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\title{Perfect models and Gelfand $W$-graphs}

\author{
Eric Marberg\thanks{\href{mailto:emarberg@ust.hk}{emarberg@ust.hk}. The first author was partially supported by RGC Grant ECS 26305218.
}\addressmark{1}
\and Yifeng Zhang\thanks{\href{mailto:yzhangci@connect.ust.hk}{yzhangci@connect.ust.hk}.
The second author was partially supported by RGC Grant ECS 26305218.}\addressmark{1}
}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, Hong Kong University of Science and Technology}


%\author{Eric Marberg \and Yifeng Zhang}


%% put the date of submission here
\received{December 1, 2020}

%% leave this blank until submitting a revised version
\revised{April 25, 2021}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{
A Gelfand model for an algebra is a module isomorphic to a direct sum of irreducible modules, 
with every isomorphism class  of irreducible modules represented exactly once.
We introduce and study the notion of a perfect model for a finite Coxeter group;
such a model is a certain set of discrete data parametrizing a Gelfand model for the associated Iwahori--Hecke algebra. We classify which Coxeter groups have perfect models,
and then describe explicit Gelfand models for the classical finite Coxeter groups. This generalizes separate constructions of Adin, Postnikov, and Roichman
and of Araujo and Bratten.
Our Gelfand models have interesting ``canonical bases'' that give rise to associated $W$-graphs. We classify the molecules in these $W$-graphs when $W$ is a symmetric group,
and conjecture that in type A every molecule is a cell.
}

%% put your French abstract here, or comment this out if you don't have one
%\resume{}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Coxeter systems, Iwahori--Hecke algebras, $W$-graphs, Gelfand models, perfect involutions, quasiparabolic sets}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations, ie. [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex]{biblatex}
\addbibresource{50Marberg.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands


\begin{document}
\maketitle


\setcounter{tocdepth}{2}
%\tableofcontents


 \section{Introduction}


In this article we study certain uniform ways 
of constructing multiplicity-free representations of Iwahori--Hecke algebras
that are also instances of \emph{$W$-graphs}.
We begin by explaining the definition of the models that are our primary subject.

Let $(W,S)$ be a Coxeter system with length function $\ell : W \to \NN \coloneqq \{0,1,2,\dots\}$.
Define $\Aut(W,S)$ to be the group of \emph{Coxeter automorphisms} of $W$, that is, group automorphisms $\varphi \in \Aut(W)$ with $\varphi(S) = S$.
Let $W^+ = W \rtimes \Aut(W,S)$ be the semidirect product whose elements are the pairs $(w, \varphi)$ with $w \in W$ and $\varphi \in \Aut(W,S)$, with multiplication 
\[ (v,\alpha)(w,\beta) \coloneqq (v \cdot \alpha(w), \alpha\beta).\]
Extend the length function of $W$ to $W^+$ by setting $\ell((w,\varphi)) = \ell(w)$.
We view $W$ as a subgroup of $W^+$ by identifying $w \in W$ with $(w,\id_W) \in W^+$.

Let $z = (w,\varphi) \in W^+$.
Following \cite{RV}, we define $z$ to be a \emph{perfect involution} if 
$z^2= (z t )^4 = 1 $ for all reflections
$t \in  \{ vsv^{-1} : v \in W, s \in S\}.$
Let $\sI = \sI(W,S)$ be the set of perfect involutions in $W^+$. 
The group $W$ acts on $\sI$ by conjugation
\[v : (w,\varphi) \mapsto v (w,\varphi)v^{-1} =  (v\cdot w \cdot\varphi(v)^{-1},\varphi) .\]
%
As in \cite{RV}, we say an element $z \in \sI$ is \emph{$W$-minimal}
if $\ell(sz s) \geq \ell(z)$ for all $s \in S$, and \emph{$W$-maximal}
if $\ell(sz s) \leq \ell(z)$ for all $s \in S$.
Each $W$-conjugacy class in $\sI$ contains a unique $W$-minimal element
(and a unique $W$-maximal element when $|W|<\infty$) \cite[Cor. 2.10]{RV}.


If $C_W(\theta) \coloneqq \{ w \in W: wz = z w\}$ where
$z\in \sI$ is $W$-minimal,
 then $C_W(z)$ is a \emph{quasiparabolic subgroup} of $W$ in the sense of Rains and Vazirani \cite[Thm. 4.6]{RV}. Among other consequences, this implies that the 
 permutation representation of $W$ acting on  $W/C_W(z)$ naturally deforms to a representation of the Iwahori--Hecke algebra of $(W,S)$.


\begin{example}\label{ex-1}
%Rains and Vazirani classify the perfect involutions in the three families of finite classical Coxeter groups in 
The sets $\sI(W,S)$ for classical Coxeter systems are described in
\cite[\S9]{RV}:
\ben
\item[($\A$)] The symmetric group $S_n$ of permutations of $[n] \coloneqq \{1,2,\dots,n\}$
is a Coxeter group relative to the generating set $\{s_1,s_2,\dots,s_{n-1}\}$ where $s_i\coloneqq(i,i+1)$.
The only non-identity element of $\Aut(S_n,\{s_1,s_2,\dots,s_{n-1}\})$ is the inner automorphism $\Ad(w_0)$
induced by  $w_0 = n\cdots 321$. 
When $n$ is odd every perfect involution in $S_n^+$ is central.
When $n$ is even there are three $S_n$-minimal elements in $S_n^+$: the identity element $1 \in S_n $, the fixed-point-free involution $s_1s_3s_5\cdots s_{n-1} \in S_n$, and $(1, \Ad(w_0)) \in S_n^+$.

\item[($\BC$)] The group $\W_n$ of permutations of $\{-n,\dots,-1,0,1,\dots,n\}$ that commute with the negation map
is a Coxeter group relative to the generators $\{s_0,s_1,\dots,s_{n-1}\}$ 
with $s_0 \coloneqq (-1,1)$ and $s_i \coloneqq (-i-1,-i)(i,i+1)$ for $i>0$. 
The one-line representation of $w \in \W_n$ is the word $w(1)w(2)\cdots w(n)$ where we write $\bar i$ in place of $-i$.
When $n>2$ there are only trivial Coxeter automorphisms 
and the perfect involutions in $\W_n$ are the elements $w=w^{-1}$
such that $|w| : i \mapsto |w(i)|$ is the identity map or a fixed-point-free permutation of $[n]$.
The $\W_n$-minimal perfect involutions consist of the $n+1$ elements
$\theta_i \coloneqq \bar 1 \bar 2 \cdots \bar i (i+1)\cdots n$  plus $s_1s_3s_5\cdots s_{n-1}$ if $n$ is even.

