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%%%%% Auteurs

\author{\firstname{Patrick} \lastname{Wegener}}%%%1

\address{Technische Universität Kaiserslautern\\ 
Fachbereich Mathematik\\
Postfach 3049\\
67653 Kaiserslautern\\
Germany}


\email{wegener@mathematik.uni-kl.de}

%%%%%%%

\author{\firstname{Sophiane} \lastname{Yahiatene}}%%%2

\address{Universität Bielefeld\\ 
Fakultät für Mathematik\\
Postfach 100131\\
33501 Bielefeld\\
Germany}


\email{syahiate@math.uni-bielefeld.de}

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\keywords{Coxeter groups, Hurwitz action, reflection factorizations, Coxeter element}
 
\subjclass{05E15, 05E18, 20F55}


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\DOI{10.5802/alco.99}
\datereceived{2019-01-21}
\daterevised{2019-06-25}
\dateaccepted{2019-09-24}


%%%%% Titre et résumé

\title
[Non-reduced reflection factorizations of Coxeter elements]
{A note on non-reduced reflection factorizations of Coxeter elements}

\begin{abstract}
We extend a result of Lewis and Reiner from finite Coxeter groups to Coxeter groups of finite rank by showing that two reflection factorizations of a Coxeter element lie in the same Hurwitz orbit if and only if they share the same multiset of conjugacy classes.
\end{abstract}

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\begin{document}


\maketitle



\section{Introduction}\label{sec:intro}


Given a Coxeter system $(W,S)$ with set of reflections $T$, the braid group (\eg see~\cite{BDSW14} for a definition) acts on \emph{reflection factorizations} of a given element $w \in W$, that is it acts on tuples $(t_1, \ldots , t_m) \in T^m$ of reflections such that $w=t_1 \cdots t_m$. This action is called \emph{Hurwitz action}. A standard braid group generator $\sigma_i$ (resp. its inverse $\sigma_i^{-1}$) acts by a \emph{Hurwitz move} on a reflection factorization:
\begin{align*}
\sigma_i (t_1, \ldots , t_{i-1}, t_i, t_{i+1}, t_{i+2}, \ldots , t_n) & = (t_1, \ldots , t_{i-1}, t_{i+1}^{t_i}, t_{i}, t_{i+2}, \ldots , t_n),\\
\sigma_i^{-1} (t_1, \ldots , t_{i-1}, t_i, t_{i+1}, t_{i+2}, \ldots , t_n) & = (t_1, \ldots , t_{i-1}, t_{i+1}, t_{i}^{t_{i+1}}, t_{i+2}, \ldots , t_n),
\end{align*}
where we use the notation $g^h:=hgh^{-1}$ for conjugation. 

We call a reflection factorization $(t_1, \ldots , t_m)$ of an element $w \in W$ \emph{reduced} if $w$ cannot be written as a product of less than $m$ reflections. It has been first observed by Deligne~\cite{Del} that this action is transitive on reduced reflection factorizations of a \emph{Coxeter element} if $W$ is finite. The first published proof is due to Bessis~\cite[Proposition~1.6.1]{Bes03}. Igusa and Schiffler showed that this statement is true for every Coxeter group~\cite[Theorem~1.4]{IS10}. 

The question of how these results extend to non-reduced reflection factorizations has been first addressed by Lewis and Reiner.

\begin{thm*}
[{Lewis--Reiner,~\cite[Theorem~1.1]{LR16}}] In a finite real reflection group, two reflection factorizations of a Coxeter element lie in the same Hurwitz orbit if and only if they share the same multiset of conjugacy classes.
\end{thm*}

Their proof makes heavy use of the following remarkable result for finite Coxeter groups.
\begin{lemma}[{Lewis--Reiner,~\cite[Corollary~1.4]{LR16}}] \label{lem:LewisR}
Let $(W,S)$ be a finite Coxeter system and $w\in W$. Then every reflection factorization of $w$ lies in the same Hurwitz orbit of some reflection factorization $(t_1, \ldots , t_m)$ of $w$ such that $(t_1, \ldots , t_{\ell})$ is a reduced reflection factorization of $w$ for some $\ell \leq m$ and 
\[
t_{\ell +1} = t_{\ell+2}, ~t_{\ell+3} = t_{\ell+4}, \ldots, ~t_{m-1}=t_m.
\]
\end{lemma}

This result is proved by a case-by-case analysis and seems not to extend to infinite Coxeter groups in general. We give a case-free proof of a similar (but weaker) result for all Coxeter groups of finite rank (see Lemma~\ref{lem:same_refl}). In this way, we obtain that the result of Lewis and Reiner extends to all Coxeter groups of finite rank which provides a positive answer to a question of Lewis and Reiner~\cite[Question~6.2]{LR16}.

