\documentclass{beamer}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{array}
\usepackage{setspace}
\usepackage{graphicx}
%\usepackage{tikz}
%\usetikzlibrary{matrix,arrows,backgrounds,shapes.misc,shapes.geometric,fit}
\usepackage{etex}
\usepackage{amsthm}
\usepackage{color}
%\usepackage{wasysym}
%\usepackage{allrunes}
\usepackage[all]{xy}
\usepackage{textpos}
\usepackage{ytableau}
%\usetikzlibrary{calc,through,backgrounds}
%\CompileMatrices

\definecolor{grau}{rgb}{.5 , .5 , .5}
\definecolor{dunkelgrau}{rgb}{.35 , .35 , .35}
\definecolor{schwarz}{rgb}{0 , 0 , 0}
\definecolor{violet}{RGB}{143,0,255}
\definecolor{forestgreen}{RGB}{34, 100, 34}

\newcommand{\red}{\color{red}}
\newcommand{\grey}{\color{grau}}
\newcommand{\green}{\color{forestgreen}}
\newcommand{\violet}{\color{violet}}
\newcommand{\blue}{\color{blue}}

\newcommand{\bIf}{\textbf{If} }
\newcommand{\bif}{\textbf{if} }
\newcommand{\bthen}{\textbf{then} }

\newcommand{\EE}{{\mathbf{E}}}
\newcommand{\ZZ}{{\mathbb Z}}
\newcommand{\NN}{{\mathbb N}}
\newcommand{\QQ}{{\mathbb Q}}
\newcommand{\kk}{{\mathbf k}}
\newcommand{\OO}{\operatorname {O}}
\newcommand{\Nm}{\operatorname {N}}
\newcommand{\Par}{\operatorname{Par}}
\newcommand{\Comp}{\operatorname{Comp}}
\newcommand{\Stab}{\operatorname {Stab}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\ev}{\operatorname{ev}}
\newcommand{\Sym}{\operatorname{Sym}}
\newcommand{\QSym}{\operatorname{QSym}}
\newcommand{\Invs}{\operatorname{Invs}}
\newcommand{\Mat}{\operatorname{M}}
\newcommand{\bk}{\mathbf{k}}
\newcommand{\Nplus}{\mathbb{N}_{+}}
\newcommand\arxiv[1]{\href{http://www.arxiv.org/abs/#1}{\texttt{arXiv:#1}}}
\newcommand{\Orb}{{\mathcal O}}
\newcommand{\GL}{\operatorname {GL}}
\newcommand{\SL}{\operatorname {SL}}
\newcommand{\Or}{\operatorname {O}}
\newcommand{\im}{\operatorname {Im}}
\newcommand{\Iso}{\operatorname {Iso}}
\newcommand{\Adm}{\operatorname{Adm}}
\newcommand{\Supp}{\operatorname{Supp}}
\newcommand{\rad}{\operatorname {rad}}
\newcommand{\ora}{\overrightarrow}
\newcommand{\zero}{\mathbf{0}}
\newcommand{\xx}{\mathbf{x}}
\newcommand{\ord}{\operatorname*{ord}}
\newcommand{\op}{{\operatorname{op}}}
\newcommand{\whP}{{\widehat{P}}}
%\newcommand{\Trop}{\operatorname*{Trop}}
%\newcommand{\TropZ}{{\operatorname*{Trop}\mathbb{Z}}}
\newcommand{\rato}{\dashrightarrow}
\newcommand{\lcm}{\operatorname*{lcm}}
\newcommand\arcstr{\ar@/^1pc/}
\newcommand{\exampletwo}{\[
\xymatrixrowsep{1.3pc}
\xymatrix{
&  s_2 s_4 s_1 s_2  \arcstr[r]^{ 2 }
\arcstr[dl]^{ 2 } &  s_2 s_1 s_4 s_2
\arcstr[l]^{ 2 } \arcstr[rd]^{ 2 } \\
 s_4 s_2 s_1 s_2  \arcstr[ur]^{ 2 } \arcstr
[d]^{ 3 } & & &  s_2 s_1 s_2 s_4  \arcstr
[lu]^{ 2 }
\arcstr[d]^{ 3 } \\
 s_4 s_1 s_2 s_1  \arcstr[u]^{ 3 } \arcstr
[dr]^{ 2 } & & &  s_1 s_2 s_1 s_4  \arcstr
[u]^{ 3 }
\arcstr[dl]^{ 2 } \\
&  s_1 s_4 s_2 s_1  \arcstr[ul]^{ 2 }
\arcstr[r]^{ 2 } &  s_1 s_2 s_4 s_1  \arcstr
[l]^{ 2 } \arcstr[ur]^{ 2 }
}
\]}

