A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements

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 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. 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 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. 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 $(W,S)$.

Remark 1.1.

Observe that this definition of parabolic Coxeter elements and parabolic subgroups is more general than usual. The simplest Coxeter group for which this definition is indeed more general than considering conjugates of a fixed simple system is the finite Coxeter group of type $H_2 = I_2(5)$ given by all linear transformations of the plane that leave a regular pentagon invariant. One choice for a simple system is given by two reflections through two consecutive vertices of the pentagon, another choice is the product of two reflections through two vertices with distance two. Both choices of simple systems generate the same set of reflections, even though they are not conjugate.

The braid group on $n$ strands is the group $B_n$ with generators $\sigma _1,\ldots , \sigma _{n-1}$ subject to the relations

$$\begin{align*} \sigma _i \sigma _j & = \sigma _j \sigma _i \quad \text{for } |i-j| > 1,\\ \sigma _i \sigma _{i+1} \sigma _i & = \sigma _{i+1} \sigma _i \sigma _{i+1}. \end{align*}$$

It acts on the set $T^n$ of $n$-tuples of reflections as

$$\begin{align*} \sigma _i (t_1 ,\ldots , t_n ) &= (t_1 ,\ldots , t_{i-1} , \hspace{5pt} t_i t_{i+1} t_i, \hspace{5pt} \phantom {t_{i+1}}t_i\phantom {t_{i+1}}, \hspace{5pt} t_{i+2} , \ldots , t_n), \\ \sigma _i^{-1} (t_1 ,\ldots , t_n ) &= (t_1 ,\ldots , t_{i-1} , \hspace{5pt} \phantom {t_i}t_{i+1}\phantom {t_i}, \hspace{5pt} t_{i+1}t_it_{i+1}, \hspace{5pt} t_{i+2} , \ldots , t_n). \end{align*}$$

For example, if $n=2$, then the action of $\sigma _{1}$ is described by

$$\begin{equation*} \ldots \mapsto (srs,srsrs)\mapsto (s,srs)\mapsto (r,s)\mapsto (rsr,r)\mapsto (rsrsr,rsr)\mapsto \ldots \end{equation*}$$

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.

Lemma 1.2.

Let $W'$ be a reflection subgroup of $W$ and let $T' = T \cap W'$ be the set of reflections in $W'$. For an element $w \in W'$ with reduced $T$-factorization $w=t_1\cdots t_n$, the braid group on $n$ strands acts on $\operatorname {Red}_{T'}(w)$.

Proof.

Let $\sigma _i(t_1,\ldots ,t_n) = (t'_1,\ldots ,t'_n)$. The lemma follows from the two observations that $t_1 \cdots t_n = t'_1 \cdots t'_n$ and $\{ t_1,\ldots ,t_n \} \subseteq T'$ if and only if $\{ t'_1,\ldots ,t'_n \} \subseteq T'$.

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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 Del74. The first published proof is due to D. Bessis and can be found in Bes03. K. Igusa and R. Schiffler generalized this result to arbitrary Coxeter groups of finite rank; see IS10, Theorem 1.4. This transitivity has important applications in the theory of Artin groups, see Bes03Dig06, and as well as in the representation theory of algebras; see IS10Igu11HK13.

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 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 IS10, Theorem 1.4.

Theorem 1.3.

Let $(W,T)$ be a dual Coxeter system of finite rank $m$ and let $c = s_1\cdots s_n$ be a parabolic Coxeter element in $W$. The Hurwitz action on $\operatorname {Red}_T(c)$ is transitive. In symbols, for each $(t_1,\ldots ,t_n) \in T^n$ such that $c = t_1 \cdots t_n$, there is a braid group element $b \in B_n$ such that

$$b(t_1,\ldots ,t_n) = (s_1,\ldots ,s_n).$$

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.

Theorem 1.4.

Let $W'$ be a parabolic subgroup of $W$. Then for any $w\in W'$,

$$\operatorname {Red}_T(w) = \operatorname {Red}_{T'}(w),$$

where $T' = W' \cap T$ is the set of reflections in $W'$.

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.

Lemma 2.1.

An element $w \in W$ is a standard parabolic Coxeter element for $(W,S)$ if and only if $\ell _T(w) = \ell (w)$.

