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\title{Reflections and quotients in Coxeter groups}
\alttitle{RÃ©flexions et quotients dans les groupes de Coxeter}

\author{\firstname{Paolo} \lastname{Sentinelli}}
\address{Dipartimento di Matematica, Politecnico di Milano, Milan, Italy}
\email{paolosentinelli@gmail.com}

\begin{abstract}
We present a formula relating the set of left descents of an element of a Coxeter group with the sets of left descents of its projections on maximal quotients indexed by simple right descents. This formula is an instance of a general result involving quotients and descents.
\end{abstract}

\begin{altabstract}
Nous prÃ©sentons une formule reliant l'ensemble des descentes gauches d'un Ã©lÃ©ment d'un groupe de Coxeter aux ensembles des descentes gauches de ses projections sur les quotients maximaux indexÃ©s par les descentes droites simples. Cette formule est un exemple d'un rÃ©sultat gÃ©nÃ©ral impliquant les quotients et les descentes.
\end{altabstract}

\makeatletter
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\begin{document}
\maketitle
\vspace{-0.75em}

\section{Introduction}

A reflection $t$ in a Coxeter system $(W,S)$ is a conjugate of a generator, and $t$ is a left descent of an element $w\in W$ if $\ell(tw)&lt;\ell(w)$, where $\ell$ is the length function. Denoting by $T_L(w)$ the set of left descents of $w$, it is well known that the right weak order $\leq_R$ on $W$ is isomorphic to the set $\bigl\{T_L(w):w\in W\bigr\}$ ordered by inclusion. In this note we prove the following formula:
\begin{equation*}
T_L(v)=T_L(u)\, \cup \bigcup_{s\in D_R(u^{-1}v)}T_L\bigl(v^{S\setminus\{s\}}\bigr),
\end{equation*}
for all $u,v \in W$ such that $u\leq_R v$. Here $D_R(w)$ is the set of right simple descent (right descents of length one) of $w$, and $w^J$ is the element of minimal length in the coset $wW_J$, for any $J\subseteq S$. This formula is a special case of a general result we prove in Theorem~\ref{teorema riflessioni}. We then deduce some corollaries; for example, our result implies that the set $\bigl\{w^{S\setminus\{s\}}: s\in D_R(w)\bigr\}$ admits a join and it is precisely $w$ (Corollary~\ref{cor join}), i.e.\
\begin{equation*}
w=\bigvee_{s\in D_R(w)}w^{S\setminus \{s\}},
\end{equation*}
for all $w\in W$. We are regarding the poset $(W,\leq_R)$ as a complete meet-semilattice, for which the joins may not exist.

We now fix notation and recall some definitions useful for the rest of the paper. We refer to the book of Bj\"{o}rner and Brenti~\cite{BB} for Coxeter groups.