\newcommand{\fti}[1]{\frametitle{\ \ \ \ \ #1}}
\newenvironment{iframe}[1][]{\begin{frame} \fti{[#1]} \begin{itemize}}{\end{frame}\end{itemize}}

\setbeamertemplate{itemize/enumerate body begin}{\large}
\setbeamertemplate{itemize/enumerate subbody begin}{\large}
\setbeamertemplate{itemize/enumerate subsubbody begin}{\large}

\usepackage{url}%this line and the next are related to hyperlinks
%\usepackage[colorlinks=true, pdfstartview=FitV, linkcolor=blue, citecolor=blue, urlcolor=blur]{hyperref}

\usepackage{color}

%\usetheme{Antibes}
%\usetheme{Bergen}
%\usetheme{Berkeley}
%\usetheme{Berlin}
%\usetheme{Boadilla}
%\usetheme{Copenhagen}
%\usetheme{Darmstadt}
%\usetheme{Dresden}
\usetheme{Frankfurt}
%\usetheme{Goettingen}
%\usetheme{Hannover}
%\usetheme{Ilmenau}
%\usetheme{JuanLesPins}
%\usetheme{Luebeck}
%\usetheme{Madrid}
%\usetheme{Malmoe}
%\usetheme{Marburg}
%\usetheme{Montpellier}
%\usetheme{PaloAlto}
%\usetheme{Pittsburgh}
%\usetheme{Rochester}
%\usetheme{Singapore}
%\usetheme{Szeged}
%\usetheme{Warsaw}

\usefonttheme[onlylarge]{structurebold}
\setbeamerfont*{frametitle}{size=\normalsize,series=\bfseries}
\setbeamertemplate{navigation symbols}{}
\setbeamertemplate{footline}[frame number]
\setbeamertemplate{itemize/enumerate body begin}{}
\setbeamertemplate{itemize/enumerate subbody begin}{\normalsize}
%\setbeamertemplate{section in head/foot shaded}[default][60]
%\setbeamertemplate{subsection in head/foot shaded}[default][60]
\beamersetuncovermixins{\opaqueness<1>{0}}{\opaqueness<2->{15}}

%\usepackage{beamerthemesplit}
\usepackage{epsfig,amsfonts,bbm,mathrsfs}
\usepackage{verbatim} 



\newcommand{\STRUT}{\vrule width 0pt depth 8pt height 0pt}
\newcommand{\ASTRUT}{\vrule width 0pt depth 0pt height 11pt}


\theoremstyle{plain}
\newtheorem{conj}[theorem]{Conjecture}


\setbeamertemplate{headline}{}
%This removes a black stripe from the top of the slides.
\author{Darij Grinberg (UMN)
\\ \textit{joint work with Alexander Postnikov (MIT)}
}
\title[Sign functions for reduced expressions]{Sign functions for reduced expressions in Coxeter groups: proof of a
conjecture of Bergeron, Ceballos and Labb\'e}

\date{30 October 2016 \\ University of St. Thomas, Minneapolis}

\begin{document}

\frame{\titlepage
\textbf{slides:} \\ {\red \url{http://mit.edu/~darij/www/algebra/october06.pdf}} \\
\textbf{paper:} {\red \arxiv{1603.03138}} or \\ {\red \url{http://mit.edu/~darij/www/algebra/bcl.pdf} }
}

\begin{frame}
\fti{A motivating example, part 1}

\begin{itemize}

\item Fix a positive integer $n$.