Proof.

Given a reduced $S$-factorization $w = s_{i_1} \ldots s_{i_k}$, it was shown in Dye01, Theorem 1.1 that $\ell _T(w)$ is given by the minimal number of simple generators that can be removed from $s_{i_1} \ldots s_{i_k}$ to obtain the identity. This yields that $\ell _T(w) = k = \ell (w)$ if and only if $s_{i_1} \ldots s_{i_k}$ does not contain any generator twice.

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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

$$x \mathbin{\vcenter{\img[][11pt][2pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5404b2a85ac92439f447736fb2f6a292.svg}}}xt_1 \mathbin{\vcenter{\img[][11pt][2pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5404b2a85ac92439f447736fb2f6a292.svg}}}xt_1t_2 \mathbin{\vcenter{\img[][11pt][2pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5404b2a85ac92439f447736fb2f6a292.svg}}}\ldots \mathbin{\vcenter{\img[][11pt][2pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5404b2a85ac92439f447736fb2f6a292.svg}}}xt_1\cdots t_n = xw$$

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.

Proposition 2.2.

Let $(W,S)$ be a Coxeter system. Moreover, let $w = t_1 \cdots t_n \in W$ be a reduced $T$-factorization of an element in $W$, and let

$$x \mathbin{\vcenter{\img[][11pt][2pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5404b2a85ac92439f447736fb2f6a292.svg}}}xt_1 \mathbin{\vcenter{\img[][11pt][2pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5404b2a85ac92439f447736fb2f6a292.svg}}}xt_1t_2 \mathbin{\vcenter{\img[][11pt][2pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5404b2a85ac92439f447736fb2f6a292.svg}}}\ldots \mathbin{\vcenter{\img[][11pt][2pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5404b2a85ac92439f447736fb2f6a292.svg}}}xt_1\cdots t_n = xw$$

be the corresponding path in $\Omega$ starting at an element $x \in W$. Then there is a $T$-factorization $w = t'_1 \cdots t'_n$ in the Hurwitz orbit of the $T$-factorization $t_1 \cdots t_n$ such that the corresponding path in $\Omega _{\operatorname {dir}}$ starting at $x$ is first decreasing in length, then increasing; more precisely, it is of the form

$$x\mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[<-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img40a6a1e62faf9be3cac192e3452f2154.svg}}}xt'_1 \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[<-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img40a6a1e62faf9be3cac192e3452f2154.svg}}}xt'_1t'_2\mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[<-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img40a6a1e62faf9be3cac192e3452f2154.svg}}}\ldots \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[<-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img40a6a1e62faf9be3cac192e3452f2154.svg}}}xt_{1}'\cdots t_{i}'\mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}\ldots \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}xt'_1 \cdots t'_n = xw$$

for some (unique) integer $i$ with $0\leq i\leq n$. In the special case $x=e$, this gives a directed path

$$e\mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}t'_1 \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}t'_1t'_2 \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}\ldots \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}t'_1 \cdots t'_n = w$$

in $\Omega _{\operatorname {dir}}$.

Proof.

First consider two distinct reflections $t_1$ and $t_2$ and an element $z \in W$ such that $z \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}zt_1 \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[<-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img40a6a1e62faf9be3cac192e3452f2154.svg}}}zt_1t_2$ in $\Omega _{\operatorname {dir}}$. We claim that there exist reflections $t'_1,t'_2 \in \langle t_1,t_2 \rangle$ with $t_1t_2 = t'_1t'_2$ such that $z \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}zt'_1 \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}zt'_1t'_2$ or $z \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[<-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img40a6a1e62faf9be3cac192e3452f2154.svg}}}zt'_1 \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[<-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img40a6a1e62faf9be3cac192e3452f2154.svg}}}zt_{1}'t_{2}'$ or $z \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[<-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img40a6a1e62faf9be3cac192e3452f2154.svg}}}zt'_1\mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}zt_{1}'t_{2}'$. This implies, by the comment before Lemma 1.2, that one can get from the factorization $t_1 t_2$ to the factorization $t'_1 t'_2$ inside $W' = \langle t_1,t_2 \rangle$ by braid moves, and hence in particular that $W' = \langle t'_1,t'_2 \rangle$. Moreover, one has $\ell (zt_{1}')< \max (\ell (z),\ell (zt_{1}t_{2}))<\ell (zt_{1})$.