\item[($\D$)] The subgroup $\WD_n$ of $w \in \W_n$ such that $|\{ i \in [n] : w(i) < 0\}|$ is even 
is a Coxeter group relative to the generators $\{s_{-1},s_1,s_2,\dots,s_{n-1}\}$ where $s_{-1} \coloneqq (-2,1)(-1,2)$
and $s_i $ for $i\geq 0$ is as in ($\BC$).
For $n>4$ there is just one nontrivial Coxeter automorphism  $w\mapsto w^\ast \coloneqq s_0ws_0$.
The perfect involutions in $(\WD_n)^+$ are the pairs $(w, \varphi)$ with $\varphi \in \{1,\ast\}$ and $w^{-1} = \varphi(w)$ such that $|w| \in S_n$ is $1$ or fixed-point-free.
The $\WD_n$-minimal perfect involutions are 
$\theta_i$ for $i$ even, $\tilde \theta_i \coloneqq (1 \bar2 \bar3 \cdots \bar i (i+1)\cdots n, \ast)$ for $i$ odd,   plus both $s_1s_3s_5\cdots s_{n-1}$ and $s_{-1}s_3s_5\cdots s_{n-1}$ if $n$ is even.
\een
\end{example}







Given  $J\subset S$, let $W_J \coloneqq \langle s \in J\rangle$. % be the corresponding parabolic subgroup. 
Then $(W_J, J)$ is a Coxeter system
whose length function is $\ell|_{W_J}$.
Write $\sI_J \coloneqq \sI(W_J,J)$ for the set of perfect involutions in $(W_J)^+$.

\begin{definition}
A \emph{model triple} $(J, z_{\min}, \sigma)$ for a finite Coxeter group $W$ consists of
a set $J\subset S$, a $W_J$-minimal element $z_{\min} \in  \sI_J$,
and a linear character $\sigma : W_J \to \{\pm 1\}$.
A set $\sP$ of model triples for $W$ is a \emph{perfect model}   if
$ \sum_{(J,z_{\min},\sigma) \in \sP} \Ind_{C_{W_J}(z_{\min})}^W \Res^{W_J}_{C_{W_J}(z_{\min})}(\sigma) = \sum_{\chi \in \Irr(W)} \chi$,
where $\Irr(W)$ is the set of complex irreducible characters of $W$
and $\Ind_H^G$ and $\Res_H^G$ denote the usual operations of induction and restriction
for groups $H\subset G$.
\end{definition}

Our first objective is to classify which finite Coxeter groups have perfect models. We will then investigate how to use such models to construct some interesting representations 
of the Iwahori--Hecke algebras of finite classical Coxter systems.
Since space is limited, we have omitted all proofs in this extended abstract; for 
these details, see \cite{MZ}.

\section{Classification of perfect models}

A Coxeter system $(W,S)$ is \emph{irreducible} if the only disjoint decomposition $S=S'\sqcup S''$ 
in which every $s' \in S'$ commutes with every $s'' \in S''$ has either $S'=\varnothing\neq S''$ or $S' \neq \varnothing=S''$.
An \emph{irreducible factor} of $(W,S)$ is a parabolic subsystem $(W_J,J)$ that is irreducible.

\begin{theorem}\label{main-thm1}
A finite Coxeter group $W$ 
has a perfect model if and only if each of its irreducible factors has a perfect model.
The irreducible Coxeter systems
with perfect models are those of type $\A_{n-1}$, $\BC_n$,
$\D_{2n+1}$, $\mathsf{H}_3$, and $\mathsf{I}_2(n)$ for all $n\geq 2$.
The irreducible Coxeter systems without perfect models are
those of type $\D_{2n}$, $\E_6, \E_7, \E_8$, $\mathsf{F}_4$, and $\mathsf{H}_4$.
\end{theorem}

Perfect models for the finite classical Coxeter groups are given below.
The model for type $\A_{n-1}$ is well-known; see \cite{APR2007,IRS,KV2004}.
What we describe for type $\BC_n$ (respectively, $\D_n$)
is a coarser version of the models in
\cite{APR2010,Baddeley,M2012}
(respectively, \cite{Caselli2009,CF, CM,M2011}).


\begin{definition}\label{main-thm1-def}
 Let $W \in \{ S_n, \W_n, \WD_n\}$ be a classical Weyl group.
 Write $\one=\one_W : w \mapsto 1$ and $\sgn=\sgn_W : w \mapsto (-1)^{\ell(w)}$ for the trivial and sign 
 representations.
%of type $\A$, $\BC$, or $\D$.
\ben

\item[($\A$)] Assume $W=S_n$ with $n\geq 1$.  For each integer $0\leq k\leq \lfloor\frac{n}{2}\rfloor$, form a triple $(J,z_{\min},\sigma)$ 
by taking $J = \{s_1,s_2,\dots,s_{n-1}\} \setminus\{s_{2k}\}$
so that  $ \langle J\rangle  = S_{2k} \times S_{n-2k} $,
and then setting
\[z_{\min} = s_1s_3s_5\cdots s_{2k-1}
\quand \sigma = \one_{S_{2k}} \times \sgn_{S_{n-2k}}.\]



\item[($\BC$)] Assume $W=\W_n$ with $n\geq 2$.  For each $0\leq k\leq \lfloor\frac{n}{2}\rfloor$, form a triple
  $(J,z_{\min},\sigma)$ by taking
$J = \{s_0,s_1,s_2,\dots,s_{n-1}\} \setminus\{s_{2k}\}$
 so that  $ \langle J\rangle  = \W_{2k} \times S_{n-2k}  $,
 and then setting
\[z_{\min} = s_1s_3s_5\cdots s_{2k-1}
\quand \sigma = \one_{\W_{2k}} \times \sgn_{S_{n-2k}}.\]

\item[($\D$)] Assume $W=\WD_n$ with $n\geq 3$. For each $0<k\leq \lfloor\frac{n}{2}\rfloor$, form a triple
  $(J,z_{\min},\sigma)$ by taking
$J = \{s_{-1},s_1,s_2,\dots,s_{n-1}\} \setminus\{s_{2k}\}$
 so that $ \langle J\rangle  = \WD_{2k} \times S_{n-2k}  $,
 and then setting
\[z_{\min} = s_1s_3s_5\cdots s_{2k-1}
\quand\sigma = \one_{\WD_{2k}} \times \sgn_{S_{n-2k}}.\]
Also include one additional triple $(J,z_{\min},\sigma)=(\{s_1,s_2,\dots,s_{n-1}\}, 1, \sgn_{S_n})$.