\item The symmetric group $S_n$ can be presented by
generators $s_1, s_2, \ldots, s_{n-1}$ and relations
\begin{itemize}
\item $s_i^2 = 1$ (the \textit{quadratic relations});
\item $s_i s_j = s_j s_i$ whenever $\left|i-j\right| > 1$ (the \textit{2-braid relations});
\item $s_i s_j s_i = s_j s_i s_j$ whenever $\left|i-j\right| = 1$ (the \textit{3-braid relations}).
\end{itemize}
(\textit{Coxeter presentation}, aka \textit{Moore presentation}).

\pause

\item An \textit{expression} for $w \in S_n$ is a way to write
$w$ as $s_{i_1} s_{i_2} \cdots s_{i_k}$.

\item A \textit{reduced expression} for $w \in S_n$ is an expression
for $w$ having minimum length (i.e., minimum $k$).

\item \textbf{Example:} In $S_5$, the permutation
$\left(\begin{array}{ccccc} 1 & 2 & 3 & 4 & 5 \\ 3 & 2 & 1 & 5 & 4
\end{array}\right)$ has reduced expressions
\[
s_2 s_4 s_1 s_2, \qquad s_2 s_1 s_4 s_2, \qquad s_1 s_4 s_2 s_1,
\qquad \text{and 5 others}.
\]

\end{itemize}

\end{frame}

\begin{frame}
\fti{A motivating example, part 2}

\begin{itemize}

\item Fix a positive integer $n$ and a permutation $w \in S_n$.

\item The braid relations give ways to transform reduced expressions
into other reduced expressions:
\begin{itemize}
\item $\cdots s_i s_j \cdots \mapsto \cdots s_j s_i \cdots$ for
$\left|i-j\right| > 1$ \\
\qquad (a \textit{2-braid move});
\item $\cdots s_i s_j s_i \cdots \mapsto \cdots s_j s_i s_j \cdots$ for
$\left|i-j\right| = 1$ \\
\qquad (a \textit{3-braid move}).
\end{itemize}
These are called \textit{braid moves}.

\item \textbf{Example:}
\[
\xymatrixcolsep{10pc}
\xymatrix{
s_2 s_4 s_1 s_2 \ar@{|->}[r]^{\text{2-braid move with } i = 4 \text{ and }
j = 1}_{\text{at positions } 2 \text{ and } 3} & s_2 s_1 s_4 s_2 .
}
\]

\end{itemize}

\end{frame}

\begin{frame}
\fti{A motivating example, part 3}

\begin{itemize}

\item The natural thing to do: Define an edge-colored directed graph
$\mathcal{R}_0 \left( w\right)$ with
\begin{itemize}
\item vertices = reduced expressions for $w$;
\item an arc going from one expression $\ora{a}$ to another
expression $\ora{b}$
whenever a braid move takes $\ora{a}$ to $\ora{b}$;
\item color each arc with a $2$ if we used a 2-braid move, and a 3
if we used a 3-braid move.
\end{itemize}

\end{itemize}

\end{frame}

\begin{frame}
\fti{A motivating example, part 4}

\begin{itemize}

\item \textbf{Example:} In $S_5$, the permutation
$\left(\begin{array}{ccccc} 1 & 2 & 3 & 4 & 5 \\ 3 & 2 & 1 & 5 & 4
\end{array}\right)$ has the following $\mathcal{R}_0 \left( w\right)$:
\\ (The number over any edge is its color.)
\exampletwo

