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%%%%% Auteur
\author{\firstname{Christos} \middlename{A.} \lastname{Athanasiadis}}
\address{Department of Mathematics\\
National and Kapodistrian University of Athens\\
Panepistimioupolis\\
Athens 15784, Greece}
\email{caath@math.uoa.gr}
%%%%% Sujet
\keywords{Rees product, poset homology, group action, Schur gamma-positivity, local face module}
\subjclass{05E05, 05E18, 05E45, 06A07}
%%%%% Gestion
%\DOI{10.5802/alco.85}
%\datereceived{2018-07-19}
%\daterevised{2019-04-30}
%\dateaccepted{2019-05-06}
%%%%% Titre et résumé
\title[Rees products of posets and equivariant gamma-positivity]
{Some applications of Rees products of posets to equivariant
gamma-positivity}
\begin{abstract}\
The Rees product of partially ordered sets was
introduced by Bj{\"o}rner and Welker. Using the theory
of lexicographic shellability, Linusson, Shareshian
and Wachs proved formulas, of significance in the theory
of gamma-positivity, for the dimension of the homology
of the Rees product of a graded poset $P$ with a certain
$t$-analogue of the chain of the same length as $P$.
Equivariant generalizations of these formulas are
proven in this paper, when a group of automorphisms
acts on $P$, and are applied to establish the
Schur gamma-positivity of certain symmetric functions
arising in algebraic and geometric combinatorics.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\maketitle
\section{Introduction}
\label{sec:intro}
The Rees product $P \ast Q$ of two partially ordered
sets (posets, for short) was introduced and studied by
Bj\"orner and Welker~\cite{BW05} as a combinatorial
analogue of the Rees construction in commutative
algebra (a precise definition of $P \ast Q$ can be
found in Section~\ref{sec:pre}). The connection of
the Rees product of posets to enumerative
combinatorics was hinted in~\cite[Section~5]{BW05},
where it was conjectured that the dimension of the
homology of the Rees product of the truncated
Boolean algebra $B_n \sm \{ \varnothing \}$ of rank
$n-1$ with an $n$-element chain equals the number of
permutations of $[n] \coloneqq \{1, 2,\ldots,n\}$ without
fixed points. This statement was generalized in
several ways in~\cite{SW09}, using enumerative and
representation theoretic methods, and
in~\cite{LSW12}, using the theory of lexicographic
shellability.
One of the results of~\cite{LSW12} proves formulas
\cite[Corollary~3.8]{LSW12} for the dimension of the
homology of the Rees product of an EL-shellable poset
$P$ with a contractible poset which generalizes
the chain of the same rank as $P$. This paper provides
an equivariant analogue of this result which seems
to have enough applications on its own to be of
independent interest. To state it, let $P$ be a
finite bounded poset, with minimum element $\hat{0}$
and maximum element $\hat{1}$, which is graded of
rank $n+1$, with rank function $\rho: P \to
\{0, 1,\ldots,n+1\}$ (for basic terminology on posets,
see~\cite[Chapter~3]{StaEC1}). Fix a field $\kk$ and
let $G$ be a finite group which
acts on $P$ by order preserving bijections. Then, $G$
defines a permutation representation $\alpha_P(S)$
over $\kk$ for every $S \subseteq [n]$, induced by
the action of $G$ on the set of maximal chains of
the rank-selected subposet
%
\begin{equation} \label{eq:P_S}
P_S = \{ x \in P: \rho(x) \in S \} \cup
\{\hat{0}, \hat{1} \}
\end{equation}
%
of $P$. Following Stanley's seminal work~\cite{Sta82},
we may consider the virtual $G$-representation
%
\begin{equation} \label{eq:beta(S)1}
\beta_P(S) = \sum_{T \subseteq S} (-1)^{|S-T|}
\, \alpha_P(T),
\end{equation}
%
defined equivalently by the equations
%
\begin{equation} \label{eq:beta(S)2}
\alpha_P(T) = \sum_{S \subseteq T} \beta_P(S)
\end{equation}
%
for $T \subseteq [n]$. The dimensions of $\alpha_P(S)$
and $\beta_P(S)$ are important enumerative invariants
of $P$, known as the entries of its flag $f$-vector and
flag $h$-vector, respectively. When $P$ is Cohen--Macaulay
over $\kk$, $\beta_P(S)$ is isomorphic to the
non-virtual $G$-representation $\widetilde{H}_{|S|-1}
(\bar{P}_S; \kk)$ induced on the top homology group of
$\bar{P}_S \coloneqq P_S \sm \{\hat{0}, \hat{1}\}$
\cite[Theorem~1.2]{Sta82}. As discussed and illustrated
in various situations in~\cite{Sta82}, the decomposition
of $\beta_P(S)$ as a direct sum of
irreducible $G$-representations often leads to very
interesting refinements of the flag $h$-vector of $P$.
As in references~\cite{LSW12, SW09}, we write
$\beta(\bar{P})$ in place of $\beta_P([n])$ and note,
as just mentioned, that this $G$-representation is
isomorphic to $\widetilde{H}_{n-1} (\bar{P}; \kk)$ if
$P$ is Cohen--Macaulay over $\kk$. We
denote by $T_{t,n}$ the poset whose Hasse diagram
is a complete $t$-ary tree of height $n$, rooted at
the minimum element. We denote by $P^-$, $P_{-}$
and $\bar{P}$ the poset obtained from $P$ by
removing its maximum element, or minimum element,
or both, respectively, and recall
from~\cite{SW09} (see also Section~\ref{sec:pre})
that the action of $G$ on $P$ induces actions on
$P^- \ast T_{t,n}$ and $\bar{P} \ast T_{t,n-1}$
as well. We also write $[a, b] \coloneqq \{a,
a+1,\ldots,b\}$ for integers $a \le b$ and denote
by $\stab (\Theta)$ the set of all subsets, called
stable, of $\Theta \subseteq \ZZ$ which do not
contain two consecutive integers. The following
result reduces to~\cite[Corollary~3.8]{LSW12},
proven in~\cite{LSW12} under additional
shellability assumptions on $P$, in the special
case of a trivial action.
\begin{theo} \label{thm:main}
Let $G$ be a finite group acting on a finite bounded
graded poset $P$ of rank $n+1$ by order preserving
bijections. Then,
\begin{multline} \label{eq:main1}
\beta ((P^- \ast T_{t,n})_{-})
\cong_G
\sum_{S \in \stab ([n-1])}
\beta_P ([n] \sm S) \ t^{|S|} (1+t)^{n-2|S|}
\\
+\sum_{S \in \stab ([n-2])} \beta_P
([n-1] \sm S) \ t^{|S|+1} (1+t)^{n-1-2|S|}
\end{multline}
and
\begin{multline} \label{eq:main2}
\beta (\bar{P} \ast T_{t,n-1})
\cong_G
\sum_{S \in \stab ([2, n-2])} \beta_P
([n-1] \sm S) \ t^{|S|+1} (1+t)^{n-2-2|S|}
\\
+\sum_{S \in \stab ([2, n-1])}
\beta_P ([n] \sm S) \ t^{|S|} (1+t)^{n-1-2|S|}
\end{multline}
for every positive integer $t$, where $\cong_G$
stands for isomorphism of $G$-representations.