Let $(W,S)$ be a Coxeter system, i.e.\ a presentation of the group $W$ given by a set $S$ of involutive generators and relations encoded by a \emph{Coxeter matrix} $m\colon S\times S \rightarrow \{1,2,\dots,\infty\}$. A Coxeter matrix over $S$ is a symmetric matrix which satisfies the following conditions for all $s,t\in S$:
\begin{enumerate}
\item $m(s,t)=1$ if and only if $s=t$;
\item $m(s,t)\in \{2,3,\dots,\infty\}$ if $s\neq t$.
\end{enumerate}
The presentation $(W,S)$ of the group $W$ is then the following:
\[
\begin{cases}
\text{generators:} 	&amp;S, \\
\text{relations:} 	&amp;(st)^{m(s,t)}=e,
\end{cases}
\]
for all $s,t\in S$, where $e$ denotes the identity in $W$. The Coxeter matrix $m$ attains the value $\infty$ at $(s,t)$ to indicate that there is no relation
between the generators $s$ and $t$. The class of words expressing an element of $W$ contains words of minimal length; the \emph{length function} $\ell\colon W \rightarrow
\mathbb{N}$ assigns to an element $w\in W$ such minimal length. The identity $e$ is represented by the empty word and then $\ell(e)=0$. A \emph{reduced word} or \emph{reduced expression} for an element $w\in W$ is a word of minimal length representing $w$. The set of \emph{reflections} of $(W,S)$ is defined by $T\coloneqq\bigl\{wsw^{-1}:w\in W, \, s\in S\bigr\}$. For $J\subseteq S$ and $v\in W$, we let
\begin{align*}
W^J&amp;\coloneqq\bigl\{w\in W:\ell(w)&lt;\ell(ws)\;\forall s\in J\bigr\},
\\ D_R(v)&amp;\coloneqq\bigl\{s\in S:\ell(vs)&lt;\ell(v)\bigr\}.
\end{align*}
With $W_J$ we denote the subgroup of $W$ generated by $J\subseteq S$; in particular, $W_S=W$ and $W_\varnothing = \{e\}$. We let $\leqslant_R$ and $\leqslant$ be the \emph{right weak order} and the \emph{Bruhat order} on $W$, respectively. The covering relations of the right weak order are characterized as follows: $u \var_R v$ if and only if $\ell(u)&lt;\ell(v)$ and $u^{-1}v \in S$.
For $w\in W$, define
\[
T_L(w)\coloneqq\bigl\{t\in T: \ell(tw)&lt;\ell(w)\bigr\},
\]
\[
T_R(w)\coloneqq\bigl\{t\in T: \ell(wt)&lt;\ell(w)\bigr\}.
\]
The right weak order has the following property (see~\cite[Proposition~3.1.3]{BB}):
\[
u \leq_R v \; \Longleftrightarrow \; T_L(u)\subseteq T_L(v).
\]
The covering relations of the Bruhat order are characterized as follows: $u \var v$ if and only if $\ell(u)=\ell(v)-1$ and $u^{-1}v \in T$. The posets $(W,\leqslant_R)$ and $(W,\leqslant)$ are graded with rank function $\ell$ and $(W,\leqslant_R) \hookrightarrow (W,\leqslant)$. For $J\subseteq S$, each element $w\in W$ factorizes uniquely as $w=w^Jw_J$, where $w^J\in W^J$, $w_J\in W_J$ and $\ell(w)=\ell(w_J)+\ell(w^J)$; see~\cite[Proposition~2.4.4]{BB}. We consider the idempotent function $P^J\colon W
\rightarrow W$ defined by
\begin{equation*}
P^J(w)=w^J,
\end{equation*}
for all $w\in W$. This function is order-preserving for the Bruhat order
(see~\cite[Proposition~2.5.1]{BB}). Moreover $I\subseteq J \subseteq S$ implies $P^J\circ P^I = P^J$ (see~\cite[Lemma~5.4]{deodhar}). We let
\[
P^{(s)}\coloneqq P^{S\setminus \{s\}},
\]
for all $s\in S$.

For the following proposition see~\cite[Corollary~2.6.2]{BB}.
\begin{prop}\label{deodhar}
Let $u,v\in W$. Then
\[
u\leq v \; \Longleftrightarrow \; P^{(s)}(u) \leq P^{(s)}(v) \;\forall s \in D_R(u).
\]
\end{prop}

Now we recall some results about the standard geometric representation of a finitely generated Coxeter group. Let $(W,S)$ be a Coxeter system with $\abs{S}&lt;\infty$ and Coxeter matrix $m$. Let $V$ be the free $\R$-vector space on the set $\bigl\{\alpha_s:s \in S\bigr\}$ and $\sigma_s \in \End(V)$ defined by $\sigma_s(v)=v-2(\alpha_s \vert v)\alpha_s$, where the bilinear form $(\,\cdot\,\vert\,\cdot\,)$ is defined by setting $(\alpha_s\vert\alpha_t)\coloneqq -\cos\biggl(\frac{\pi}{m(s,t)}\biggr)$, for all $s,t\in S$; then the assignment $w \mapsto \sigma_w\coloneqq\sigma_{s_1}\cdots \sigma_{s_k}$, where $s_1\cdots s_k$ is any reduced word of $w\in W$, provides a faithful representation $W \rightarrow \GL(V)$ (see, e.g.~\cite[Chapter~4]{BB}). Then we let $w(v)\coloneqq\sigma_w(v)$, for all $v\in V$, $w\in W$. Let $\Phi\coloneqq\bigl\{w(\alpha_s): w\in W, \, s \in S\bigr\}$, $V^+\coloneqq\bigl\{\sum_{s\in S}a_s\alpha_s: \, a_s \geqslant 0 \; \forall s\in S\bigr\} \setminus \{0\}$ and $V^-\coloneqq\bigl\{\sum_{s\in S}a_s\alpha_s: \, a_s \leqslant 0 \; \forall s\in S\bigr\} \setminus \{0\}$; then it is well known that $\Phi=\Phi^+ \uplus \Phi^-=\bigl\{\alpha_t : t \in T\bigr\} \uplus \bigl\{-\alpha_t : t \in T\bigr\}$, $\Phi^+ \subseteq V^+$ and $\Phi^- \subseteq V^-$. It holds that $t(v)=v-2(\alpha_t\vert v)\alpha_t$ and $w(\alpha_t) \in \Phi^-$ if and only if $wt&lt;w$, for all $w\in W$, $t\in T$ (see, e.g.~\cite[Section~4.4]{BB}). Moreover, if $s\in S$ and $t\in T\setminus \{s\}$, then $s(\alpha_t)=\alpha_{sts}$.