\end{itemize}

\end{frame}

\begin{frame}
\fti{A motivating example, part 5}

\begin{itemize}

\item What do we see on the example?
\exampletwo

\begin{itemize}
\pause
\item A single bidirected cycle.
\pause
(Does not generalize.)
\pause
\item Strongly connected.
\pause
(Generalizes to arbitrary Coxeter groups: Matsumoto-Tits theorem.)
\end{itemize}

\end{itemize}

\end{frame}

\begin{frame}
\fti{A motivating example, part 6}

\begin{itemize}

\item What do we see on the example?
\exampletwo

\begin{itemize}
\item Walk down the long cycle counterclockwise.
\begin{itemize}
\pause
\item The total number of 2-braid moves used is even.
\item The total number of 3-braid moves used is even.
\end{itemize}
\end{itemize}

\end{itemize}

\end{frame}

\begin{frame}
\fti{A motivating example, part 7}

\begin{itemize}

\item These latter observations do generalize: \\
For any $n \geq 1$ and any $w \in S_n$, any directed cycle in
$\mathcal{R}_0 \left(w\right)$ uses an even number of 2-braid relations
and an even number of 3-braid relations.

\item This was found by Bergeron, Ceballos and Labb\'e
({\red \arxiv{1404.7380v2}}). Their proof used hyperplane arrangement geometry.

\end{itemize}

\end{frame}

\begin{frame}
\fti{Coxeter groups: recalling definitions}

\begin{itemize}

\item Let $\left(W, S\right)$ be a Coxeter group with Coxeter matrix
$\left(m_{s,t}\right)_{\left(s, t\right) \in S \times S}$.

\item Set
\[
\mathfrak{M} = \left\{\left(s,t\right) \in S \times S
\mid s \neq t \text{ and } m_{s,t} < \infty \right\} .
\]

\item Recall that
$W$ has generators $s$ (for $s \in S$) and relations
\begin{itemize}
\item $s^2 = 1$ for all $s \in S$ (the \textit{quadratic relations});
\item $s t s \cdots = t s t \cdots$ (where both sides have $m_{s, t}$
factors) \\
\qquad for all $\left(s, t\right) \in \mathfrak{M}$ (the
\textit{braid relations}).
\end{itemize}

\pause

\item An \textit{expression} for $w \in W$ is a way to write $w$ as a
product $a_1 a_2 \cdots a_k$ where $a_1, a_2, \ldots, a_k \in S$.

\item A \textit{reduced expression} for $w \in W$ is an expression
for $w$ having minimum length (i.e., minimum $k$).

\end{itemize}

\end{frame}

\begin{frame}
\fti{Coxeter groups: braid moves}

\begin{itemize}

\item The braid relations give ways to transform reduced expressions
into other reduced expressions:
\[
        \cdots \left(s t s \cdots \right) \cdots
\mapsto \cdots \left(t s t \cdots \right) \cdots
\]
(where both parenthesized products have $m_{s, t}$ factors) for
$\left(s, t\right) \in \mathfrak{M}$. \\
These are called \textit{braid moves}.

\pause

\item We can again assemble the reduced expressions of a given
$w \in W$ into an edge-colored directed graph. \\
\textbf{Examples:} (courtesy Rob Edman, Victor Reiner)
\begin{itemize}
\item \href{http://www.math.umn.edu/~reiner/H3_reducedwords_graph.pdf}{$H_3$ (longest element)};
\item \href{http://www.math.umn.edu/~reiner/B3_reducedwords_graph.pdf}{$B_3$ (longest element)};
\item \href{http://www.math.umn.edu/~reiner/A3_reducedwords_graph.pdf}{$A_3$ (longest element)}.
\end{itemize}
But we can do better than take $m_{s,t}$'s as colors.