\een
Let $\sP=\sP(W)$ be the set of
 $1+\lfloor \frac{n}{2}\rfloor$ triples $(J,z_{\min},\sigma)$ listed in each case.
 \end{definition}
 


\begin{theorem}
\label{main-thm1b}
Assume $W$ is one of the classical Weyl groups $S_{n-1}$, $\W_n$, or $\WD_{2n+1}$ for an integer $n\geq 2$. 
Then the set  $\sP(W)$ given in Definition~\ref{main-thm1-def} is a perfect model.
\end{theorem}


Let $G$ be a finite group. Choose representatives for the distinct conjugacy classes  
of involutions $g=g^{-1} \in G$. For each representative involution $g$, form the centralizer $H = C_G(g) = \{x \in G : xg = gx\}$
and choose a linear character $\lambda : H \to \CC$.
The resulting set of pairs $(H,\lambda)$ is 
an \emph{involution model} for $G$ if $\sum_{(H,\lambda)} \Ind_H^G(\lambda) = \sum_{\chi \in \Irr(G)} \chi$.  
The existence of an involution model implies
that $ |\{ g \in G : g=g^{-1}\}| = \sum_{\chi \in \Irr(G)} \deg \chi$, which holds if and only if all representations of $G$ are realizable over $\RR$ (see \cite[\S2]{BumpGinzburg}).

The involution models for finite Coxeter groups
are classified in \cite{BaddeleyThesis,Vinroot}.
Surprisingly, although 
we do not know of any general procedure for converting
involution models to perfect models or vice versa,
this classification is identical to the one in
Theorem~\ref{main-thm1}:

\begin{corollary}
A finite Coxeter group has a perfect model if and only if it has an involution model.
\end{corollary}




\section{Gelfand models for Iwahori--Hecke algebras}

Let $x$ be an indeterminate.
The \emph{(single-parameter) Iwahori--Hecke algebra} of a Coxeter system $(W,S)$ is
the free $\QQ[x,x^{-1}]$-module $\H = \H(W)$ with basis $\{H_w : w \in W\}$, equipped with the unique algebra
structure in which $H_wH_s = H_{sw}$ if $\ell(sw) > \ell(w)$
and $H_wH_s = H_{sw} +(x-x^{-1})H_w$ if $\ell(sw) < \ell(w)$ for all $s \in S$ and $w \in W$.
%\[ H_s H_w  =\begin{cases} H_{sw} &\text{if }\ell(sw) > \ell(w) \\
%H_{sw} + (x-x^{-1}) H_w &\text{if }\ell(sw) <\ell(w)
%\end{cases}
%\quad\text{for all $s \in S$ and $w \in W$.} \]

A \emph{Gelfand model} for $\H$ is an $\H$-module
 isomorphic to the direct sum of all irreducible $\H$-modules.
Any perfect model for $W$ gives rise to  
 a Gelfand model for $\H$.
Our results in this section are formal consequences of this fact and
Theorem~\ref{main-thm1b}.

 

For integers $i > 0$, define permutations 
$
s_i \coloneqq (-i-1,-i)(i,i+1),
$
$
s_0 \coloneqq (-1,1),
$
and
$ s_{-i} = (-i-1,i)(-i,i+1).
$
We realize the Coxeter groups of type $\A_{n-1}$, $\BC_n$, and $\D_n$
as 
$
S_n \coloneqq \langle s_1,s_2,\dots, s_{n-1}\rangle
\subset \WD_n \coloneqq \langle s_{-1},s_1,s_2,\dots,s_{n-1}\rangle
\subset \W_n \coloneqq \langle s_0,s_1,s_2,\dots,s_{n-1}\rangle
.$
Assume below that $W$  is one of these  groups
with $S$ the set of simple generators just listed.
Then the length function of $W$ is determined by the following properties:
\begin{itemize} 
\item If $w\in W$  and $i >0$ 
then $\ell(ws_i) < \ell(w) $ if and only if $w(i) > w(i+1)$.

\item If $w \in W=\W_n$ then $\ell(ws_0) <\ell(w) $ if and only if $w(1) < 0$.

\item If $w \in W= \WD_n$ then $\ell(ws_{-1}) < \ell(w) $ if and only if 
$-w(2) > w(1)$.
% we have either
%$w(1) <0>w(2)$ or $ 0<w(2)<-w(1) $ or $ 0 < w(1) < -w(2).$
\end{itemize}
%
Let $\cF_{2n} \coloneqq \left\{ z \in \W_{2n} : z=z^{-1}\text{ and }|z(i)| \neq i \text{ for all }i \in [2n]\right\}\subset \WD_{2n}.$
We say that an integer $i>0$ is a \emph{visible descent} of $z \in \cF_{2n}$
if $z(i+1) < \min\{i,z(i)\}$ or $ z(i) < -i.$ Next:
\begin{itemize}
\item Define $\cK_{n}^\BC$ to be the set of $z \in \cF_{2n}$ with no visible descents greater than $n$.
\item Define $\cK^\A_{n-1} \coloneqq \cK_n^\BC \cap S_{2n}$.
\item Define $\cK^\D_n$ to  be the subset of 
 $z \in \cK_n^\BC$ for which $|\{ i\in [n] : z(i) < -i\}|$
is even.
\end{itemize}
%
Define $\cK = \cK(W)$ to be $\cK^\A_{n-1}$, $\cK_n^\BC$, or $\cK_n^\D$
according to whether $W$ is $S_n$, $\W_n$, or $\WD_n$.
When evaluating $\ell(z)$ for $z \in \cK$, we consider
$
\cK^\A_{n-1} \subset S_{2n},
$
$
\cK^\BC_{n} \subset \W_{2n} ,
$
and
$
\cK^\D_{n} \subset \WD_{2n}.
$
\begin{remark}\label{notation-rmk}
One can characterize the elements of $\cK$ 
more explicitly. 
Namely,  $z \in \cF_{2n}$ belongs to $ \cK_n^\BC$
if and only if 
there is an integer $0 \leq i \leq n$ with 
$ i \equiv n \modu 2)$ such that
\[
-n \leq z(n+1) < z(n+2) < \dots < z(n+i) \leq n
\quad\text{and}\quad
z(n+i+2j) =n+i+2j-1\] 
for 
%
each 
 integer $j>0$ with $n+i+2j \leq 2n$. 
This suggests the following notation.
Suppose $w =w^{-1} \in W^\BC_n$. Let 
$a_1 >a_2  > \dots >a_p$
(respectively, $ b_1<b_2<\dots<b_q$)
be the numbers $a \in [n]$ with $w(a) = -a$ (respectively, $b \in [n]$ with $w(b) = b$).
Define $\underline w \in \cK^\BC_n$ to be the unique element mapping 
$a_i \mapsto -n -i$ and $b_i \mapsto n + p + i$ and $c \mapsto w(c)$ for $c \in [n] \setminus \{ a_1,a_2,\dots,a_p, b_1,b_2,\dots,b_q\}$.
For example, we would have
\[
\underline{2134} = 21563487 \in \cK^\A_3
\quand 
\underline{\bar 3 2 \bar 1 \bar 4 \bar 5} = \bar 3, 8, \bar 1, \bar 7, \bar 6, \bar 5, \bar 4, 2,10,9 \in \cK^\BC_5
\]
When $W \in \{S_n, \W_n\}$, the map $w \mapsto \underline w$ is a bijection $\{ w=w^{-1} \in W\} \to \cK.$
In type $\D$, 
if we set  $e(w) \coloneqq | \{ i \in [n]  : w(i) < -i\}|$ for $w \in \W_n$,
then the same map is a bijection
$%\be\label{dun-eq}
\{ w=w^{-1} \in \WD_n : e(w)\text{ is even}\} \sqcup
\{ w=w^{-1} \in \W_n \setminus \WD_n : e(w)\text{ is odd}\}
\to \cK_n^\D.
$%\ee