If $P$ is Cohen--Macaulay over $\kk$, then the
left-hand sides of \eqref{eq:main1} and \eqref{eq:main2}
may be replaced by the $G$-representations
$\widetilde{H}_{n-1} ((P^- \ast T_{t,n})_{-}; \kk)$
and $\widetilde{H}_{n-1} (\bar{P} \ast T_{t,n-1};
\kk)$, respectively, and all representations which
appear in these formulas are non-virtual.
\end{theo}
Several applications of~\cite[Corollary~3.8]{LSW12}
to $\gamma$-positivity appear in~\cite[Section~3]{Ath14}~\cite{LSW12}
and are summarized in
\cite[Section~2.4]{Ath17}. Theorem~\ref{thm:main}
has non-trivial applications to Schur
$\gamma$-positivity, which we now briefly discuss.
A polynomial in $t$ with real coefficients is said
to be \emph{$\gamma$-positive} if for some $m \in
\NN$, it can be written as a nonnegative linear
combination of the binomials $t^i (1+t)^{m-2i}$ for
$0 \le i \le m/2$. Clearly, all such polynomials
have symmetric and unimodal coefficients. Two
symmetric function identities due to Gessel
(unpublished), stated without proof
in~\cite[Section~4]{LSW12}~\cite[Theorem~7.3]{SW10},
can be written in the form
%
\begin{equation} \label{eq:Ge2}
{\displaystyle
\frac{1-t} {E(\bx; tz) - tE(\bx; z)} =
1 + \sum_{n \ge 2} z^n
\sum_{k=0}^{\lfloor (n-2)/2 \rfloor}
\xi_{n,k}(\bx) \, t^{k+1} (1 + t)^{n-2k-2}}
\end{equation}
%
and
%
\begin{equation} \label{eq:Ge1}
{\displaystyle
\frac{(1-t) E(\bx; tz)} {E(\bx; tz) - tE(\bx; z)}
= 1 + \sum_{n \ge 1} z^n
\sum_{k = 0}^{\lfloor (n-1)/2 \rfloor}
\gamma_{n,k}(\bx) \, t^{k+1} (1 + t)^{n-1-2k}},
\end{equation}
where $E(\bx; z) = \sum_{n \ge 0} e_n(\bx) z^n$
is the generating function for the elementary
symmetric functions in $\bx = (x_1, x_2,\ldots)$
and the $\xi_{n,k}(\bx)$ and $\gamma_{n,k}(\bx)$
are Schur-positive symmetric functions, whose
coefficients in the Schur basis can be explicitly
described (see Corollary~\ref{cor:Gessel}). The
coefficients of $z^n$ in the right-hand sides of
Equations~\eqref{eq:Ge2} and~\eqref{eq:Ge1} are
Schur $\gamma$-positive symmetric functions, in
the sense that their coefficients in the Schur
basis are $\gamma$-positive polynomials in $t$
with the same center of symmetry. Their Schur
$\gamma$-positivity refines the $\gamma$-positivity
of derangement and Eulerian polynomials, respectively;
see~\cite[Section~2.5]{Ath17} and Corollary~\ref{cor:Gessel} for more information.
We will show (see Section~\ref{sec:app}) that
Gessel's identities
can in fact be derived from the special case of
Theorem~\ref{thm:main} in which $P^-$ is the
Boolean algebra $B_n$, endowed with the natural
symmetric group action. Moreover, applying the
theorem when $P^-$ is a natural signed analogue
of $B_n$, endowed with a hyperoctahedral group
action, we obtain new identities of the form
%
\begin{equation} \label{eq:Ath2+}
{\displaystyle \frac{E(\bx; tz) - t E(\bx; z)}
{E(\bx; tz) E(\by; tz) - tE(\bx; z) E(\by; z)}
= \sum_{n \ge 0} z^n \,
\sum_{k=0}^{\lfloor n/2 \rfloor}
\xi^+_{n,k}(\bx, \by) \, t^k (1 + t)^{n-2k}},
\end{equation}
%
\begin{equation} \label{eq:Ath2-}
{\displaystyle \frac{E(\bx; z) - E(\bx; tz)}
{E(\bx; tz) E(\by; tz) - tE(\bx; z) E(\by; z)}
= \sum_{n \ge 1} z^n
\sum_{k=0}^{\lfloor (n-1)/2 \rfloor}
\xi^{-}_{n,k}(\bx, \by) \, t^k (1 + t)^{n-1-2k}},
\end{equation}
%
\begin{equation} \label{eq:Ath1+}
{\displaystyle \frac{E(\bx; z) E(\bx; tz)
\left( E(\by; tz) - t E(\by; z) \right)}
{E(\bx; tz) E(\by; tz) - tE(\bx; z) E(\by; z)}
= \sum_{n \ge 0} z^n
\sum_{k=0}^{\lfloor n/2 \rfloor} \gamma^{+}_{n,k}
(\bx, \by) \, t^k (1 + t)^{n-2k}}
\end{equation}
%
and
%
\begin{equation} \label{eq:Ath1-}
{\displaystyle \frac{t E(\bx; z) E(\bx; tz)
\left( E(\by; z) - E(\by; tz) \right)}
{E(\bx; tz) E(\by; tz) - tE(\bx; z) E(\by; z)} \ = \
\sum_{n \ge 1} z^n \sum_{k=1}^{\lfloor (n+1)/2 \rfloor}
\gamma^{-}_{n,k}(\bx, \by) \, t^k (1 + t)^{n+1-2k}},
\end{equation}
where the $\xi^{\pm}_{n,k}(\bx, \by)$ and
$\gamma^{\pm}_{n,k}(\bx, \by)$ are Schur-positive
symmetric functions in the sets of variables $\bx =
(x_1, x_2,\ldots)$ and $\by = (y_1, y_2,\ldots)$
separately. Their Schur positivity refines
the $\gamma$-positivity of type $B$ analogues or
variants of derangement and Eulerian polynomials;
this is explained and
generalized in the sequel~\cite{Ath1x} to this
paper. Note that for $\bx = 0$, the left-hand side
of Equation~\eqref{eq:Ath2+} specializes to that
of~\eqref{eq:Ge2} (with $\bx$ replaced by $\by$)
and the sum of the left-hand sides of
Equations~\eqref{eq:Ath1+} and~\eqref{eq:Ath1-}
specializes to that of~\eqref{eq:Ge1} (again with
$\bx$ replaced by $\by$).