\end{itemize}

\end{frame}

\begin{frame}
\fti{Coxeter groups: the edge-colored digraph $\mathcal{R}\left(w\right)$}

\begin{itemize}

\item Define an equivalence relation $\sim$ (``simultaneous
conjugation'') on $\mathfrak{M}$ as follows:
\begin{align*}
& \left(  s,t\right)  \sim\left(  s^{\prime},t^{\prime}\right)
\\
 \Longleftrightarrow\ & \text{there exists a }q\in W\text{ such that }%
qsq^{-1}=s^{\prime}\text{ and }qtq^{-1}=t^{\prime}  .
\end{align*}

\pause

\item For each $\left(s, t\right) \in \mathfrak{M}$, we get an
equivalence class $\left[\left(s, t\right)\right] \in
\mathfrak{M} / \sim$.

\pause

\item Define an edge-colored directed graph $\mathcal{R}\left(w\right)$
as follows:
\begin{itemize}
\item vertices = reduced expressions for $w$;
\item an arc going from one expression $\ora{a}$ to another
expression $\ora{b}$
whenever a braid move takes $\ora{a}$ to $\ora{b}$;
\item color each arc with the equivalence class
$\left[\left(s, t\right)\right]$ if the braid move used was
\[
        \cdots \left(s t s \cdots \right) \cdots
\mapsto \cdots \left(t s t \cdots \right) \cdots .
\]
\end{itemize}

\end{itemize}

\end{frame}

\begin{frame}
\fti{Main Theorem}

\begin{itemize}

\item \textbf{Theorem (Postnikov, G.).} Let $C$ be a directed cycle
in the graph $\mathcal{R}\left(w\right)$ for some $w \in W$. \\
Let $c \in \mathfrak{M} / \sim$ be an equivalence class (under
simultaneous conjugation). \\
Let $c^\op$ denote the equivalence class
of the opposite pair (i.e., if $c = \left[\left(s,t\right)\right]$,
then $c^\op = \left[\left(t,s\right)\right]$).

\pause
\begin{enumerate}
\item[(a)]
\begin{align*}
& \left(\text{the number of arcs colored } c \text{ in } C \right) \\
= & \left(\text{the number of arcs colored } c^\op \text{ in } C \right) .
\end{align*}

\pause
\item[(b)]
\begin{align*}
\left(\text{the number of arcs colored } c \text{ or } c^\op \text{ in } C \right)
\equiv 0 \mod 2 .
\end{align*}

\end{enumerate}

\pause

\item \textbf{Note:} Neither of \textbf{(a)} and \textbf{(b)} implies
the other!

\item Bergeron, Ceballos, Labb\'e proved a special case of
\textbf{(b)}.

\end{itemize}

\end{frame}

\begin{frame}
\fti{Strategy of the proof, part 1}

\begin{itemize}

\item Let $T = \bigcup\limits_{w \in W} w S w^{-1}$ (the set of \textit{reflections}
in $W$).

\item Extend the relation $\sim$ to $T$ (same definition).

\item Every reduced expression $\ora{a} = a_1 a_2 \cdots a_k$ for $w$ gives
rise to a list (``inversion word'', aka ``reflection order'')
\begin{align*}
& \Invs \ora{a} = \left(t_1, t_2, \ldots, t_k\right) \in T^k,
\qquad \text{where} \\
& t_i = a_1 a_2 \cdots a_i \cdots a_2 a_1 
\qquad \left(\text{``up to } a_i \text{ and then down''}\right) .
\end{align*}

\pause

\item If \ \ \ \ \ \ \ \ \ 
$\xymatrixcolsep{12pc}
\xymatrix{\ora{a} \ar[r]^{\text{braid move involving } s \text{ and } t } & \ora{b}}$,
\\ then
$\xymatrixcolsep{12pc}
\xymatrix{\Invs \ora{a} \ar[r]^{\text{revert a certain factor}} & \Invs \ora{b}}$.

\end{itemize}

\end{frame}

\begin{frame}
\fti{Strategy of the proof, part 2}

\begin{itemize}

\item If \ \ \ \ \ \ \ \ \ 
$\xymatrixcolsep{12pc}
\xymatrix{\ora{a} \ar[r]^{\text{braid move involving } s \text{ and } t } & \ora{b}}$,
\\ then
$\xymatrixcolsep{12pc}
\xymatrix{\Invs \ora{a} \ar[r]^{\text{revert a certain factor}} & \Invs \ora{b}}$.