\section{Reflections and quotients}\label{sezione Coxeter}

In this section we prove the main result of this paper, i.e.\ the formula mentioned in the introduction, as a corollary of the following theorem.

\begin{theo}\label{teorema riflessioni}
Let $(W,S)$ be a Coxeter system and $u,v\in W$ such that
$u\leq_R v$. If $E\subseteq \mathcal{P}(S)$ is such that $\bigcap_{J\in E}J=S\setminus D_R(u^{-1}v)$, then
\[
T_L(v)=T_L(u)\, \cup \bigcup_{J\in E}T_L(v^J).
\]
\end{theo}

\begin{proof}
It is known that
\begin{equation}\label{eq0}
T_L(xy)=T_L(x)+xT_L(y)x^{-1},
\end{equation}
for all $x,y \in W$, where $+$ is the symmetric difference. Hence, for any $v\in W$, we have that $\varnothing = T_L(v^{-1}v) =T_L(v^{-1})+v^{-1}T_L(v)v$; this implies $T_R(v)=T_L(v^{-1})=v^{-1}T_L(v)v$, for all $v\in W$. Consider $J\subseteq S$. Then
\begin{equation}\label{eq1}
(v^J)^{-1}T_L(v^J)v^J \cap W_J = T_R(v^J)\cap W_J= \varnothing.
\end{equation}
Notice that, if $\ell(xy)=\ell(x)+\ell(y)$, the symmetric difference in~\eqref{eq0} is a disjoint union. Therefore we have that
\begin{equation}\label{eq2}
T_L(v) = T_L(v^J) \uplus v^JT_L(v_J)(v^J)^{-1},
\end{equation}
for all $v\in W$, $J\subseteq S$. Equalities~\eqref{eq1} and~\eqref{eq2} imply
\begin{equation}\label{eq3}
\begin{split}
T_L(v^J)&amp;=T_L(v)\setminus v^JT_L(v_J)(v^J)^{-1} \\
&amp;= T_L(v)\setminus v^JW_J(v^J)^{-1} \\
&amp;= T_L(v)\setminus vW_Jv^{-1}.
\end{split}
\end{equation}
If $u\leq_R v$ and $w\coloneqq u^{-1}v$, we have that $T_L(v)=T_L(u)\uplus uT_L(w)u^{-1}$. Moreover, by~\eqref{eq3},
\begin{equation}\label{eq5}
\begin{split}
uT_L(w^J)u^{-1} &amp;= u\bigl(T_L(w)\setminus wW_Jw^{-1}\bigr)u^{-1} \\
&amp;= uT_L(w)u^{-1}\setminus vW_Jv^{-1} \\
&amp; \subseteq T_L(v) \setminus vW_Jv^{-1} = T_L(v^J).
\end{split}
\end{equation}
Therefore, by~\eqref{eq3} and~\eqref{eq5}, if we let $K\coloneqq\bigcap_{J\in E}J$, we obtain
\begin{align*}
\bigcup_{J\in E} T_L(v^J) &amp; \supseteq \bigcup_{J\in E} uT_L(w^J)u^{-1}\\
&amp;= u\left(\bigcup_{J\in E} \bigl(T_L(w)\setminus wW_Jw^{-1}\bigr)\right)u^{-1} \\
&amp;= u\bigl(T_L(w)\setminus wW_Kw^{-1}\bigr)u^{-1} \\
&amp;= u\bigl(T_L(w)\setminus wW_{S\setminus D_R(w)}w^{-1}\bigr)u^{-1} \\
&amp;= uT_L\bigl(w^{S\setminus D_R(w)}\bigr)u^{-1}\\
&amp;= uT_L(w)u^{-1}.
\end{align*}
This implies that
\[
T_L(u)\, \cup \bigcup_{J\in E}T_L(v^J) \, \supseteq \, T_L(u) \cup uT_L(w)u^{-1} \, = \, T_L(v).
\]
On the other hand, since $u\leq_R v$ and $v^J\leq_R v$ for all $J\in E$, we also have that $T_L(v) \supseteq T_L(u)\, \cup \, \bigcup_{J\in E}T_L(v^J)$ and this concludes the proof.
\end{proof}

By the fact that $\bigcap_{s\in D_R(u^{-1}v)}\bigl(S\setminus \{s\}\bigr)=S\setminus D_R(u^{-1}v)$, we obtain the following corollary, which is the finest instance of the statement of Theorem~\ref{teorema riflessioni}.

