s_1\cdots s_n$ for some $n \leq m$. We call the reflection subgroup generated by $\{s_1,\ldots ,s_n\}$ a parabolic subgroup. The element $c$ is moreover called a standard parabolic Coxeter element for the Coxeter system $(W,S)$.
The braid group on $n$ strands is the group $B_n$ with generators $\sigma _1,\ldots , \sigma _{n-1}$ subject to the relations
for any $r,s\in T$. Note that in this case, the $B_{2}$-orbit of $(r,s)$ is the set of all pairs $(t_{1},t_{2})$ of reflections of the subgroup $\langle r,s\rangle$, such that $t_{1}t_{2}=rs$.
The following lemma is a direct consequence of the definition.
This action on $\operatorname {Red}_{T'}(w)$ is also known as the Hurwitz action. For finite Coxeter systems, the Hurwitz action was first shown to act transitively on $\operatorname {Red}_T(c)$ for a Coxeter element $c$ in a letter from P. Deligne to E. Looijenga Reference Del74. The first published proof is due to D. Bessis and can be found in Reference Bes03. K. Igusa and R. Schiffler generalized this result to arbitrary Coxeter groups of finite rank; see Reference IS10, Theorem 1.4. This transitivity has important applications in the theory of Artin groups, see Reference Bes03Reference Dig06, and as well as in the representation theory of algebras; see Reference IS10Reference Igu11Reference HK13.
The aim of this note is to provide a simple proof of K. Igusa and R. Schiffler’s theorem, based on arguments similar to those in Reference Dye01. We moreover emphasize that the condition on the Coxeter element $c \in W$ in this note is slightly relaxed from the condition in the original theorem; compare Reference IS10, Theorem 1.4.
By the observation in Lemma 1.2, this theorem has the direct consequence that the parabolic subgroup $\langle s_1,\ldots ,s_n\rangle$ of $W$ does indeed not depend on the particular $S$-factorization$c = s_1 \cdots s_n$ but only on the parabolic Coxeter element $c$ itself. We thus denote this parabolic by $W_c := \langle t_1,\ldots ,t_n \rangle$ for any $T$-factorization$c = t_1 \cdots t_n$. We moreover obtain that $\operatorname {Red}_T(c) = \operatorname {Red}_{T'}(c)$ with $T' = W_c \cap T$ being the set of reflections in the parabolic subgroup $W_c$. The main argument in the proof of this theorem (see Proposition 2.2 below) will also imply the following theorem that extends this direct consequence to all elements in a parabolic subgroup.
2. The proof
For the proof of the two theorems, we fix a Coxeter system $(W,S)$. Denote by $\ell = \ell _S$ and by $\ell _T$ the length function on $W$ with respect to the simple generators $S$ and with respect to the generating set $T$, respectively. Since $S \subseteq T$, we have that $\ell _T(w) \leq \ell (w)$ for all $w \in W$.
The following lemma provides an alternative description of standard parabolic Coxeter elements.
Define the Bruhat graph$\Omega$ for the dual Coxeter system $(W,T)$ as the undirected graph on vertex set $W$ with edges given by $w \mathbin{\vcenter{\img[][11pt][2pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5404b2a85ac92439f447736fb2f6a292.svg}}}wt$ for $t \in T$. For any factorization $w = t_1 \cdots t_n \in W$ with $t_{i}\in T$ and any $x\in W$, there is a corresponding path
from $x$ to $xw$ in $\Omega$. It is clear that this $T$-factorization of $w$ is reduced if and only if the corresponding path from $x$ to $xw$ has minimal length among paths from $x$ to $xw$ for some (equivalently, every) $x\in W$. The simple system $(W,S)$ induces an orientation on $\Omega$ given by $w \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}wt$ if $\ell (w) < \ell (wt)$. We denote the resulting directed Bruhat graph by $\Omega _{\operatorname {dir}}$.
The proof of the two main results is based on the case $x=e$ of the following proposition.
Given this proposition, we are finally in the position to prove the two main results of this note.
As an example of the construction in the previous proof, consider the path
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Show rawAMSref\bib{3294251}{article}{
author={Baumeister, Barbara},
author={Dyer, Matthew},
author={Stump, Christian},
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title={A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements},
journal={Proc. Amer. Math. Soc. Ser. B},
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pages={149-154},
issn={2330-1511},
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}
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