\pause

\item Which factor? Let's say the braid move replaces some
$a_{i+1} a_{i+2} \cdots a_{i+k} = s t s \cdots $ in $\ora{a}$ by
$b_{i+1} b_{i+2} \cdots b_{i+k} = t s t \cdots $. \\
Then, the factor that gets reverted is in positions
$i+1, i+2, \ldots, i+k$ again. \\
\pause
\item
The dihedral subgroup $\left<s,t\right>$ has $m_{s,t}$ reflections,
and two canonical ways to list them:
\begin{align*}
\rho_{s,t} &= \left(s, sts, ststs, \cdots, \underbrace{sts\cdots s}_{2m_{s,t}-1 \text{ factors}}\right) , \\
\rho_{t,s} &= \left(t, tst, tstst, \cdots, \underbrace{tst\cdots t}_{2m_{s,t}-1 \text{ factors}}\right) .
\end{align*}
(These are mutually reverse.)

\end{itemize}

\end{frame}

\begin{frame}
\fti{Strategy of the proof, part 3}

\begin{itemize}

\item If \ \ \ \ \ \ \ \ \ 
$\xymatrixcolsep{12pc}
\xymatrix{\ora{a} \ar[r]^{\text{braid move involving } s \text{ and } t }_{\text{in positions } i+1, i+2, \ldots, i+k} & \ora{b}}$,
\\ then
$\xymatrixcolsep{12pc}
\xymatrix{\Invs \ora{a} \ar[r]^{\text{revert the word }q \rho_{s, t} q^{-1}}_{\text{in positions } i+1, i+2, \ldots, i+k} & \Invs \ora{b}}$,
where
$
q = a_1 a_2 \cdots a_i
$.
(Words are conjugated letter-wise. Reverting $q \rho_{s, t} q^{-1}$
gives $q \rho_{t, s} q^{-1}$.)

\pause

\item This is why we had to take $\sim$-conjugacy classes (and not
plain pairs) as colors!

\pause

\item Thus we can try a parity argument: Count how often a
$q \rho_{s, t} q^{-1}$ appears as a subword in $\Invs\ora{a}$ (either
never or once), and notice that our braid move changes this count
by $1 \mod 2$.

\end{itemize}

\end{frame}

\begin{frame}
\fti{Strategy of the proof, part 4}

\begin{itemize}

\item Complications:
\begin{itemize}
\item Be careful with redundant counts (counting everything twice
makes $\mod 2$ useless).
\item Subwords start out as factors, but can get broken apart by
other braid moves.
\item Need to show that other braid moves never mutate our subword
(even though they can spread its letters apart / move them together).
Need some subtle descent/length/parabolic-coset arguments.
\item The $c = c^\op$ and $c \neq c^\op$ cases need separate proofs
at the end.
\end{itemize}

See paper for details.

\end{itemize}

\end{frame}

\begin{frame}
\fti{Conjecture 1}

\begin{itemize}

\item What happens if we replace ``reduced expression'' by
``expression'' everywhere?

\item \textbf{Conjecture 1.} Let $C$ be a directed cycle
in the graph $\mathcal{E}\left(w\right)$ (defined as
$\mathcal{R}\left(w\right)$, but using all expressions)
for some $w \in W$. \\
Let $c \in \mathfrak{M} / \sim$ be an equivalence class (under
simultaneous conjugation). \\
Let $c^\op$ denote the equivalence class
of the opposite pair (i.e., if $c = \left[\left(s,t\right)\right]$,
then $c^\op = \left[\left(t,s\right)\right]$).