\end{remark}

Fix $z \in \cK$. Let 
$ \Des(z): =\{ s \in S : \ell(zs) < \ell(z)\}
$ and 
$
\Asc(z) \coloneqq\{ s \in S : \ell(zs) > \ell(z)\}
$
where $\ell$ is the length function of $S_{2n}$, $\W_{2n}$, or $\WD_{2n}$,
according to whether the Coxeter group $W$ is $S_n$, $\W_n$, or $\WD_n$,
respectively.
Next let
\[ \Des^=(z) \coloneqq \{ s \in S : sz=zs\}
\quand
\Asc^=(z) \coloneqq \{ s \in S : zsz \in \{ s_i : i > n\} \},\]
and define $\Des^< (z) $ and $ \Asc^< (z)$ to be the complements of 
$\Des^=(z)\sqcup \Asc^=(z)$
in $ \Des(z)$ and $\Asc(z)$, respectively.


Finally, define 
$\cM = \cM(W)$ and $\cN = \cN(W)$ to be the
 free $\LL$-modules with respective bases
  $\{M_z : z \in \cK \}$ and $ \{N_z : z \in \cK \}$.
  Our third main result is the following:


\begin{theorem}\label{main-thm2}
Assume $(W,S)$ is a classical finite Coxeter system of type $\A_{n-1}$, $\BC_n$, or $\D_n$ for some $n\geq 2$.
Then there is a unique $\H$-module structure on $\cM$ in which
\[
H_s  M_z = \begin{cases} 
M_{s  zs}  &\text{if }s \in \Asc^<(z) \\
M_{s  zs} + (x - x^{-1}) M_z &\text{if }s\in \Des^<(z) \\
xM_z & \text{if }s \in \Des^=(z) \\
-x^{-1} M_z & \text{if }s \in \Asc^=(z)
\end{cases}
\quad\text{for all $s \in S$ and $z \in \cK$},
\]
and there is a unique $\H$-module structure on $\cN$ in which
\[
H_s  N_z = \begin{cases} 
N_{s  zs}  &\text{if }s \in \Asc^<(z) \\
N_{s  zs} + (x - x^{-1}) N_z &\text{if }s\in \Des^<(z) \\
-x^{-1}N_z & \text{if }s \in \Des^=(z) \\
x N_z & \text{if }s \in \Asc^=(z)
\end{cases}
\quad\text{for all $s \in S$ and $z \in \cK$}.
\]
These $\H$-modules are multiplicity-free direct sums of irreducible submodules.
If
$(W,S)$ is not of type
 $ \D_{n}$
 with $n$ even,
then $\cM$ and $\cN$ are both Gelfand models
for $\H$.
\end{theorem}

The module $\cM(S_n)$ can be identified with the Gelfand model for $\H(S_n)$
that Adin, Postnikov, and Roichman study in \cite{APR2007}.
On the other hand, 
$\cN(W)$ for each group $W \in\{S_n,\W_n,\WD_n\}$ is a deformation
%\footnote{
%In the notation of \cite[\S1.2]{APR2007}, one has $q=x^2$ and $T_i = -x H_{s_i}$.
%On fixing $z \in \cG^\BC_{n} \supset \cG^\A_{n-1}$, define $w \in \W_n$ to have
% $w(i) = z(i)$ if $|z(i)| \leq n$,  $w(i) = i$ if $z(i) > n$, and $w(i) = -i$ if $z(i) < n$.
%Then an isomorphism from $\cM_\QQ(S_n)$ to the module $V_n$ in \cite[\S1.2]{APR2007}
%is  given by the linear map with $M_z \mapsto C_w$.
%Similarly, if one sets $x=1$, then
% the linear map sending $N_z \mapsto e_w$ 
%  is an isomorphism from $\cN(W)$ to the representation in \cite[\S2]{AraujoBratten}
%  for each $W \in\{S_n,\W_n,\WD_n\}$.
%} 
of the $W$-representation 
described by Araujo and Bratten in \cite{AraujoBratten}.
%
Theorem~\ref{main-thm2} implies that $\cM(S_n)$ and $\cN(S_n)$ are both abstractly isomorphic 
%after an appropriate extension of scalars)
to the type $\A_{n-1}$ version of the $\H$-module studied by Lusztig and Vogan in \cite{LV};
see \cite[\S5.3]{LV}. It is not clear how to write down these isomorphisms in a concrete way.
%
Our construction for types $\BC_n$ and $\D_{2n+1}$ 
partly resolves an open problem mentioned in \cite[\S7]{APR2010}.


\section{Gelfand $W$-graphs}


Let $p \mapsto \overline p$ denote the ring automorphism of $\LL$ sending $x \mapsto x^{-1}$.
A map $\phi : \cA \to \cB$ between $\LL$-modules is \emph{antilinear}
if $\phi(pa) = \overline p \cdot \phi(a)$ for all $a \in \cA$. 
%
%
For the Iwahori--Hecke algebra $\H$ of any Coxeter system $(W,S)$,
there is a unique antilinear map $\H \to \H$, written $H \mapsto \overline H$ and called the \emph{bar operator},
such that $\overline{H_w} = (H_{w^{-1}})^{-1}$ for all $w \in W$.
This map is a ring involution.