\begin{thm} \label{thm:Main}
Let $(W,S)$ be a Coxeter system of finite rank. Then two reflection factorizations of a Coxeter element in $W$ lie in the same Hurwitz orbit if and only if they share the same multiset of conjugacy classes.
\end{thm}



\section{The proof}
Throughout this note let $(W,S)$ be a \emph{Coxeter system} of finite rank $n \in \mathbb{N}$ with set of \emph{reflections} $T=\{ wsw^{-1} \mid w \in W,~s \in S\}$. All necessary definitions and facts about Coxeter groups we will use are covered by~\cite{BDSW14, Dye90, Dye91, Hu90}.

A subgroup $W'$ of $W$ is called a \emph{reflection subgroup} if $W'= \langle W' \cap T \rangle$. Each reflection subgroup $W'$ admits a canonical set of generators $\chi(W')$ such that $(W', \chi(W'))$ is a Coxeter system and the set of reflections for $(W', \chi(W'))$ is given by $W' \cap T = \bigcup_{w \in W'} w \chi(W') w^{-1}$ (see~\cite[(3.3)~Theorem]{Dye90}). A reflection subgroup of the form $\langle I \rangle$ for some $I \subseteq S$, is called \emph{parabolic subgroup}.

Let $S =\{ s_1, \ldots , s_n \}$. For each permutation $\pi$ of the numbers $\{1, \ldots, n\}$, the element $c= s_{\pi(1)} \cdots s_{\pi(n)}$ is called a \emph{Coxeter element}. A Coxeter element of a parabolic subgroup is called \emph{parabolic Coxeter element}.

We denote by $\ell_S$ (\resp $\ell_T$) the length function on $W$ with respect to the generating set $S$ (\resp $T$).



\begin{defi} \label{def:BruhatGraph}
We define the \emph{Bruhat graph} of $(W,S)$ to be the directed graph $\Omega_{(W,S)}$ on vertex set $W$ with a directed edge from $x$ to $y$ if there exists $t \in T$ such that $y=xt$ and $\ell_S(x) < \ell_S(y)$.

Moreover, we denote by $\overline{\Omega}_{(W,S)}$ the corresponding undirected graph and for a subset $X \subseteq W$ we denote by $\Omega_{(W,S)}(X)$ the full subgraph of $\Omega_{(W,S)}$ on the vertex set $X$.

Note that $\overline{\Omega}_{(W,S)}$ is exactly the Cayley graph of $W$ with generating set $T$. 
\end{defi}

We use the notation $(t_1, \ldots, t_m) \sim (r_1, \ldots , r_m)$ to indicate that both tuples lie in the same orbit under the Hurwitz action.

\medskip
The following fact is already part of the proof of~\cite[Proposition~2.2]{BDSW14}. For sake of completeness we include a proof (which can also be found in the first author's Ph.D. thesis~\cite[Proposition~2.3.6]{Weg17}).

\begin{prop} \label{prop:BruhatGraphDihedral}
Let $w \in W$ and $t_1,t_2 \in T$ with $t_1 \neq t_2$ such that 
\[ 
w \edgedir wt_1 \edgedirback wt_1t_2
\]
in $\Omega_{(W,S)}$. Then there exist $t_1', t_2 ' \in \langle t_1, t_2 \rangle \cap T$ with $(t_1, t_2) \sim (t_1', t_2')$ such that one of the following cases hold:
\begin{enumerate}
\item\label{prop2.2_1} $ w \edgedir wt_1' \edgedir wt_1't_2'= wt_1t_2$;
\item\label{prop2.2_2} $ w \edgedirback wt_1' \edgedirback wt_1't_2'= wt_1t_2$;
\item\label{prop2.2_3} $ w \edgedirback wt_1' \edgedir wt_1't_2'= wt_1t_2$;
\end{enumerate}
Furthermore, in each of these cases we have $\ell_S(wt_1')<\ell_S(wt_1)$.
\end{prop}