Various combinatorial and algebraic-geometric
interpretations of the left-hand sides of
Equations~\eqref{eq:Ge2} and~\eqref{eq:Ge1} are
discussed in~\cite[Section~4]{LSW12}~\cite[Section~7]{SW10}
~\cite{SW16b}. For instance,
by~\cite[Proposition~4.20]{Sta92}, the coefficient
of $z^n$ in the left-hand side of~\eqref{eq:Ge2}
can be interpreted as the Frobenius characteristic
of the symmetric group representation on the local
face module of the barycentric subdivision of the
$(n-1)$-dimensional simplex, twisted by the sign
representation. Thus, the Schur $\gamma$-positivity
of this coefficient, manifested by
Equation~\eqref{eq:Ge2}, is an instance of the
local equivariant Gal phenomenon, as discussed in
\cite[Section~5.2]{Ath17}. Section~\ref{sec:face}
shows that another instance of this phenomenon
follows from the specialization $\bx = \by$ of
Equation~\eqref{eq:Ath2+}. Similarly, setting
$\by=0$ to~\eqref{eq:Ath1+} yields another
identity, recently proven by Shareshian and Wachs
(see Proposition~3.3 and Theorem~3.4 in~\cite{SW17})
in order to establish the equivariant Gal phenomenon
for the symmetric group action on the $n$-dimensional
stellohedron and Section~\ref{sec:toric} combines
Equation~\eqref{eq:Ge2} with~\eqref{eq:Ath1+} to
establish the same phenomenon for the hyperoctahedral
group action on its associated Coxeter complex. Further
applications of Theorem~\ref{thm:main} are given
in~\cite{Ath1x}. It would be interesting to find
direct combinatorial proofs of
Equations~\eqref{eq:Ath2+}--\eqref{eq:Ath1-} and
to generalize other known interpretations of the
left-hand sides of Equations~\eqref{eq:Ge2}
and~\eqref{eq:Ge1} to those of~\eqref{eq:Ath2+}--\eqref{eq:Ath1-}.
\subsection*{Outline} The proof of Theorem~\ref{thm:main}
is given in Section~\ref{sec:proof}, after the
relevant background and definitions are explained in
Section~\ref{sec:pre}. This proof is fairly
elementary and different from that
of~\cite[Corollary~3.8]{LSW12}. Section~\ref{sec:app}
derives Equations~\eqref{eq:Ge2}--\eqref{eq:Ath1-}
from Theorem~\ref{thm:main} and provides explicit
combinatorial interpretations, in terms of standard
Young (bi)tableaux and their descents, for the
Schur-positive symmetric functions which appear
there. Sections~\ref{sec:face} and~\ref{sec:toric}
provide the promised applications of
Equations~\eqref{eq:Ath2+} and~\eqref{eq:Ath1+} to
the equivariant $\gamma$-positivity of the symmetric
group representation on the local face module of
a certain triangulation of the simplex and the
hyperoctahedral group representation on the cohomology
of the projective toric variety associated to the
Coxeter complex of type $B$.
\section{Preliminaries}
\label{sec:pre}
This section briefly records definitions and
background on posets, group representations and
(quasi)symmetric functions. For basic notions and
more information on these topics, the reader is
referred to the sources
\cite{Sa01,Sta82}~\cite[Chapter~7]{StaEC2}~\cite[Chapter~3]{StaEC1}~\cite{Wa07}. The
symmetric group of permutations of the set $[n]$
is denoted by $\fS_n$ and the cardinality of a
finite set $S$ by $|S|$.
\subsection{Group actions on posets and Rees products.}
All groups and posets considered here are assumed
to be finite. Homological notions for posets always
refer to those of their order complex;
see~\cite[Lecture~1]{Wa07}. A poset $P$ has the
structure of a \emph{$G$-poset}
if the group $G$ acts on $P$ by order preserving
bijections. Then, $G$ induces a representation on
every reduced homology group $\widetilde{H}_i (P;
\kk)$, for every field $\kk$.
Suppose that $P$ is a $G$-poset with minimum
element $\hat{0}$ and maximum element $\hat{1}$.
Sundaram~\cite{Su94} (see also
\cite[Theorem~4.4.1]{Wa07}) established the
isomorphism of $G$-representations
%
\begin{equation} \label{eq:sun}
\bigoplus_{k \ge 0} \ (-1)^k \bigoplus_{x \in P/G}
\widetilde{H}_{k-2} ((\hat{0}, x); \kk)
\uparrow^G_{G_x} \ \, \cong_G \ 0.
\end{equation}
Here $P/G$ stands for a complete set of $G$-orbit
representatives, $(\hat{0}, x)$ denotes the open
interval of elements of $P$ lying strictly between
$\hat{0}$ and $x$, $G_x$ is the stabilizer of
$x$ and $\uparrow$ denotes induction. Moreover,
$\widetilde{H}_{k-2} ((\hat{0}, x); \kk)$ is
understood to be the trivial representation
$1_{G_x}$ if $x = \hat{0}$ and $k=0$, or $x$
covers $\hat{0}$ and $k = 1$.
The \emph{Lefschetz character} of a finite
$G$-poset $P$ (over the field $\kk$) is defined
as the virtual $G$-representation
\[
L(P; G) \coloneqq \bigoplus_{k \ge 0} \, (-1)^k \,
\widetilde{H}_k (P; \kk). \]
%
Note that $L(P; G) = (-1)^r \widetilde{H}_r
(P; \kk)$, if $P$ is Cohen--Macaulay over $\kk$
of rank $r$.
Given finite graded posets $P$ and $Q$ with rank
functions $\rho_P$ and $\rho_Q$, respectively,
their \emph{Rees product} is defined
in~\cite{BW05} as $P \ast Q = \{ (p, q) \in P
\times Q: \rho_P (p) \ge \rho_Q (q) \}$, with
partial order defined by setting $(p_1, q_1)
\preceq (p_2, q_2)$ if all of the following
conditions are satisfied:
\begin{itemize}
%\itemsep=0pt
\item %[$\bullet$]
$p_1 \preceq p_2$ holds in $P$,
\item %[$\bullet$]
$q_1 \preceq q_2$ holds in $Q$ and
\item %[$\bullet$]
$\rho_P (p_2) - \rho_P (p_1) \ge \rho_Q
(q_2) - \rho_Q (q_1)$.
\end{itemize}
Equivalently, $(p_1, q_1)$ is covered by $(p_2, q_2)$
in $P \ast Q$ if and only if (a) $p_1$ is covered by
$p_2$ in $P$; and (b) either $q_1 = q_2$, or $q_1$ is
covered by $q_2$ in $Q$. We note that the Rees product
$P \ast Q$ is graded with rank function given by
$\rho(p,q) = \rho_P(p)$ for $(p,q) \in P \ast Q$, and
that if $P$ is a $G$-poset, then so is $P \ast Q$
with the $G$-action defined by setting $g \cdot (p,
q) = (g \cdot p, q)$ for $g \in G$ and $(p, q) \in
P \ast Q$.