\begin{coro}\label{corollario teorema riflessioni}
Let $(W,S)$ be a Coxeter system and $u,v\in W$ such that
$u\leq_R v$. Then
\[
T_L(v)=T_L(u)\, \cup \bigcup_{s\in D_R(u^{-1}v)}T_L\bigl(v^{S\setminus\{s\}}\bigr).
\]
\end{coro}

In the following example we consider a Coxeter system of type $B_4$. The set of generators is $S=\{s_0,s_1,s_2,s_3\}$; the Coxeter matrix takes the values $m(s_0,s_1)=4$ and $m(s_1,s_2)=m(s_2,s_3)=3$.
\begin{exem}
In type $B_4$, let $u=s_2s_3s_2$ and $v=s_2s_3s_2s_1s_0s_2s_3$. Then $u^{-1}v=s_1s_0s_2s_3$, $D_R(v)=\{s_0,s_2,s_3\}$ and $D_R(u^{-1}v)=\{s_0,s_3\}$. We have that
\begin{itemize}
\item $T_L(u)=\bigl\{s_2,\,s_3,\,s_2s_3s_2\bigr\}$,
\item $T_L\bigl(v^{S\setminus \{s_3\}}\bigr)=T_L(s_1s_2s_3)=\bigl\{s_1,\,s_1s_2s_1,\, s_1s_2s_3s_2s_1\bigr\}$,
\item $T_L\bigl(v^{S\setminus \{s_0\}}\bigr)=T_L(s_3s_2s_1s_0)=\bigl\{s_3,\,s_2s_3s_2,\,s_1s_2s_3s_2s_1,\,s_3s_2s_1s_0s_1s_2s_3\bigr\}$,
\item $T_L(v)=\bigl\{s_1,\,s_2,\,s_3,\,s_1s_2s_1,\, s_2s_3s_2,\, s_1s_2s_3s_2s_1,\, s_3s_2s_1s_0s_1s_2s_3\bigr\}$.
\end{itemize}
\end{exem}

An element $w\in W$ is uniquely determined by its descent set; hence, by Theorem~\ref{teorema riflessioni} for $u=e$, we have that $w$ is uniquely determined by the set $\bigl\{w^{S\setminus \{s\}}: s\in D_R(w)\bigr\}$. This fact can be also deduced from Proposition~\ref{deodhar}. An additional information provided by Theorem~\ref{teorema riflessioni} is the following one, which concerns the join operation in the right weak order.

\begin{coro}\label{cor join}
Let $w\in W$. Then $\displaystyle w= \bigvee_{s\in D_R(w)}w^{S\setminus \{s\}}$.
\end{coro}
\begin{proof}
Clearly $T_L(w^J) \subseteq T_L(w)$, for all $J\subseteq S$, or, equivalently, $w^J\leq_R w$, for all $J\subseteq S$. Hence the set $\bigl\{w^{S\setminus \{s\}}: s\in D_R(w)\bigr\}$ is bounded and then it admits a join $w'$ (see~\cite[Theorem~1.5]{dyer}). Moreover, by~\cite[Proposition~2.14]{hohlweg} we have that $T_L(w')=T_L(w)$ and this concludes the proof.
\end{proof}

By repeated use of Corollary~\ref{corollario teorema riflessioni}, we obtain the following result.
\begin{coro}
Let $s_1s_2\cdots s_k$ be a reduced expression for $w\in W$. Then
\begin{equation*}
T_L(w) = \bigcup_{i=0}^{k-1}T_L\biggl((s_1\cdots s_{k-i})^{S\setminus \{s_{k-i}\}}\biggr).
\end{equation*}
\end{coro}

We now deduce from Theorem~\ref{teorema riflessioni} the statement of~\cite[Lemma~4.2]{io2}. Such result has inspired our main formula and it is related to the shellability of Coxeter complexes.

\begin{coro}\label{corollario riflessioni}
Let $u,v,w\in W$ be such that $u\leq_R v$ and $u\leq_R w$. If $P^{(s)}(v)=P^{(s)}(w)$ for all $s\in D_R(u^{-1}v)$, then $v\leq_R w$.
\end{coro}

\begin{proof}
Since $u\leq_R w$, we have that $T_L(u)\subseteq T_L(w)$. Moreover $T_L\bigl(v^{S\setminus \{s\}}\bigr)=T_L\bigl(w^{S\setminus \{s\}}\bigr)\subseteq T_L(w)$ for all $s\in D_R(u^{-1}v)$. Hence by Corollary~\ref{corollario teorema riflessioni} we obtain $T_L(v) \subseteq T_L(w)$, which is equivalent to $v\leq_R w$.
\end{proof}
For $n\in \N$, let $B_n$ be the power set $\mathcal{P}\bigl(\{1,\dots,n\}\bigr)$ ordered by inclusion and, for $w\in W$ and $K\subseteq S\setminus D_R(w)$,
\[
\mathcal{T}^K_w\coloneqq\biggl\{T_L\bigl(w^{K\cup J}\bigr) : J \subseteq D_R(w)\biggr\} \subseteq \mathcal{P}(T),
\]
ordered by inclusion.