\pause
\begin{enumerate}
\item[(a)]
\begin{align*}
& \left(\text{the number of arcs colored } c \text{ in } C \right) \\
= & \left(\text{the number of arcs colored } c^\op \text{ in } C \right) .
\end{align*}

\pause
\item[(b)]
\begin{align*}
\left(\text{the number of arcs colored } c \text{ or } c^\op \text{ in } C \right)
\equiv 0 \mod 2 .
\end{align*}

\end{enumerate}

\end{itemize}

\end{frame}

\begin{frame}
\fti{Conjecture 2}

\begin{itemize}

\only<1>{\item An attempt to explain at least part \textbf{(b)}...} \pause

\item \textbf{Conjecture 2.} For every $\left(  s,t\right)  \in\mathfrak{M}$, let
$c_{s,t} \in \left\{  1,-1\right\}  $. 
\begin{align*}
\text{Assume that } c_{s,t} &= c_{s^{\prime},t^{\prime}} \qquad \text{ whenever }
\left(  s,t\right)  \sim\left(  s^{\prime},t^{\prime}\right) ; \\
c_{s,t} &= c_{t,s} \qquad \text{ for all } \left(s, t\right)
\in \mathfrak{M} .
\end{align*}
Let $W^{\prime}$ be the group given by:
\begin{itemize}
\item \textit{Generators:} the elements $s\in S$ and an extra generator $q$.
\item \textit{Relations:} \only<3>{{\blue (think spin symmetric groups!)}}
\begin{align*}
s^{2}  &  =1\ \ \ \ \ \ \ \ \ \ \text{for every }s\in S;\\
q^{2}  &  =1;\\
qs  &  =sq\ \ \ \ \ \ \ \ \ \ \text{for every }s\in S;\\
\left(  st\right)  ^{m_{s,t}}  &  =1\ \ \ \ \ \ \ \ \ \ \text{for every
}\left(  s,t\right)  \in\mathfrak{M}\text{ satisfying }c_{s,t}=1;\\
\left(  st\right)  ^{m_{s,t}}  &  =q\ \ \ \ \ \ \ \ \ \ \text{for every
}\left(  s,t\right)  \in\mathfrak{M}\text{ satisfying }c_{s,t}=-1.
\end{align*}
\end{itemize}

Then, $q \neq 1$ in $W^\prime$. Equivalently, this sequence is exact:
\xymatrixcolsep{4pc}
\xymatrix{
1 \ar[r] & \mathbb{Z} / 2 \mathbb{Z} \ar[r]^{\overline{1} \mapsto q} &
W^\prime \ar[r]^{s_i \mapsto s_i}_{q \mapsto 1} & W \ar[r] & 1
}

\end{itemize}

\end{frame}

\begin{frame}
\fti{References}

\begin{itemize}

\item \href{https://arxiv.org/abs/1603.03138v2}{Darij Grinberg, Alexander Postnikov,
\textit{Proof of a conjecture of Bergeron, Ceballos and Labb\'e},
{\red arXiv:1603.03138v2}}.

\item \href{http://arxiv.org/abs/1404.7380v2}{Nantel
Bergeron, Cesar Ceballos, Jean-Philippe Labb\'{e}, \textit{Fan realizations of
subword complexes and multi-associahedra via Gale duality},
{\red arXiv:1404.7380v2}.}

\item \href{http://www.ams.org/journals/tran/2013-365-05/S0002-9947-2012-05719-9/S0002-9947-2012-05719-9.pdf}{Victor
Reiner, Yuval Roichman, \textit{Diameter of graphs of reduced words and
galleries}, Trans. Amer. Math. Soc. 365 (2013), pp. 2779--2802.}\newline
Preprint: \href{http://arxiv.org/abs/0906.4768v3}{{\red arXiv:0906.4768v3}}.

\end{itemize}

\pause

\textbf{Thanks to}

\begin{itemize}

\item Nantel Bergeron for an invitation to York U, and Cesar Ceballos
for advertising the results of 1404.7380v2 to me;

\item Franco Saliola for fixing a bug in these slides;

\item you for the attention!

\end{itemize}

\end{frame}

\end{document}