A bounded graded $G$-poset $P$, with maximum element
$\hat{1}$, is said to be \emph{$G$-uniform}
\cite[Section~3]{SW09} if the following hold:
\begin{itemize}
%\itemsep=0pt
\item %[$\bullet$]
the intervals $[x, \hat{1}]$ and $[y, \hat{1}]$ in
$P$ are isomorphic for all $x, y \in P$ of the same
rank,
\item %[$\bullet$]
the stabilizers $G_x$ and $G_y$ are isomorphic for
all $x, y \in P$ of the same rank, and
\item %[$\bullet$]
there is an isomorphism between $[x, \hat{1}]$ and
$[y, \hat{1}]$ that intertwines the actions of $G_x$
and $G_y$, for all $x, y \in P$ of the same rank.
\end{itemize}
\newcommand{\Mkk}{\ensuremath{\mkern-0.45mu}}
G\Mkk{}i\Mkk{}v\Mkk{}e\Mkk{}n\Mkk{} a\Mkk{} s\Mkk{}e\Mkk{}q\Mkk{}u\Mkk{}e\Mkk{}n\Mkk{}c\Mkk{}e\Mkk{} o\Mkk{}f\Mkk{} g\Mkk{}r\Mkk{}o\Mkk{}u\Mkk{}p\Mkk{}s\Mkk{} $G \Mk =\Mk (G_0,
G_1,\ldots,G_n)$\Mkk{},\Mkk{} a\Mkk{} s\Mkk{}e\Mkk{}q\Mkk{}u\Mkk{}e\Mkk{}n\Mkk{}c\Mkk{}e\Mkk{} o\Mkk{}f\Mkk{} p\Mkk{}o\Mkk{}s\Mkk{}e\Mkk{}t\Mkk{}s\Mkk{}~$(P_0,
P_1,\ldots,P_n)$ is said to be \emph{$G$-uniform}
\cite[Section~3]{SW09} if
\begin{itemize}
%\itemsep=0pt
\item %[$\bullet$]
$P_k$ is $G_k$-uniform of rank $k$ for all $k$,
\item %[$\bullet$]
$G_k$ is isomorphic to the stabilizer $(G_n)_x$
for every $x \in P_n$ of rank $n-k$ and all $k$,
and
\item %[$\bullet$]
\looseness-1
there is an isomorphism between $P_k$ and the
interval $[x, \hat{1}]$ in $P_n$ that intertwines
the actions of $G_k$ and $(G_n)_x$ for every $x
\in P_n$ of rank $n-k$ and all~$k$.
\end{itemize}
Under these assumptions, Shareshian and Wachs
\cite[Proposition~3.7]{SW09} established the
isomorphism of $G$-representations
%
\begin{equation} \label{eq:uniform}
1_{G_n} \oplus \, \bigoplus_{k=0}^n \,
W_k(P_n;G_n) \, [k+1]_t \, L((P_{n-k} \ast
T_{t,n-k})_{-}; G_{n-k}) \uparrow^{G_n}_{G_{n-k}}
\ \, \cong_G \ 0
\end{equation}
for every positive integer $t$, where $W_k(P_n;G_n)$
is the number of $G_n$-orbits of elements of $P_n$
of rank $k$ and $[k+1]_t \coloneqq 1 + t + \cdots + t^k$.
\begin{exam} \label{ex:uniform}
The Boolean algebra $B_n$ consists of all subsets
of $[n]$, partially ordered by inclusion. When endowed
with the standard $\fS_n$-action, it becomes a
prototypical example of an $\fS_n$-uniform poset.
Every element $x \in B_n$ of rank $k$ is a set of
cardinality $k$; its stabilizer $(\fS_n)_x = \{w \in
\fS_n: w x = x\}$ is isomorphic to the Young
subgroup $\fS_k \times \fS_{n-k}$ of $\fS_n$, which
can be defined as the stalilizer of $[k]$. The sequence
$(B_0, B_1,\ldots,B_n)$ can easily be verified to be
$(G_0, G_1,\ldots,G_n)$-uniform for $G_k \coloneqq \fS_k
\times \fS_{n-k}$.
\end{exam}
\subsection{Permutations, Young tableaux and symmetric
functions.} Our notation concerning these topics
follows mostly that of~\cite{Sa01}~\cite[Chapter~7]{StaEC2}~\cite[Chapter~1]{StaEC1}.
In particular, the set of standard Young tableaux
of shape $\lambda$ is denoted by $\SYT(\lambda)$,
the descent set $\{ i \in [n-1]: w(i) > w(i+1) \}$
of a permutation $w \in \fS_n$ is denoted by $\Des
(w)$ and that of a tableau $Q \in
\SYT(\lambda)$, consisting of those entries $i$ for
which $i+1$ appears in $Q$ in a lower row than $i$,
is denoted by $\Des(Q)$. We recall that the
Robinson--Schensted correspondence is a
bijection from the
symmetric group $\fS_n$ to the set of pairs
$(\pP, Q)$ of standard Young tableaux of the same
shape and size $n$. This correspondence has the
property~\cite[Lemma~7.23.1]{StaEC2} that $\Des(w)
= \Des(Q(w))$ and $\Des(w^{-1}) = \Des(\pP(w))$,
where $(\pP(w), Q(w))$ is the pair of tableaux
associated to $w \in \fS_n$.
We will consider symmetric functions in the
indeterminates $\bx = (x_1, x_2,\ldots)$ over the
complex field $\CC$. We denote by $E(\bx; z) \coloneqq
\sum_{n \ge 0} e_n(\bx) z^n$ and $H(\bx; z) \coloneqq
\sum_{n \ge 0} h_n(\bx) z^n$ the generating
functions for the elementary and complete
homogeneous symmetric functions, respectively,
and recall the identity $E(\bx; z) H(\bx; -z) = 1$.
The (Frobenius) characteristic map, a $\CC$-linear
isomorphism of fundamental importance from the
space of virtual $\fS_n$-representations to that
of homogeneous symmetric functions of degree $n$,
will be denoted by $\ch$. This map sends the
irreducible $\fS_n$-representation corresponding
to $\lambda \vdash n$ to the Schur function
$s_\lambda(\bx)$ associated to $\lambda$ and thus
it sends non-virtual $\fS_n$-representations to
Schur-positive symmetric functions.
The \emph{fundamental quasisymmetric function}
associated to $S \subseteq [n-1]$ is defined as
\begin{equation} \label{eq:defF(x)}
F_{n, S} (\bx) \ =
\sum_{\substack{1 \le i_1 \le i_2 \le \cdots
\le i_n \\ j \in S \,\Rightarrow\,
i_j < i_{j+1}}}
x_{i_1} x_{i_2} \cdots x_{i_n}.