The famous \emph{Kazhdan--Lusztig basis} of $\H$ \cite{KL} consists of the unique elements $\underline H_w$ for $w \in W$ satisfying
$\overline{\underline H_w} = \underline H_w \in H_w + \sum_{ \ell(y) < \ell(w)} x^{-1} \ZZ[x^{-1}] H_y.$
Our third main result constructs analogous ``canonical bases'' for the modules $\cM$ and $\cN$
from Theorem~\ref{main-thm2}. 
%
An \emph{$\H$-compatible bar operator} on an $\H$-module $\cA$
is an antilinear map $A \mapsto\overline A$ with $\overline{HA} = \overline H \cdot \overline A$ 
for all $ H \in \H$ and $A \in \cA$.
For the rest of this section,
assume $(W,S)$ is a classical finite Coxeter system and define $\cK$, $\cM$, and $\cN$ as in Theorem~\ref{main-thm2}.

\begin{theorem}\label{main-thm3}
There are unique $\H$-compatible bar operators on $\cM$ and $\cN$
with
$\overline{M_z} = M_z$ and $\overline{N_z} = N_z$ for all $z \in \cK$ with 
$\Des^{<}(z) = \varnothing$.
Each of these bar operators is an involution.
%
Additionally, 
the module $\cM$ has a unique basis $\{ \underline M_z : z \in \cK\}$ satisfying
$\overline{\underline M_z} = \underline M_z \in M_z + \sum_{\substack{%y \in \cK \\ 
\ell(y) < \ell(z)}} x^{-1} \ZZ[x^{-1}] M_y.$
The module $\cN$ likewise has a unique basis 
$\{ \underline N_z : z \in \cK\}$ satisfying
$\overline{\underline N_z} = \underline N_z \in N_z + \sum_{\substack{%y \in \cK \\ 
\ell(y) < \ell(z)}} x^{-1} \ZZ[x^{-1}] N_y.$
\end{theorem}

The structure constants for multiplication $\H \times \cM \to \cM$ and $\H \times \cN \to \cN$
in the bases $\{ \underline M_z\}$ and $\{\underline N_z\}$
may be encoded using Kazhdan and Lusztig's notion of a \emph{$W$-graph} from \cite{KL}.
We review the definition below, following the conventions in \cite{Stembridge}.

\begin{definition}
An \emph{$S$-labeled graph}
  $\Gamma= (V,\omega,I)$ is a set $V$ with maps
 $\omega : V\times V \to \ZZ[x,x^{-1}]$
and $I : V \to \{\text{ subsets of $S$ }\}.$
%
We often think of this structure as a weighted directed graph on $V$
with edges $u\xrightarrow{\omega(u,v)} v$ for each $u,v \in V$ with $\omega(u,v) \neq 0$.
\end{definition}
\begin{definition}
  An $S$-labeled graph $\Gamma=(V,\omega,I)$ 
  is a \emph{$W$-graph}
if
   the free $\LL$-module $\cY(\Gamma)$ with basis $\{ Y_v : v \in V \}$ has an $\H$-module structure where
  \be\label{hy-eq}
    H_s Y_u = \begin{cases} x Y_u &\text{if }s \notin I(u) \\ 
 -x^{-1} Y_u +  \ds\sum_{\substack{v \in V\\ s\notin I(v)}} \omega(u,v) Y_v &\text{if }s \in I(u)
  \end{cases}
\quad\text{for all $s \in S$ and $ u \in V$.}\ee
\end{definition}
\begin{definition}
A $W$-graph  $\Gamma=(V,\omega,I)$ is \emph{quasi-admissible} 
  if (1) the corresponding directed graph is bipartite,
(2) we always have $\omega(u,v) \in \ZZ$,
(3)  we have
  $\omega(u,v) =0$ 
 whenever  $I(u)\subset I(v)$,
 and 
 (4) we have
$\omega(u,v) = \omega(v,u)$
  whenever
   $I(u)\not\subset I(v) \not\subset I(u)$.
A quasi-admissible $W$-graph is \emph{admissible}
if we always have $\omega(u,v) \in \NN$.
\end{definition}
%

%Assume $(W,S)$ is a classical finite Coxeter system and continue from Theorem~\ref{main-thm3}.
Define $\m_{yz}, \n_{yz} \in \ZZ[x^{-1}]$ for $y,z \in \cK$
to be the polynomials with
$\underline M_z = \sum_{y \in \cK} \m_{yz} M_y
$
and
$
\underline N_z = \sum_{y \in \cK} \n_{yz} N_y.$
Write
$ \mu^\m_{yz} \coloneqq [x^{-1}]  \m_{yz}$
and
$\mu^\n_{yz} \coloneqq [x^{-1}]  \n_{yz}$
for the coefficients of $x^{-1}$ in these polynomials. % $\m_{yz}$ and $\n_{yz}$.
For $z \in \cK$, define
\[\Asc^\m(z) \coloneqq \Asc^<(z) \sqcup \Asc^=(z)\quand \Asc^\n(z) \coloneqq \Asc^<(z) \sqcup \Des^=(z).\] 
Let
$\omega^\m(y,z) \coloneqq \mu^\m_{yz} + \mu^\m_{zy}$ if $\Asc^\m(y)\not\subset \Asc^\m(z)$
and $\omega^\m(y,z) \coloneqq 0$ otherwise for $y,z\in\cK$.
Likewise define
$\omega^\n(y,z) \coloneqq \mu^\n_{yz} + \mu^\n_{zy}$ if $\Asc^\n(y)\not\subset \Asc^\n(z)$
and $\omega^\n(y,z) \coloneqq 0$ otherwise.
%
Now let 
$\Gamma^\m=\Gamma^\m(W) \coloneqq (\cK,  \omega^\m, \Asc^\m)
$
and
$
\Gamma^\n=\Gamma^\n(W) \coloneqq (\cK,  \omega^\n, \Asc^\n).
$



\begin{theorem}\label{main-thm4}
%When $W$ is a finite classical Coxeter group,
The $S$-labeled graphs $\Gamma^\m$ and $\Gamma^\n$ are
quasi-admissible $W$-graphs. The linear maps $Y_z \mapsto \underline M_z$ and $Y_z \mapsto \underline N_z$ 
are isomorphisms $\cY(\Gamma^\m) \cong \cM$ and $\cY(\Gamma^\n)\cong \cN$.
\end{theorem}

These $W$-graphs are not usually admissible.
% the first negative edge weights occur when $W\in\{S_8,W^\BC_4,W^\D_7\}$.
%%and for all higher rank groups in each type.
%To actually compute 
%$\underline M_z$ and $\underline N_z$, one can use \eqref{hy-eq}, which becomes an inductive formula after substituting 
%$Y_z \mapsto \underline M_z$ and $Y_z \mapsto \underline N_z$.
%
Define a \emph{Gelfand $W$-graph} %for a finite Coxeter system $(W,S)$
to be a $W$-graph $\Gamma$ such that $\cY(\Gamma)$ is a Gelfand model for $\H(W)$.
Theorem~\ref{main-thm4} constructs a pair of Gelfand $W$-graphs $\Gamma^\m$ and $\Gamma^\n$
for the groups $W \in \{S_n ,\W_n, \WD_{2n+1}\}$. 
In the next section, we explain a precise sense in which these Gelfand $W$-graphs should be considered
as the canonical ones for classical types.