\begin{proof}
Consider the dihedral reflection subgroup $W':= \langle t_1, t_2 \rangle$ and let $S':= \chi(W')$. We have $w, wt_1, wt_1t_2 \in wW'$. Therefore we just have to consider the coset $wW'$ to prove the claim. By~\cite[(1.4)~Theorem]{Dye91} we have 
\[
\Omega_{(W,S)}(W') \cong \Omega_{(W,S)}(wW') \cong \Omega_{(W',S')},
\]
where $(W',S')$ is dihedral and we can check the claim there directly. Any reflection (element of odd $S'$-length) of $W'$ is joined by an edge to a rotation (element of even $S'$-length) which in $\Omega_{(W',S')}$ is oriented towards the element of greater $S'$-length. For $x \in W'$ there are three possible situations:
\begin{itemize}
\item $\ell_{S'}(x)< \ell_{S'}(xt_1t_2)$
\item $\ell_{S'}(x)> \ell_{S'}(xt_1t_2)$
\item $\ell_{S'}(x)= \ell_{S'}(xt_1t_2)$ (in particular $x \neq e$ since $t_1 \neq t_2$).
\end{itemize}
We can choose $t_1', t_2' \in W' \cap T$ with $t_1't_2' = t_1t_2$ in the three situations such that we have one of the following situations:
\begin{itemize}
\item $x \edgedir xt_1' \edgedir xt_1' t_2'$
\item $x \edgedirback xt_1' \edgedirback xt_1' t_2'$
\item $x \edgedirback xt_1' \edgedir xt_1' t_2'$
\end{itemize}
To see this, note that $x$ and $xt_1t_2$ are both either reflections or rotations. Therefore both are either of odd or even $S'$-length. Thus $\ell_{S'}(x) < \ell_{S'}(xt_1t_2)$ implies $\ell_{S'}(x) +2 \leq \ell_{S'}(xt_1t_2)$ and we find $t_1'$ with $\ell_{S'}(x) < \ell_{S'}(xt_1') < \ell_{S'}(xt_1t_2)$. By setting $t_2' := t_1't_1t_2$ we obtain $x \edgedir xt_1' \edgedir xt_1' t_2'$ and $t_1't_2'=t_1t_2$. The remaining cases are similar.

It is easy to see that the Hurwitz orbit of $(t_1,t_2)$ is the set of all pairs $(r_1,r_2)$ of reflections of $W'$ such that $t_1t_2=r_1r_2$ (see also~\cite{BDSW14}). Hence we have $(t_1, t_2) \sim (t_1', t_2')$.

It remains to show that $\ell_S(wt_1')<\ell_S(wt_1)$ in each of the cases~\eqref{prop2.2_1}-\eqref{prop2.2_3}. Since the initial path is of the form $w \edgedir wt_1 \edgedirback wt_1t_2$, we have:
\begin{enumerate}[(\roman*)]
\item\label{proof_prop2.2_i} %[(i)] 
$\ell_S(w) < \ell_S(wt_1)$, and
\item\label{proof_prop2.2_ii} %[(ii)] 
$\ell_S(wt_1t_2) < \ell_S(wt_1) $.
\end{enumerate}
In case~\eqref{prop2.2_1} we have an edge $wt_1' \edgedir wt_1't_2'= wt_1t_2$, thus 
\[
\ell_S(wt_1') < \ell_S(wt_1't_2') = \ell_S(wt_1t_2) \stackrel{\ref{proof_prop2.2_ii}}{<} \ell_S(wt_1).
\]
In cases~\eqref{prop2.2_2} and~\eqref{prop2.2_3} we have an edge $w \edgedirback wt_1'$, thus 
\[
\ell_S(wt_1') < \ell_S(w) \stackrel{\ref{proof_prop2.2_i}}{<} \ell_S(wt_1).\qedhere
\]
\end{proof}




\begin{lemma}\label{lem:same_refl}
Let $w\in W$ with $\ell_{S}(w)=m$ and $w=t_{1}\cdots t_{m+2k}$ with $t_{i}\in T$ for $1\leq i \leq m+2k$ and some $k \in \mathbb{Z}_{\geq 0}$. Then there exists a braid $\sigma \in \mathcal{B}_{m+2k}$ such that 
\[
\sigma(t_{1},\ldots,t_{m+2k})=(r_{1},\ldots,r_{m},r_{i_1},r_{i_1}, \ldots , r_{i_k},r_{i_k}).
\]
\end{lemma}