\end{equation}
%
The following well known expansion
\cite[Theorem~7.19.7]{StaEC2}
\begin{equation} \label{eq:sFexpan}
s_\lambda(\bx) = \sum_{Q \in \SYT (\lambda)}
F_{n, \Des(Q)} (\bx)
\end{equation}
%
will be used in Section~\ref{sec:app}.
For the applications given there,
we need the analogues of these concepts in the
representation theory of the hyperoctahedral group
of signed permutations of the set $[n]$, denoted
here by $\bB_n$. We will keep this discussion rather
brief and refer to~\cite[Section~2]{AAER17} for
more information.
A \emph{bipartition} of a positive integer $n$,
written $(\lambda, \mu) \vdash n$, is any pair
$(\lambda, \mu)$ of integer partitions of total
sum $n$. A \emph{standard Young bitableau} of
shape $(\lambda, \mu) \vdash n$ is any pair $Q
= (Q^+, Q^{-})$ of Young tableaux such that
$Q^+$ has shape $\lambda$, $Q^{-}$ has shape
$\mu$ and every element of $[n]$ appears
exactly once as an entry of $Q^+$ or $Q^{-}$.
The tableaux $Q^+$ and $Q^{-}$ are called the
\emph{parts} of $Q$ and the number $n$ is its
\emph{size}. The Robinson--Schensted
correspondence of type $B$, as described
in~\cite[Section~6]{Sta82} (see
also~\cite[Section~5]{AAER17}), is a
bijection from the group $\bB_n$ to the set
of pairs $(\pP, Q)$ of standard Young bitableaux
of the same shape and size $n$.
The (Frobenius) characteristic map for the
hyperoctahedral group, denoted by $\ch_\bB$,
is a $\CC$-linear isomorphism from the space of
virtual $\bB_n$-representations to that of
homogeneous symmetric functions of degree $n$ in
the sets of indeterminates $\bx = (x_1, x_2,\ldots)$
and $\by = (y_1, y_2,\ldots)$ separately; see, for
instance,~\cite[Section~5]{Ste92}. The map $\ch_\bB$
sends the irreducible $\bB_n$-representation
corresponding to $(\lambda, \mu) \vdash n$ to the
function $s_\lambda(\bx) s_\mu(\by)$ and thus
it sends non-virtual $\bB_n$-representations to
Schur-positive functions, meaning nonnegative
integer linear combinations of the functions
$s_\lambda(\bx) s_\mu(\by)$. The following basic
properties of $\ch_\bB$ will be useful in
Section~\ref{sec:app}:
%\smallskip
\begin{itemize}
%\itemsep=1pt
\item %[$\bullet$]
$\ch_\bB(1_{\bB_n}) = h_n(\bx)$, where $1_{\bB_n}$
is the trivial $\bB_n$-representation,
\item %[$\bullet$]
$\ch_\bB (\sigma \otimes \tau
\uparrow^{\bB_n}_{\bB_k \times \bB_{n-k}}) =
\ch_\bB(\sigma) \cdot \ch_\bB (\tau)$ for all
representations $\sigma$ and $\tau$ of $\bB_k$
and $\bB_{n-k}$, respectively, where $\bB_k
\times \bB_{n-k}$ is the Young subgroup of $\bB_n$
consisting of all signed permutations which preserve
the set $\{ \pm 1, \pm 2,\ldots,\pm k\}$,
\item %[$\bullet$]
$\ch_\bB (\uparrow^{\bB_n}_{\fS_n} \rho) = \ch
(\rho) (\bx, \by)$ for every $\fS_n$-representation
$\rho$, where $\fS_n \subset \bB_n$ is the standard
embedding.
\end{itemize}
We denote by $E(\bx, \by; z) \coloneqq \sum_{n \ge 0} e_n
(\bx, \by) z^n$ and $H(\bx, \by; z) \coloneqq \sum_{n \ge 0}
h_n(\bx, \by) z^n$ the generating function for the
elementary and complete homogeneous, respectively,
symmetric functions in the variables $(\bx, \by) =
(x_1, x_2,\ldots,y_1, y_2,\ldots)$ and note that
$E(\bx, \by; z) = E(\bx; z) E(\by; z)$, since $e_n
(\bx, \by) = \sum_{k=0}^n e_k(\bx) e_{n-k}(\by)$,
and similarly that $H(\bx, \by; z) = H(\bx; z)
H(\by; z)$.
The \emph{signed descent set}
\cite[Section~2]{AAER17}~\cite{Poi98} of $w \in
\bB_n$, denoted $\sDes(w)$, records the positions
of the increasing (in absolute value) runs of
constant sign in the sequence $(w(1),
w(2),\ldots,w(n))$. Formally, we may define
$\sDes(w)$ as the pair $(\Des(w), \varepsilon)$,
where $\varepsilon = (\varepsilon_1,
\varepsilon_2,\ldots,\varepsilon_n) \in \{-, +\}^n$
is the sign vector with $i$th coordinate equal to
the sign of $w(i)$ and $\Des(w)$ consists of the
indices $i \in [n-1]$ for which either $\varepsilon_i
= +$ and $\varepsilon_{i+1} = -$, or else
$\varepsilon_i = \varepsilon_{i+1}$ and $|w(i)| >
|w(i+1)|$ (this definition is slightly different
from, but equivalent to, the ones given in
\cite{AAER17,Poi98}). The fundamental
quasisymmetric function associated to $w$,
introduced by Poirier~\cite{Poi98} in a more
general setting, is defined as
\begin{equation} \label{eq:defF(xy)}
F_{\sDes(w)} (\bx, \by) \ =
\sum_{\substack{i_1 \le i_2 \le \cdots
\le i_n \\ j \in \Des(w) \,
\Rightarrow \, i_j < i_{j+1}}}
z_{i_1} z_{i_2} \cdots z_{i_n},
\end{equation}
where $z_{i_j} = x_{i_j}$ if $\varepsilon_j = +$, and
$z_{i_j} = y_{i_j}$ if $\varepsilon_j = -$. Given a
standard Young bitableau $Q$ of size $n$, one defines
the signed descent set $\sDes(Q)$ as the pair
$(\Des(Q), \varepsilon)$, where $\varepsilon =
(\varepsilon_1, \varepsilon_2,\ldots,\varepsilon_n)
\in \{-, +\}^n$ is the sign vector with $i$th
coordinate equal to the sign of the part of $Q$ in
which $i$ appears and $\Des(Q)$ is the set of indices
$i \in [n-1]$ for which either $\varepsilon_i = +$
and $\varepsilon_{i+1} = -$, or else $\varepsilon_i
= \varepsilon_{i+1}$ and $i+1$ appears in $Q$ in a
lower row than $i$. The function $F_{s\Des(Q)} (\bx,
\by)$ is then defined by the right-hand side of
Equation~\eqref{eq:defF(xy)}, with $w$ replaced by
$Q$; see~\cite[Section~2]{AAER17}. The analogue
\begin{equation} \label{eq:sxyFexpan}
s_\lambda(\bx) s_\mu(\by) \ = \sum_{Q \in \SYT
(\lambda,\mu)} F_{s\Des(Q)} (\bx, \by)
\end{equation}
of the expansion~\eqref{eq:sFexpan} holds
(\cite[Proposition~4.2]{AAER17}) and the
Robinson--Schensted correspondence of type $B$ has
the properties that $\sDes(w) = \sDes(Q^B(w))$ and
$\sDes(w^{-1}) = \sDes(\pP^B(w))$, where $(\pP^B(w),
Q^B(w))$ is the pair of bitableaux associated to $w
\in \bB_n$; see~\cite[Proposition~5.1]{AAER17}.