\section{Model equivalence}

Let $\TT = (J,z_{min},\sigma)$ be a model triple for a finite Coxeter system $(W,S)$.
Each map $\alpha \in \Aut(W,S)$ extends to an automorphism of $W^+$ that preserves $\sI(W,S)$  by the formula $\alpha : (w,\varphi) \mapsto (\alpha(w), \alpha \varphi\alpha^{-1})$. Using this convention, we define
\[ \TT^\alpha \coloneqq (\alpha(J), \alpha(z_{min}), \sigma \alpha^{-1})
\text{ for $\alpha \in \Aut(W,S)$}
\quand 
\overline{\TT} \coloneqq (J, z_{min}, \sigma \cdot \sgn).\]
Both
 $\TT^\alpha$ and  $\overline{\TT}$ are again model triples for $(W,S)$.
 Write $w_J$ for the longest element in $W_J$ and set $w_0 \coloneqq w_S$.
 If $z = (y,\theta) \in \sI_J$ for $y \in W_J$ and $\theta \in \Aut(W_J,J)$ then 
we set 
\be\label{wz-vee-eq}
z^\vee \coloneqq (y^\vee,\theta^\vee)\text{ where } \begin{cases} y^\vee \coloneqq w_0\cdot w_J \cdot  y\cdot  w_0,  \\ 
\theta^\vee \coloneqq \Ad(w_0)\circ \Ad(w_J)\circ  \theta \circ \Ad(w_0),
\end{cases}
\ee
writing $\Ad(g) : w \mapsto gwg^{-1}$ for the inner automorphism induced by $g \in W$.
One can check that $z^\vee \in \sI_{w_0 J w_0}$ % for $J^\vee \coloneqq w_0 J w_0$,
and that if $z$ is $W_J$-maximal then $z^\vee$ is $W_{w_0 J w_0}$-minimal.
Thus, if $z_{\max}$ is the unique $W_J$-maximal element  in the $W_J$-orbit of $z_{\min}$,
then 
\be \TT^\vee \coloneqq (w_0 J w_0, z_{\max}^\vee,\sigma \circ \Ad(w_0))\ee
is another model triple.
Finally, 
if $\TT' = (J',z_{\min}',\sigma')$
is a model triple such that 
$ J=J',$  $C_{W_J}(z_{\min}) = C_{W_J}(z'_{\min})$, and $\Res_{C_{W_J}(z_{\min})}^{W_J}(\sigma) = \Res_{C_{W_J}(z_{\min})}^{W_J}(\sigma')$,
then we write $\TT \equiv\TT'$.


\begin{definition}
Let $\sim$ be the transitive closure of the relation on model triples that has 
$\TT \sim \overline{\TT} \sim \TT^\vee$, $\TT \sim \TT^\alpha$ for all $\alpha \in \Aut(W,S)$,
and $\TT \sim \TT'$ whenever $\TT \equiv \TT'$.
Two perfect models $\sP$ and $\sP'$ for the same finite Coxeter group $W$ are \emph{equivalent}
if there is a bijection $ \sP \to \sP'$ such that if $ (J,z_{min},\sigma) \mapsto (J',z'_{min},\sigma')$ then $(J,z_{min},\sigma) \sim (J',z'_{min},\sigma')$.
\end{definition}

Let us explain what makes this is a sensible definition.
Results in \cite{Marberg,RV}
associate to any model triple $\TT = (J,z_{min},\sigma)$ a pair of 
 $W_J$-graphs with vertex set $W_J/C_{W_J}(z_{min})$.
Howlett and Yin's method of $W$-graph induction
\cite{HowlettYin,HowlettYin2}
transforms these structures into certain $W$-graphs, which we call $\Gamma^\m(\TT)$ and $\Gamma^\n(\TT)$,
with vertex set $W^J \times W_J/C_{W_J}(z_{min})$. Here $W^J$ is the set of minimal length left coset representatives of $W_J$ in $W$.


 



We do not have room here to fully explain these constructions (see \cite{MZ}).
 We comment, however, that the $W$-graphs in Theorem~\ref{main-thm4}
 can be identified with the disjoint unions of 
 $\Gamma^\m(\TT)$ and $\Gamma^\n(\TT)$ as $\TT$ ranges over  the  model triples 
 in Theorem~\ref{main-thm1b}, and our notion of equivalence for model triples 
 interacts nicely with  these smaller $W$-graphs: % $\Gamma^\m(\TT)$ and $\Gamma^\n(\TT)$:
\begin{itemize}
\item Replacing $\TT$ by $ \TT^\alpha$
corresponds to applying $\alpha$ to the vertices of  
$\Gamma^\m(\TT)$ and $\Gamma^\n(\TT)$.

\item It holds that $\Gamma^\m(\overline\TT) = \Gamma^\n(\TT)$ and $\Gamma^\n(\overline\TT) = \Gamma^\m(\TT)$.


\item If $\TT \equiv \TT'$  then $\Gamma^\m(\TT) \cong \Gamma^\m(\TT')$ and $\Gamma^\n(\TT) \cong \Gamma^\n(\TT')$.

\item The $W$-graph 
$\Gamma^\m(\TT^\vee)$ is
 dual to $\Gamma^\n(\TT)$, and $\Gamma^\n(\TT^\vee)$ is dual to $\Gamma^\m(\TT)$.

\end{itemize}
There is a precise notion of \emph{duality} for $W$-graphs \cite[Prop. 1.2]{Stembridge}, but
here it suffices to define this as meaning that the underlying directed graphs are anti-isomorphic,
that is, they are isomorphic after all of the directed edges in one graph are reversed. 

%Concerning the last item: if $\Gamma = (V,\omega,I)$ and
%$ \Gamma' = (V',  \omega',  I')$ are $W$-graphs with a 
% bijection $V \to V'$, written $v\mapsto v'$,
%then $\Gamma$ and $\Gamma'$ are \emph{dual} via $v\mapsto v'$ if 
% $\omega'(u',v') = \overline{\omega(v,u)}$ and $I'(v') = S \setminus I(v)$; see \cite[Prop. 1.2]{Stembridge}.


 The \emph{cells} of a $W$-graph $\Gamma = (V,\omega,I)$ are the subsets of $V$ that make up the
  strongly connected components in the associated directed graph. 
  If $C \subset V$ is a cell then 
  $\Gamma|_C \coloneqq (C, \omega|_{C\times C}, I|_C)$
   is itself a $W$-graph, and so defines a \emph{cell representation}
   $\cY(\Gamma|_C)$ of the Iwahori--Hecke algebra $\H$.
Identifying the cells in any particular $W$-graph
   is a natural problem of interest.
   %
The remarks above imply that if we can describe the cells in
$\Gamma^\m(\TT)$ and $\Gamma^\n(\TT)$
then we can also describe the cells in
$\Gamma^\m(\TT')$ and $\Gamma^\n(\TT')$
when $\TT\sim \TT'$.