\begin{prop}\label{prop minimale}
Let $w\in W$, $K\subseteq S\setminus D_R(w)$ and $n\coloneqq\abs{D_R(w)}$. Then, as posets, $\mathcal{T}^K_w \simeq B_n$.
\end{prop}

\begin{proof}
If $w=e$ the result is trivial. So let $w\neq e$ and $H\coloneqq\bigl\{s\in S: s\leq w\bigr\}$. We consider the standard geometric representation of the finitely generated Coxeter group $W_H$. Then $w\in W_H$ and assume $T_L\bigl(w^{K\cup I}\bigr) = T_L\bigl(w^{K\cup J}\bigr)$, for some $I,J\subseteq D_R(w)$. Consider the factorization $w=w^{K \cup J}w_{K \cup J}$. We have that $s \in J\subseteq D_R(w)$ if and only if $s\in D_R\bigl(w_{K \cup J}\bigr)$, i.e.\ if and only if $w_{K \cup J}(\alpha_s) \in \Phi^-$. Let $t_s\coloneqq wsw^{-1}$; then $\alpha_{t_s}=-w(\alpha_s)$. We have that
\[
\bigl(w^{K \cup J}\bigr)^{-1}(\alpha_{t_s})= -\bigl(w^{K \cup J}\bigr)^{-1}w(\alpha_s)
=-w_{K \cup J}(\alpha_s).
\]
Hence
\begin{align*}
s \in J &amp;\Longleftrightarrow s\in D_R\bigl(w_{K \cup J}\bigr) \\
&amp;\Longleftrightarrow \bigl(w^{K \cup J}\bigr)^{-1}(\alpha_{t_s}) \in \Phi^+ \\
&amp;\Longleftrightarrow t_s \not \in T_L\bigl(w^{K \cup J}\bigr)\\
&amp;\Longleftrightarrow t_s \not \in T_L\bigl(w^{K \cup I}\bigr) \\
&amp;\Longleftrightarrow s\in D_R\bigl(w_{K \cup I}\bigr)\\
&amp;\Longleftrightarrow s \in I.
\end{align*}
Therefore $I=J$. We have proved that $\abs{\mathcal{T}_w}=\abs{B_n}=2^n$. This also proves that $T_L\bigl(w^{K \cup I}\bigr) \subseteq T_L\bigl(w^{K \cup J}\bigr)$ implies $J \subseteq I$. On the other hand, if $I \subseteq J$, then $K \cup I \subseteq K \cup J$ and we have that $w^{K \cup J}=\bigl(P^{K \cup J}\circ P^{K \cup I}\bigr)(w) \leq_R w^{K \cup I}$, i.e.\ $T_L\bigl(w^{K \cup J}\bigr) \subseteq T_L\bigl(w^{K \cup I}\bigr)$. Then the poset $\mathcal{T}_w$ is isomorphic to the dual of $B_n$ and the result follows since $B_n$ is self-dual.
\end{proof}

Proposition~\ref{prop minimale} implies that the atoms of the poset $\mathcal{T}^{S\setminus D_R(w)}_w$ are the sets $T_L\bigl(w^{S\setminus \{s\}}\bigr)$, with $s\in D_R(w)$. The proof of this proposition also shows that if $s\in D_R(w)$ then
\[
wsw^{-1} \in T_L\bigl(w^{S\setminus \{s\}}\bigr) \setminus \bigcup_{r\in D_R(w)\setminus \{s\}} T_L\bigl(w^{S\setminus \{r\}}\bigr),
\]
i.e.\ the union in Corollary~\ref{corollario riflessioni} is minimal with respect to its indexing set.

\section*{Acknowledgments}

I thank the anonymous referee for having generalized the original main result (now Corollary~\ref{corollario teorema riflessioni}) and proposing a more elegant proof. I also thank Christophe Hohlweg for some useful conversations and for having pointed out the result of Corollary~\ref{cor join}.

\section*{Declaration of interests}

The author does not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and has declared no affiliations other than their research organizations.

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