\section{Proof of Theorem~\ref{thm:main}}
\label{sec:proof}
This section proves Theorem~\ref{thm:main} using
only the definition of Rees product and the defining
equation~\eqref{eq:beta(S)1} of the representations
$\beta_P(S)$. For $S = \{s_1, s_2,\ldots,s_k\}
\subseteq [n]$ with $s_1 < s_2 < \cdots < s_k$ we
set
\begin{align*}
\varphi_t(S) & \coloneqq [s_1 + 1]_t \, [s_2 - s_1 + 1]_t
\cdots [s_k - s_{k-1} + 1]_t \\
\psi_t(S) & \coloneqq [s_1]_t \, [s_2 - s_1 + 1]_t \cdots
[s_k - s_{k-1} + 1]_t
\end{align*}
where, as mentioned already, $[m]_t = 1 + t + \cdots +
t^{m-1}$ for positive integers $m$.
\begin{lemm} \label{lem:alpha}
Let $G$ be a finite group, $P$ be a finite bounded
graded $G$-poset of rank $n+1$, as in
Theorem~\ref{thm:main}, and $Q, R$ be the posets
defined by $\bar{Q} = (P^- \ast T_{t,n})_{-}$ and
$\bar{R} = \bar{P} \ast T_{t,n-1}$. Then,
\begin{align*}
\alpha_Q (S) & \cong_G \varphi_t(S) \,
\alpha_P (S) \\
\alpha_R (S) & \cong_G \psi_t(S) \,
\alpha_P (S)
\end{align*}
for all positive integers $t$ and $S \subseteq [n]$.
\end{lemm}
\begin{proof}
Let $S = \{s_1, s_2,\ldots,s_k\} \subseteq [n]$ with
$s_1 < s_2 < \cdots < s_k$ and let $\rho: T_{t,n}
\to \NN$ be the rank function of $T_{t,n}$. By the
definition of Rees product, the maximal chains in
$Q_S$ have the form
%
\begin{equation} \label{Qchain}
\hat{0} \prec (p_1, \tau_1) \prec
(p_2, \tau_2) \prec \cdots \prec
(p_k, \tau_k) \prec \hat{1}
\end{equation}
%
where $\hat{0} \prec p_1 \prec p_2 \prec \cdots
\prec p_k \prec \hat{1}$ is a maximal chain in
$P_S$ and $\tau_1 \preceq \tau_2 \preceq \cdots
\preceq \tau_k$ is a multichain in $T_{t,n}$ such
that
\begin{itemize}
%\itemsep=0pt
\item %[$\bullet$]
$0 \le \rho(\tau_1) \le s_1$ and
\item %[$\bullet$]
$\rho(\tau_j) - \rho(\tau_{j-1}) \le s_j - s_{j-1}$
for $2 \le j \le k$.
\end{itemize}
Let $m_t(S)$ be the number of these multichains.
Since the elements of $G$ act on~\eqref{Qchain} by
fixing the $\tau_j$ and acting on the corresponding
maximal chain of $P_S$, we have $\alpha_Q(S)
\cong_G m_t(S) \, \alpha_P(S)$. To choose such a
multichain $\tau_1 \preceq \tau_2 \preceq \cdots
\preceq \tau_k$, we need to specify the sequence
$i_1 \le i_2 \le \cdots \le i_k$ of ranks of its
elements so that $i_j - i_{j-1} \le s_j - s_{j-1}$
for $1 \le j \le k$, where $i_0 \coloneqq s_0 \coloneqq 0$, and
choose its maximum element $\tau_k$ in $t^{i_k}$
possible ways. Summing over all such sequences,
we get
\[
m_t (S) = \sum_{(i_1, i_2,\ldots,i_k)}
t^{i_k} = \sum_{0 \le a_j \le s_j -
s_{j-1}} t^{a_1 + a_2 + \cdots + a_k} =
\varphi_t (S)
\]
and the result for $\alpha_Q(S)$ follows. The same
argument applies to $\alpha_R(S)$; one simply has to
switch the condition for the rank of $\tau_1$ to $0
\le \rho(\tau_1) \le s_1 - 1$.
\end{proof}
%\medskip
The proof of the following technical lemma will be
given after that of Theorem~\ref{thm:main}.
\begin{lemm} \label{lem:phipsi}
For every $S \subseteq [n]$ we have
%
\begin{equation} \label{eq:chi1}
\sum_{S \subseteq T \subseteq [n]} (-1)^{n-|T|}
\, \varphi_t(T) = \begin{cases}
0, & \text{if $[n] \sm S$ is not stable}, \\
t^r(1+t)^{n-2r}, & \text{if $[n] \sm S$ is
stable and $n \in S$}, \\
t^r(1+t)^{n+1-2r}, & \text{if $[n] \sm S$ is
stable and $n \not\in S$}
\end{cases}
\end{equation}
and
%
\begin{equation} \label{eq:chi2}
\sum_{S \subseteq T \subseteq [n]} (-1)^{n-|T|}
\, \psi_t(T) = \begin{cases}
0, & \text{if $1 \not\in S$}, \\
0, & \text{if $[n] \sm S$ is not stable}, \\
t^r(1+t)^{n-1-2r}, & \text{if $[n] \sm S$ is
stable and $1, n \in S$}, \\
t^r(1+t)^{n-2r}, & \text{if $[n] \sm S$ is
stable, $1 \in S$ and $n \not\in S$},
\end{cases}
\end{equation}
where $r = n - |S|$.
\end{lemm}
\begin{proof}[Proof of Theorem~\ref{thm:main}] Using
Equations~\eqref{eq:beta(S)1} and~\eqref{eq:beta(S)2},
as well as Lemma~\ref{lem:alpha}, we compute that
\begin{align*}
\beta_Q ([n]) & = \sum_{T \subseteq [n]}
(-1)^{n-|T|} \, \alpha_Q (T) \ \cong_G \
\sum_{T \subseteq [n]} (-1)^{n-|T|} \,
\varphi_t(T) \, \alpha_P (T) \\
& = \sum_{T \subseteq [n]} (-1)^{n-|T|} \,
\varphi_t(T) \, \sum_{S \subseteq T} \beta_P(S)
\\
& = \sum_{S \subseteq [n]} \beta_P(S)
\sum_{S \subseteq T \subseteq [n]} (-1)^{n-|T|}
\, \varphi_t(T)
\end{align*}
and find similarly that
%
\[ \beta_R ([n]) \ \cong_G \ \sum_{S \subseteq [n]}
\beta_P(S) \sum_{S \subseteq T \subseteq [n]}
(-1)^{n-|T|} \, \psi_t(T).