 


Say that a model triple $(J,z_{min},\sigma)$ is \emph{strict} if 
$\sigma(s) = \sigma(t)$ whenever $s,t \in J$ have $st\neq ts$.
A perfect model is \emph{strict} if all of its model triples are strict.
This always holds if there are only odd edges in the Coxeter graph of $(W,S)$,
as happens in types $\A_n$ and $\D_n$.

  \begin{theorem}
 Suppose $(W,S)$ is an irreducible finite Coxeter system. 
 If $|S| \neq 3$
 then $(W,S)$ has at most one equivalence class of strict perfect models.
 Moreover, if $(W,S)$
  has a perfect model and 
  is not of type $\mathsf{I}_2(2n)$,
then $(W,S)$ also has a strict perfect model.
 \end{theorem}

\begin{remark*}
In type $\mathsf{I}_2(2n)$ there are no strict perfect models.
In type $\BC_n$ for $n\neq 3$,
there are only 4 perfect models. Two of these are strict and the others are  uninteresting ``refinements'' obtained by replacing a single strict model triple by a pair of non-strict ones.
\end{remark*}

To sum up, each perfect model gives rise to two Gelfand $W$-graphs,
and for equivalent models the cells in these $W$-graphs are essentially the same.
The Gelfand $W$-graphs for the classical groups in Theorem~\ref{main-thm4} are derived from representatives of a unique equivalence classes of (strict) perfect models.
In types $\BC_n$ and $\D_n$, there is one more property that underscores how the Gelfand $W$-graphs from Theorem~\ref{main-thm4}  are canonical:
%Recall the notation $\underline w $ for elements of $ \cK$ from Remark~\ref{notation-rmk}.
%For $w \in \W_{n}$, let $-w\in \W_{n}$ be the permutation $i \mapsto -w(i) = w(-i)$, and when $w=w^{-1}$ define $\iota_n^\BC(\underline w) = \underline{-w}$. This map is an involution of $\cK^\BC_n$.
%For $z \in \cK^\D_n$ define $\iota_n^\D(z) = \iota_n^\BC(z)$ 
%if $n + |\{ i \in [n] : |z(i)| > n\}|$ is divisible by $4$ and otherwise
%set $\iota_n^\D(z) = s_0\cdot \iota_n^\BC(z) \cdot s_0$ where $s_0=(-1,1)$.
%This is an involution of $\cK^\D_n$.

\begin{theorem}\label{bcd-dual-thm}
If $W \in \{W^\BC_n,W^\D_n\}$, then 
 $\Gamma^\m(W)$ and $\Gamma^\n(W)$
are dual in the sense of being anti-isomorphic as directed graphs (via an explicit bijection given in \cite{MZ}).
\end{theorem}

%\begin{theorem}\label{bc-dual-thm}
%The $W^\BC_n$-graphs $\Gamma^\m(W^\BC_n)$ and $\Gamma^\n(W^\BC_n)$
%are dual to each other via $\iota_n^\BC$.
%\end{theorem}
%
%
%
%\begin{theorem}\label{d-dual-thm}
%The $W^\D_n$-graphs $\Gamma^\m(W^\D_n)$ and $\Gamma^\n(W^\D_n)$
%are isomorphic after reversing the arrow of each edge.
%\end{theorem}

The analogous property for $\Gamma^\m(S_n)$ and $\Gamma^\n(S_n)$ seems to hold if and only if $n\leq 4$.

\section{Molecules in type A}

 It is an interesting open problem to identify the cells in the $W$-graphs 
 $\Gamma^\m(W)$ and $\Gamma^\n(W)$ for each classical Weyl group $W$. For $W \in \{\W_n, \WD_n\}$, it follows by Theorem~\ref{bcd-dual-thm} that the cells in $\Gamma^\m(W)$ determine the cells in $\Gamma^\n(W$ and vice-versa.
The cell classification problems for $\Gamma^\m(S_n)$ and $\Gamma^\n(S_n)$ are genuinely different.

  The \emph{molecules} in a $W$-graph $\Gamma = (V,\omega,I)$
  are the equivalence classes in $V$ under the
  transitive closure of the relation with $u \sim v$ if $\omega(u,v)\neq 0 \neq \omega(v,u)$.
%  If $C \subset V$ is a cell then 
%  $\Gamma(C) \coloneqq (C, \omega|_{C\times C}, I|_C)$
%   is itself a $W$-graph.
%   Each molecule is naturally viewed as an undirected graph and is a subset of exactly one cell.
   These subset  do not inherit a $W$-graph structure, but every cell is a disjoint union of molecules.   





 
 
 %\subsection{Cells in type A}

Computations suggest that there are fewer cells than molecules
 in $\Gamma^\m(\W_n)$ for all $n\geq 6$ and in $\Gamma^\m(\WD_n)$ for all $n\geq 4$.
 Moreover, the cell representations for both of these $W$-graphs are often reducible.
  In type $\A$, however, the following holds for at least $n \leq 10$:
  



\begin{conjecture}
Each molecule in $\Gamma^\m(S_n)$ and $\Gamma^\n(S_n)$
is a cell, and the cell representations associated to these $S_n$-graphs are all irreducible.
\end{conjecture}

As a first step toward proving this conjecture, 
 we describe the molecules in  $\Gamma^\m(S_n)$ and $\Gamma^\n(S_n)$.
 The \emph{(Young) diagram} of a partition $\lambda =(\lambda_1\geq \lambda_2 \geq \dots \geq \lambda_k>0)$
is the set $\D_\lambda = \{ (i,j) \in [k]\times \ZZ: 1 \leq j \leq \lambda_i\}$. For us, a \emph{tableau} of shape $\lambda$
is just a map $T : \D_\lambda \to \ZZ$, which we envision as an assignment of numbers
to some set of positions in a matrix. 
%A tableau is \emph{semistandard} if its rows are weakly increasing and its columns are strictly increasing.
A tableau is \emph{standard} if its rows and columns are strictly increasing and its entries
are the numbers $1,2,3,\dots,m$ for some $m\geq 0$ without any repetitions.


Suppose $T$ is a tableau and $a \in \ZZ$.
Define $T \fromRSK a$ to be the tableau formed by the familiar \emph{Schensted insertion} process.
 In detail, start by inserting $a$ into the first row of $T$. 
At each stage, a number $x$ is inserted into a row.
Let $y$ be 
the first entry in the row with $x< y$.
If no such $y$ exists then $x$ is added to the end of the row;
otherwise, $x$ replaces $y$ and  
$y$ is inserted into
the next row.
Once this process terminates, the result is $T\fromRSK a$.

 \begin{definition}
 Suppose $T$ is a tableau and  $a\leq b$ are integers. 
 Let  $T \fromMRSK (a,b)$ and $T \fromNRSK (a,b)$ be the tableaux given as follows.
 \ben


\item[(a)] %Next, let $T \fromMRSK (a,b)$ be the tableau given as follows.
If $a=b$ then  $T \fromMRSK (a,b) \coloneqq T\fromRSK a$.
If $a<b$ and $(i,j)$ is the box of $T \fromRSK a$   not in $T$,
then form $T \fromMRSK (a,b)$ from $T \fromRSK a$ by adding $b$ to the end of row $i+1$.