\begin{proof}
We proceed by induction on $k$. The case $k=0$ is trivially satisfied. Therefore let $k\geq 1$ and assume that all factorizations in $\mathcal{B}_{m+2k}(t_{1},\ldots,t_{m+2k})$ consist of pairwise different factors. Consider the path of $\Omega_{(W,S)}$ starting in $e$ and ending in $w$ induced by $(t_{1},\ldots,t_{m+2k})$. Then Proposition~\ref{prop:BruhatGraphDihedral} allows us to replace successively the parts of the path of shape $\star \edgedir \star \edgedirback \star$ by
\[
\star \edgedir \star \edgedir \star, ~\star \edgedirback \star \edgedirback \star, ~\text{or } \star \edgedirback \star \edgedir {} \star
\] 
only using the Hurwitz action. The latter is possible since the reflections of the factorizations in the Hurwitz orbit are pairwise different. Since by Proposition~\ref{prop:BruhatGraphDihedral} each replacement reduces the sum of the length of the vertices, eventually we get after finitely many replacements a path of the form
\[
e\edgedirback t'_{1}\edgedirback t'_{1}t'_{2}\edgedirback \ldots \edgedirback t'_{1}t'_{2}\cdots t'_{p}\edgedir t'_{1}t'_{2}\cdots t'_{p}t'_{p+1}\edgedir \ldots \edgedir t'_{1}\cdots t'_{m+2k}=w
\]
with $t'_{i}\in T$ for $1\leq i \leq m+2k$, that is, the path is first decreasing, then increasing. Since the path starts with $e$, it holds $p=0$ and therefore it has no decreasing part. Altogether the initial path can be transformed to
\[
e \edgedir t'_{1}\edgedir t'_{1}t'_{2}\edgedir \ldots \edgedir t'_{1}\cdots t'_{m+2k}=w
\]
by using the Hurwitz action. Since the length of $w$ is $m$ and $k\geq 1$, there cannot be an increasing path of length $m+2k$, so we arrive at a contradiction. Thus there exists a factorization $(t_{1}',\ldots,t_{m+2(k-1)}',r_{i_{k}},r_{i_{k}})$ in $\mathcal{B}_{m+2k}(t_{1},\ldots,t_{m+2k})$. From the induction hypothesis follows 
\[
(t_{1}',\ldots,t_{m+2(k-1)}')\sim (r_{1},\ldots,r_{m},r_{i_{1}},r_{i_{1}},\ldots,r_{i_{k-1}},r_{i_{k-1}})
\] 
and the latter yields the assumption.
\end{proof}



\begin{rema}
Note that $\ell_T(w) \leq \ell_S(w)$ for all $w \in W$. Therefore the reflection factorization $w = r_1 \cdots r_m$ obtained by Lemma~\ref{lem:same_refl} does not have to be a reduced reflection factorization. This is the main difference compared with the key argument Lemma~\ref{lem:LewisR} in the proof of Lewis and Reiner. However, by~\cite[Lemma~2.1]{BDSW14} we have $\ell_S(w)=\ell_T(w)$ for an element $w \in W$ if and only if $w$ is a parabolic Coxeter element. Therefore, if $w$ is a parabolic Coxeter element, then Lemma~\ref{lem:same_refl} generalizes Lemma~\ref{lem:LewisR}. In particular, a reflection factorization of a parabolic Coxeter element can be reduced by just using Hurwitz moves and deleting matching neighbors.
\end{rema}


%\medskip
A proof of the following fact already implicitly appears in the proof of~\cite[Theorem~1.1]{LR16}.

\begin{lemma} \label{le:Reduction}
Let $t_1, \ldots, t_{n}, t \in T$. Then $(t_1, \ldots, t_{n},t,t) \sim (t_1, \ldots, t_{n}, t^w,t^w)$ for all $w \in \langle t_1, \ldots, t_{n} \rangle$. 
\end{lemma}

\begin{proof}
Let $i \in \{ 1,\ldots , n \}$ be arbitrary and assume $w=t_i$ (the general assertion follows by induction). Denoting an omitted entry by $\widehat{t_i}$, we obtain
\begin{align*}
(t_1, \ldots, t_{n},t,t) & \sim (t_1, \ldots, t_{i-1},\widehat{t_i}, t_{i+1}^{t_i} \ldots , t_{n}^{t_i}, t^{t_i},t^{t_i}, t_i)\\
{} & \sim (t_1, \ldots, t_{i-1},\widehat{t_i}, t_{i+1}^{t_i} \ldots , t_{n}^{t_i}, t_i, t^{t_i},t^{t_i})\\
{} & \sim (t_1, \ldots,t_{i-1},t_{i}, t_{i+1},\ldots, t_{n},t^{t_i},t^{t_i}). \qedhere
\end{align*}
\end{proof}