To prove the claim, consider the coset $z\langle t_1,t_2 \rangle$ in $W$. By Dye01, Theorem 2.1, the proof immediately reduces to the case $z = e$ and $W' = \langle t_1,t_2 \rangle$ dihedral. We check this case directly. To this end, let $s'_1,s'_2$ be the Coxeter generators of $W'$, and observe that any reflection (element of odd length) and any rotation (element of even length) in $W'$ are joined by an edge in $\Omega$, which in $\Omega _{\operatorname {dir}}$ is oriented towards the element of greater length with respect to the generating set $s'_1,s'_2$. Given $z \in W'$, there are three situations: either $\ell (z) < \ell (zt_1t_2)$, or $\ell (z) > \ell (zt_1t_2)$, or $\ell (z) = \ell (zt_1t_2)$. This implies that one can choose $t'_1$ and $t'_2$ with $t'_1 t'_2 = t_1t_2$ in the three situations such that $z \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}zt'_1 \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}zt'_1t'_2$, $z \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[<-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img40a6a1e62faf9be3cac192e3452f2154.svg}}}zt'_1 \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[<-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img40a6a1e62faf9be3cac192e3452f2154.svg}}}zt'_1t'_2$ or $z \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[<-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img40a6a1e62faf9be3cac192e3452f2154.svg}}}zt'_1 \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}zt'_1t'_2$ respectively (note that in the third case, one has $z\neq e$ or else $l(zt_{1}t_{2})=l(z)$ implies $t_{1}=t_{2}$, contrary to assumption).

Consider the path in $\Omega$ attached to $w=t_{1}\cdots t_{n}$ and beginning at $x$. Any subpath $z\mathbin{\vcenter{\img[][11pt][2pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5404b2a85ac92439f447736fb2f6a292.svg}}}zt_{1}\mathbin{\vcenter{\img[][11pt][2pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5404b2a85ac92439f447736fb2f6a292.svg}}}zt_{1}t_{2}$ as in the claim may be replaced by a path $z\mathbin{\vcenter{\img[][11pt][2pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5404b2a85ac92439f447736fb2f6a292.svg}}}zt'_{1}\mathbin{\vcenter{\img[][11pt][2pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5404b2a85ac92439f447736fb2f6a292.svg}}}zt'_{1}t'_{2}$ as there, to give a new path from $x$ to $xw$ of the same length $n$; we call this a “replacement.” Apply to the original path a sequence of successive replacements. Any path so obtained corresponds to the path beginning at $x$ attached to some reduced $T$-factorization of $w$ in the same Hurwitz orbit as $t_{1}\cdots t_{n}$, and is a shortest path in $\Omega$ from $x$ to $xw$. Note that a replacement of any subpath $z \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}zt_{1}' \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[<-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img40a6a1e62faf9be3cac192e3452f2154.svg}}}zt'_{1}t'_{2}$ of such a path is possible since the path’s minimal length implies that $t_{1}'\neq t_{2}'$. Each replacement decreases the total sum of the $\ell$-lengths of the vertices of the path, so eventually one obtains a path in which no further replacements are possible i.e. of the desired decreasing-then-increasing form. Finally, if $x=e$, then $i=0$ since there are no paths $e\mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[<-] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img40a6a1e62faf9be3cac192e3452f2154.svg}}}t$.

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Given this proposition, we are finally in the position to prove the two main results of this note.

Proof of Theorem 1.4.

Consider a Coxeter system $(W,S)$, a reflection subgroup $W'$ of $W$. It is known that the directed Bruhat graph for $W'$ corresponding to the simple system of $W'$ induced by $S$ is the full subgraph $\Omega _{\operatorname {dir}}\big |_{W'}$ of $\Omega _{\operatorname {dir}}$ on vertex set $W'$; see Dye01, Theorem 2.1.