\item[(b)] 
If $a=b$ then  $T \fromNRSK (a,b)\coloneqq T\fromRSK a$.
If $a<b$ and $(i,j)$ is the box of $T \fromRSK a$ not in $T$,
then form $T \fromNRSK (a,b)$ from $T \fromRSK a$ by adding $b$ to the end of column $j+1$.

\een
 \end{definition}
 
 We refer to the algorithms constructing $T \fromMRSK a$  and $T \fromNRSK a$ as \emph{$\fkm$-insertion} and \emph{$\fkn$-insertion}.
 Our notion of $\fkm$-insertion is closely related to what Beissinger describes as \cite[Algorithm 3.1]{Beissinger}, while $\fkn$-insertion appears to be new.
  For example, we have
{\small \[ \ytab{
  2 & 3 
  \\4
} \fromRSK 2
   \  = \
    \ytab{
  2 & 3 
  \\4
} \fromMRSK (2,2)
   \  = \
    \ytab{
  2 & 3 
  \\4
} \fromNRSK (2,2)
   \  = \
\ytab{
   2 & 2 
   \\
   3\\
   4
} \]}
while
{\small \[
\ytab{
  2 & 3 
  \\4
} \fromMRSK (2,5)
   \  = \
\ytab{
   2 & 2  \\
   3 \\
   4 \\
   5
} 
\quand
\ytab{
  2 & 3 
  \\4
} \fromNRSK (2,5)
   \  = \
\ytab{
   2 & 2  \\
   3 & 5
    \\ 4
}. \]}

For  $w = w_1w_2\cdots w_n \in S_n$ let
$\PRSK(w) = \emptyset \fromRSK w_1 \fromRSK w_2 \fromRSK \cdots \fromRSK w_n.$
%This is a standard tableau whose shape is a partition of $n$.
The molecules in the $S_n$-graph associated to the type $\A_{n-1}$ Kazhdan--Lusztig basis (see \cite{KL}) are precisely the sets $\{ w \in S_n : \PRSK(w) = T \}$
as $T$ ranges over all standard tableaux with $n$ boxes. Each of these molecules turns out to be a cell 
with irreducible cell representation \cite{KL}.
 
We classify the molecules in   $\Gamma^\m(S_n)$ and $\Gamma^\n(S_n)$ in a related way.
 Fix  $z \in \cK^\A_{n-1}$. Let 
$
b_1 < b_2 <\dots <b_p
$
(respectively, $c_1 < c_2 < \dots <c_q$)
be the numbers $b \in [n]$ with $z(b) < b$ (respectively, the numbers $c \in [n]$ with $n < z(c)$)
and set $a_i = z(b_i)$. %Next let $c_1 < c_2 < \dots <c_q$ be the numbers $c \in [n]$ with $c \leq n < z(c)$.
Then define
\[
\ba \PMRSK(z) &=  \emptyset \fromMRSK (a_1,b_1) \fromMRSK\cdots \fromMRSK (a_p,b_p)
\fromMRSK (c_1,c_1) \fromMRSK \cdots \fromMRSK (c_q,c_q),
\\
\PNRSK(z) &=  \emptyset \fromNRSK (a_1,b_1) \fromNRSK\cdots \fromNRSK (a_p,b_p)
\fromNRSK (c_q,c_q) \fromNRSK \cdots \fromNRSK (c_1,c_1)
.
\ea\]
Let $\lambda_\fkm(z)$ and $\lambda_\fkn(z)$  be the   shapes of $\PMRSK(z)$ and $\PNRSK(z)$ for $z \in \cK^\A_{n-1}$.
Given a partition $\lambda$ of $n$, define $C^\m_\lambda = \{ z \in \cK^\A_{n-1} : \lambda_\fkm(z) = \lambda\}$ and
 $C^\n_\lambda = \{ z \in \cK^\A_{n-1} : \lambda_\fkn(z) = \lambda\}$.



\begin{example}
%Given $z =z^{-1} \in S_n$ with fixed points $c_1<c_2<\dots < c_q$ in $[n]$, let $\underline z \in \cK(S_n)$
%be the element that maps $c_i \mapsto n + i$ and $a \mapsto z(a)$ for $a \in [n] \setminus \{c_1,c_2,\dots,c_q\}$.
%For example,
%\[ \underline{2134} = (1,2)(3,5)(4,6)(7,8) \in S_8.\]
Suppose $\lambda=(3,1)$. Then, using the notation $\underline w $ from Remark~\ref{notation-rmk},
we have
$
C^\m_\lambda = \{ \underline{2134}, \underline{3214}, \underline{4231}\}
$
and 
$
C^\n_\lambda = \{ \underline{2134}, \underline{3214}, \underline{1324}\}
$
since 
{\footnotesize
\[ \PMRSK(\underline{2134})=\PNRSK(\underline{1324}) = \ytab{1 &  3 & 4 \\ 2},
\quad
\PMRSK(\underline{3214})=\PNRSK(\underline{3124}) = \ytab{1 &  2 & 4 \\ 3},
\quad
\PMRSK(\underline{4231})=\PNRSK(\underline{2134}) = \ytab{1 &  2 & 3 \\ 4}.
\]}
\end{example}

A theorem of Beissinger implies the following interesting property of $\PMRSK(z)$:

\begin{proposition}[{\cite[Thm. 3.1]{Beissinger}}]
If $z=z^{-1} \in S_n$ is fixed-point-free, then
$\PMRSK(\underline z) = \PRSK(z)$.
\end{proposition}

Here is our last main theorem, along with a more precise conjecture:

\begin{theorem}
As $\lambda$ ranges over all partitions of $n$, 
the sets $C^\m_\lambda$ and $C^\n_\lambda$ are all nonempty and give the 
distinct molecules in $\Gamma^\m(S_n)$ and $\Gamma^\n(S_n)$, respectively.
\end{theorem}

% We can also now give a more precise conjecture:

\begin{conjecture}
If $\lambda$ is any partition of $n$ then the molecules
$C^\m_\lambda$
and
$C^\n_\lambda$ in the respective $S_n$-graphs $  \Gamma^\m(S_n) $ and $ \Gamma^\n(S_n)$ are cells, and the representations associated to these cells
are both isomorphic to the usual (irreducible) Specht module $\mathbb{S}_\lambda$ of
$\H(S_n)$.
\end{conjecture}

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\end{document}