%\medskip
\begin{proof}[Proof of Theorem~\ref{thm:Main}]
Let $c \in W$ be a Coxeter element and 
\[
c=t_1' \cdots t_{n+2k}' = r_{1}' \cdots r_{n+2k}'
\]
two reflection factorizations of $c$ for some $k \in \mathbb{Z}_{\geq 0}$ such that they share the same multiset of conjugacy classes. By Lemma~\ref{lem:same_refl} we have 
\begin{align*}
(t_1', \ldots, t_{n+2k}') & \sim (t_1, \ldots , t_n, t_{i_1}, t_{i_1}, \ldots , t_{i_k}, t_{i_k})\\
\text{and }(r_1', \ldots, r_{n+2k}') & \sim (r_1, \ldots , r_n, r_{i_1}, r_{i_1}, \ldots , r_{i_k}, r_{i_k}).
\end{align*}
Since $c=t_1 \cdots t_n=r_1 \cdots r_n$ and $\ell_S(c)=\ell_T(c)=n$ by~\cite[Lemma~2.1]{BDSW14}, $(t_1, \ldots, t_n)$ and $(r_1, \ldots, r_n)$ are reduced reflection factorizations of $c$. Hence we have $(t_1, \ldots, t_n) \sim (r_1, \ldots, r_n)$ by~\cite[Theorem~1.3]{BDSW14}. In particular $(t_1, \ldots, t_n)$ and $(r_1, \ldots, r_n)$ share the same multiset of conjugacy classes. Hence $t_{i_1}, \ldots, t_{i_k}$ and $r_{i_1}, \ldots, r_{i_k}$ have to share the same multiset of conjugacy classes. Since $(t,t,r,r) \sim (r,r,t,t)$ for all $r,t \in T$, we can assume after a possible renumbering that there exists $w_j \in W$ such that $t_{i_j}^{w_j}=r_{i_j}$ for all $j \in \{1, \ldots, k \}$. We proceed by induction on $k$. As we have seen above, the case $k=0$ is precisely~\cite[Theorem~1.3]{BDSW14}. Therefore let $k>0$. By induction we have
\begin{align*}
(t_1, \ldots , t_n, t_{i_1}, t_{i_1}, \ldots,t_{i_{k\mk -\mk 1}}, t_{i_{k\mk -\mk 1}} , t_{i_k}, t_{i_k})\Mk  \sim \Mk  (r_1, \ldots , r_n, r_{i_1}, r_{i_1}, \ldots,r_{i_{k\mk -\mk 1}}, r_{i_{k\mk -\mk 1}} , t_{i_k}, t_{i_k}).
\end{align*}
As a consequence of~\cite[Theorem~1.3]{BDSW14}, we have $W=\langle r_1, \ldots , r_n \rangle$. By what we have pointed out before, there exists $w_k \in \langle r_1, \ldots , r_n \rangle $ such that $t_{i_k}^{w_k} = r_{i_k}$. We conclude
\begin{align*}
(t_1', \ldots, t_{n+2k}') & \stackrel{\phantom{\ref{le:Reduction}}}{\sim} (r_1, \ldots , r_n, r_{i_1}, r_{i_1}, \ldots,r_{i_{k-1}}, r_{i_{k-1}} , t_{i_k}, t_{i_k})\\
{} & \stackrel{\phantom{\ref{le:Reduction}}}{\sim} (r_1, \ldots , r_n,t_{i_k}, t_{i_k}, r_{i_1}, r_{i_1}, \ldots,r_{i_{k-1}}, r_{i_{k-1}})\\
{} & \stackrel{\ref{le:Reduction}}{\sim} (r_1, \ldots , r_n,t_{i_k}^{w_k}, t_{i_k}^{w_k}, r_{i_1}, r_{i_1}, \ldots,r_{i_{k-1}}, r_{i_{k-1}})\\
{} & \stackrel{\phantom{\ref{le:Reduction}}}{=} (r_1, \ldots , r_n,r_{i_k}, r_{i_k}, r_{i_1}, r_{i_1}, \ldots,r_{i_{k-1}}, r_{i_{k-1}})\\
{} & \stackrel{\phantom{\ref{le:Reduction}}}{\sim} (r_1, \ldots , r_n, r_{i_1}, r_{i_1}, \ldots,r_{i_{k-1}}, r_{i_{k-1}},r_{i_k}, r_{i_k})\\
{} & \stackrel{\phantom{\ref{le:Reduction}}}{\sim} (r_1', \ldots, r_{n+2k}').\qedhere
\end{align*}
\end{proof}


\begin{coro}
If the Coxeter graph of $(W,S)$ is connected and has a spanning tree with odd labels on all its edges, then two reflection factorizations of the same length of a Coxeter element in $W$ lie in the same Hurwitz orbit.
\end{coro}

\begin{proof}
Since the Coxeter graph of $(W,S)$ contains a spanning tree, all elements of $S$ are conjugate. Therefore all reflections in $T$ are conjugate. Thus the assertion follows by Theorem~\ref{thm:Main}. 
\end{proof}




\longthanks{The authors thank two anonymous referees for helpful comments.}


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