Let $w \in W'$. Then Lemma 1.2, Proposition 2.2 (with $x=e$), and the discussion before Proposition 2.2 imply that $\operatorname {Red}_T(w) = \operatorname {Red}_{T'}(w)$ if and only if every shortest directed path from $e$ to $w$ in $\Omega _{\operatorname {dir}}$ lies inside $\Omega _{\operatorname {dir}}\big |_{W'}$.

Now assume that $W'$ is a standard parabolic subgroup generated by some subset of $S$. Then it is well known that every reduced $S$-factorization for $w\in W'$ is actually inside $W'$. (To see this, recall that Coxeter systems have the property that all reduced $S$-factorizations for $w$ are related by braid relations.) It therefore follows that in this situation any shortest directed path from $e$ to $w$ in $\Omega _{\operatorname {dir}}$ indeed lies inside $\Omega _{\operatorname {dir}}\big |_{W'}$. The theorem follows by the above equivalence.

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Proof of Theorem 1.3.

Again, fix a parabolic Coxeter element $c = s_1\cdots s_n \in W$ and a corresponding simple system $(W,S)$, and denote by $\Omega$ and $\Omega _{\operatorname {dir}}$ the undirected and directed version of the Bruhat graph for $(W,S)$. By Proposition 2.2, it is left to show that any two directed paths from $e$ to $c$ in $\Omega _{\operatorname {dir}}$ are in the same Hurwitz orbit. Let therefore

$$e \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}t_1 \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}t_1t_2 \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}\ldots \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}t_1\cdots t_n = c$$

be such a path. We have seen in Lemma 2.1 that $\ell (c) = \ell _T(c) = n$. It thus follows that $\ell (t_1\cdots t_i) = \ell _T(t_1\cdots t_i) = i$ for any $i$. The strong exchange condition, see e.g. Hum90, Theorem 5.8, then yields that $t_1 \cdots t_{i+1}$ is obtained from $t_1\cdots t_i$ by adding a single simple generator into its position within $s_1\cdots s_n$. Therefore, such a path is (bijectively) encoded by a permutation $\pi = [\pi _1,\ldots ,\pi _n]$ where $\pi _i$ is the index of the simple generator added at the $i$-th step. Given the factorization corresponding to such a path, it is straightforward to see that the embedding of the permutation into the braid group (by sending a simple transposition $(i,i+1)$ to the generator $\sigma _i$ of $B_n$) yields a braid that turns the given factorization into the factorization $s_1\cdots s_n$. To this end, observe that given two factorizations encoded by two permutations $\pi _1$ and $\pi _2$ with $\ell (\pi _1) = \ell (\pi _2)+1$ and such that these differ only by a single simple transposition $\pi _2^{-1}\pi _1 = (i,i+1)$ for some index $i$. Then the given factorizations are obtained from each other by applying the braid group generator $\sigma _i$ to the factorization corresponding to $\pi _1$ to obtain the factorization corresponding to $\pi _2$. As the factorization corresponding to the identity permutation $[1,\ldots ,n]$ is $(s_1,\ldots ,s_n)$, the claim follows.

■

As an example of the construction in the previous proof, consider the path

$$e \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}s_2 \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}s_2 s_5 \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}s_2 s_3 s_5 \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}s_1 s_2 s_3 s_5 \mathbin{\vcenter{\img[][11pt][5pt][{\tikz[semithick, baseline=-0.2ex,-latex, ->]{\draw[->] (0pt,0.4ex) -- (1em,0.4ex);}}]{Images/img5ec91d660154588f6f9ba6123de50ac8.svg}}}s_1 s_2 s_3 s_4 s_5 = c.$$

The corresponding factorization of $c$ is given by

$$c = s_2 \cdot s_5 \cdot s_5 s_3 s_5 \cdot s_5 s_3 s_2 s_1 s_2 s_3 s_5 \cdot s_5 s_4 s_5,$$

and the permutation is $\pi = [2,5,3,1,4] = (1,2)(2,3)(4,5)(3,4)(2,3)$. On the other hand,

$$\sigma _1 \sigma _2 \sigma _4 \sigma _3 \sigma _2(s_2,s_5,s_5 s_3 s_5,s_5 s_3 s_2 s_1 s_2 s_3 s_5,s_5 s_4 s_5) = (s_1,s_2,s_3,s_4,s_5),$$

as desired.