\]
The proof follows from these formulas and
Lemma~\ref{lem:phipsi}.
For the last statement of the theorem one has to note
that, as a consequence of~\cite[Corollary~2]{BW05},
if $P$ is Cohen--Macaulay over $\kk$, then so are the
Rees products $P^{-} \ast T_{t,n}$ and $\bar{P} \ast
T_{t,n-1}$.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:phipsi}] Let us denote
by $\chi_t(S)$ the left-hand side of~\eqref{eq:chi1}
and write $S = \{s_1, s_2,\ldots,s_k\}$, with $1 \le
s_1 < s_2 < \cdots < s_k \le n$. By definition, we
have
%
\begin{equation} \label{eq:chi}
\chi_t(S) = \chi_t(s_1) \, \chi_t(s_2 - s_1)
\cdots \chi_t(s_k - s_{k-1}) \, \omega_t(n - s_k),
\end{equation}
%
where
\begin{align*}
\chi_t(n) & \coloneqq \sum_{n \in T \subseteq [n]}
(-1)^{n-|T|} \, \varphi_t(T) \\
\omega_t(n) & \coloneqq \sum_{T \subseteq [n]}
(-1)^{n-|T|} \, \varphi_t(T)
\end{align*}
for $n \ge 1$ and $\omega_t(0) \coloneqq 1$. We claim that
\[
\chi_t(n) = \begin{cases}
1+t, & \text{if $n = 1$}, \\
t, & \text{if $n = 2$}, \\
0, & \text{if $n \ge 3$} \end{cases}
\qquad \text{and} \qquad
\omega_t(n) = \begin{cases}
1, & \text{if $n = 0$}, \\
t, & \text{if $n = 1$}, \\
0, & \text{if $n \ge 2$}.
\end{cases}
\]
Equation~\eqref{eq:chi1} is a direct consequence
of~\eqref{eq:chi} and this claim. To verify the
claim, note that the defining equation for
$\chi_t(n)$ can be rewritten as
%
\[ \chi_t(n) = \sum_{(a_1, a_2,\ldots,a_k)
\vDash n} (-1)^{n-k} \, [a_1 + 1]_t \, [a_2 + 1]_t
\cdots [a_k + 1]_t, \]
%
where the sum ranges over all sequences
(compositions) $(a_1, a_2,\ldots,a_k)$ of positive
integers summing to $n$. We leave it as a simple
combinatorial exercise for the interested reader
to show (for instance, by standard generating
function methods) that $\chi_t(n) = 0$ for
$n \ge 3$. The claim follows from this fact and
the obvious recurrence $\omega_t(n) = \chi_t(n) -
\omega_t(n-1)$, valid for $n \ge 1$.
Finally, note that Equation~\eqref{eq:chi2} is
equivalent to~\eqref{eq:chi1} in the case $1 \in
S$. Otherwise, the terms in the left-hand side
can be partitioned into pairs of terms,
corresponding to pairs $\{T, T \cup \{1\}\}$ of
subsets with $1 \not\in T$, canceling each other.
This shows that the left-hand side vanishes.
\end{proof}
\section{Symmetric function identities}
\label{sec:app}
This section derives Equations~\eqref{eq:Ge2}--\eqref{eq:Ath1-} from Theorem~\ref{thm:main}
(Corollaries~\ref{cor:Gessel},~\ref{cor:Ath2}
and~\ref{cor:Ath1}) and interprets combinatorially
the Schur-positive symmetric functions which
appear there. We first explain why Gessel's
identities are special cases of this theorem.
The set of ascents of a permutation $w \in
\fS_n$ is defined as $\Asc(w) \coloneqq [n-1] \sm
\Des(w)$ and, similarly, we have $\Asc(\pP)
\coloneqq [n-1] \sm \Des(\pP)$ for every standard
Young tableau $\pP$ of size $n$. Let us recall
the fact (used in the following proof) that
the reduced homology groups of a poset with a
minimum or maximum element vanish.
\begin{coro} \label{cor:Gessel}
Equations~\eqref{eq:Ge2} and~\eqref{eq:Ge1} are
valid for the functions
\begin{equation} \label{eq:corGes1}
\xi_{n,k} (\bx) = \sum_{\lambda \vdash n}
c_{\lambda,k} \cdot s_\lambda (\bx) =
\sum_{w} F_{n,\Des(w)}(\bx)
\end{equation}
and
\begin{equation} \label{eq:corGes2}
\gamma_{n,k} (\bx) = \sum_{\lambda \vdash n}
d_{\lambda,k} \cdot s_\lambda (\bx) =
\sum_{w} F_{n,\Des(w)}(\bx),
\end{equation}
where $c_{\lambda,k}$ \textup{(}respectively,
$d_{\lambda,k}$\textup{)} stands for the number of
tableaux $\pP \in \SYT(\lambda)$ for which $\Asc(\pP)
\in \stab([2,n-2])$ \textup{(}respectively,
$\Asc(\pP) \in \stab([n-2])$\textup{)} has $k$
elements and, similarly, $w \in \fS_n$ runs through
all permutations for which $\Asc(w^{-1}) \in
\stab([2,n-2])$ \textup{(}respectively, $\Asc(w^{-1})
\in \stab([n-2])$\textup{)} has $k$ elements.
\end{coro}
\begin{proof}
We will apply Theorem~\ref{thm:main} when $P^-$ is the
Boolean lattice $B_n$, considered as an $\fS_n$-poset
as in Example~\ref{ex:uniform}. On the one hand, we have
the equality
\[
1 + \sum_{n \ge 2} \,
\ch \left( \widetilde{H}_{n-1}
((B_n \sm \{\varnothing\}) \ast T_{t,n-1};
\CC) \right) z^n
= \frac{1-t} {E(\bx; tz) - tE(\bx; z)}
\]
which, although not explicitly stated in~\cite{SW09},
follows as in the proof of its special case $t=1$
\cite[Corollary~5.2]{SW09}. On the other hand,
since $B_n$ has a maximum element, the second summand
in the right-hand side of Equation~\eqref{eq:main2}
vanishes and hence this equation gives
\begin{multline*}
\ch \left( \widetilde{H}_{n-1}
((B_n \sm \{\varnothing\}) \ast T_{t,n-1}; \CC)
\right)
\\
= \sum_{S \in \stab ([2,n-2])}
\ch \left( \beta_{B_n} ([n-1] \sm S) \right)
t^{|S|+1} \, (1+t)^{n-2-2|S|}
\end{multline*}
for $n \ge 2$. The representations $\beta_{B_n} (S)$
for $S \subseteq [n-1]$ are known to satisfy (see,
for instance,~\cite[Theorem~4.3]{Sta82})
%
\[ \ch \left( \beta_{B_n} (S) \right) =
\sum_{\lambda \vdash n} c_{\lambda,S} \cdot
s_\lambda (\bx),
\]
where $c_{\lambda,S}$ is the number of standard
Young tableaux of shape $\lambda$ and descent set
equal to $S$. Combining the previous three
equalities yields the first equality in
Equation~\eqref{eq:corGes1}. The second equality
follows from the first by expanding $s_\lambda(\bx)$
according to Equation~\eqref{eq:sFexpan} to get
%
\[ \xi_{n,k} (\bx) = \sum_{\lambda \vdash n}
\sum_\pP \sum_{Q \in \SYT (\lambda)}
F_{n, \Des(Q)} (\bx)
\]
where, in the inner sum, $\pP$ runs through all
tableaux in $\SYT(\lambda)$ for which $\Asc(\pP) \in
\pP_{\stab}([2,n-2])$ has $k$ elements, and then using
the Robinson--Schensted correspondence and its
standard properties $\Des(w) = \Des(Q(w))$ and
$\Des(w^{-1}) = \Des(\pP(w))$ to replace the
summations with one running over elements of $\fS_n$,
as in the statement of the corollary.
The proof of~\eqref{eq:corGes2} is entirely
similar; one has to use Equation~\eqref{eq:main1}
instead of~\eqref{eq:main2}, as well as the equality
%
\[ 1 + \sum_{n \ge 1} \,
\ch \left( \widetilde{H}_{n-1}
((B_n \ast T_{t,n})_{-}; \CC) \right) z^n =
\frac{(1-t) E(\bx; tz)} {E(\bx; tz) - tE(\bx; z)}.
\]
The latter follows from the proof of Equation~(3.3)
in~\cite[pp.~15--16]{SW09}, where the left-hand side
is equal to $-F_t(-z)$, in the notation used in that
proof.
\end{proof}
\begin{exam}
The coefficient of $z^4$ in the left-hand sides
of Equations~\eqref{eq:Ge2} and~\eqref{eq:Ge1}
equals
%\smallskip
\begin{itemize}
%\itemsep=3pt
\item %[$\bullet$]
$e_4(\bx) \, (t + t^2 + t^3) + e_2(\bx)^2 \, t^2$,
and
\item %[$\bullet$]
$e_4(\bx) \, (t + t^2 + t^3 + t^4) + e_1(\bx)
e_3(\bx) \, (t^2 + t^3) + e_2(\bx)^2 \, (t^2 + t^3)$,
\end{itemize}
%\noindent
respectively. These expressions may be rewritten as
%\smallskip
\begin{itemize}
%\itemsep=3pt
\item %[$\bullet$]
$s_{(1,1,1,1)} (\bx) \, t(1+t)^2 + s_{(2,1,1)} (\bx)
\, t^2 + s_{(2,2)} (\bx) \, t^2$, and
\item %[$\bullet$]
$s_{(1,1,1,1)} (\bx) \, t(1+t)^3 + 2 s_{(2,1,1)}
(\bx) \, t^2(1+t) + s_{(2,2)} (\bx) \, t^2(1+t)$,
\end{itemize}
%\smallskip
%\noindent
respectively. Hence, $\xi_{4,0} (\bx) =
s_{(1,1,1,1)} (\bx)$, $\xi_{4,1} (\bx) = s_{(2,1,1)}
(\bx) + s_{(2,2)} (\bx)$, $\gamma_{4,0} (\bx) =
s_{(1,1,1,1)} (\bx)$ and $\gamma_{4,1} (\bx) = 2
s_{(2,1,1)} (\bx) + s_{(2,2)} (\bx)$. We leave it to
the reader to verify that these formulas agree with
Corollary~\ref{cor:Gessel}.
\end{exam}
We now focus on the identities~\eqref{eq:Ath2+}--\eqref{eq:Ath1-}. We will apply
Theorem~\ref{thm:main} to the collection $sB_n$
of all subsets of $\{1, 2,\ldots,n\} \cup \{ -1,
-2,\ldots,-n\}$ which do not contain $\{i, -i\}$
for any index $i$, partially ordered by
inclusion. This signed analogue of the Boolean
algebra $B_n$ is a graded poset of rank $n$,
having the empty set as its minimum element, on
which the hyperoctahedral group $\bB_n$ acts in
the obvious way~\cite[Section~6]{Sta82}, turning
it into a $\bB_n$-poset. It is isomorphic to the
poset of faces (including the empty one) of the
boundary complex of the $n$-dimensional
cross-polytope and hence it is Cohen--Macaulay
over $\ZZ$ and any field. The left-hand sides of
Equations~\eqref{eq:main1} and~\eqref{eq:main2}
for $P^{-} = sB_n$ will be computed using the
methods of~\cite{SW09}.
Consider the $n$-element chain $C_n = \{0,
1,\ldots,n-1\}$, with the usual total order.
Following~\cite{SW09}, we denote by $I_j(B_n)$ the
order ideal of elements of $(B_n \sm \{\varnothing\})
\ast C_n$ which are strictly less than $([n], j)$.
Then $I_j(B_n)$ is an $\fS_n$-poset for every $j \in
C_n$ and one of the main results of~\cite{SW09} (see
\cite[p.~21]{SW09}~\cite[Equation~(2.5)]{SW16b})
states that
\begin{equation} \label{eq:Ij(Bn)}
1 + \sum_{n \ge 1} \, z^n \,
\sum_{j=0}^{n-1} \, t^j \, \ch \left(
\widetilde{H}_{n-2} (I_j(B_n); \CC) \right)
= \frac{(1-t) E(\bx; z)}{E(\bx; tz) - tE(\bx; z)}.
\end{equation}
\begin{prop} \label{prop:Ath2}
For the $\bB_n$-poset $sB_n$ we have
\begin{multline} \label{eq:prop2}
1 + \sum_{n \ge 1} \, \ch_\bB
\left( \widetilde{H}_{n-1} ((sB_n \sm \{\varnothing\})
\ast T_{t,n-1}; \CC) \right) z^n
\\
= \frac{(1-t) E(\by; z)}
{E(\bx; tz) E(\by; tz) - tE(\bx; z) E(\by; z)}.
\end{multline}
\end{prop}
\begin{proof}
Following the reasoning in the proof
of~\cite[Corollary~5.2]{SW09}, we apply~\eqref{eq:sun}
to the Cohen--Macaulay $\bB_n$-poset obtained from
$(sB_n \sm \{\varnothing\}) \ast T_{t,n-1}$ by adding
a minimum and a maximum element. For $0 \le j < k \le
n$, there are exactly $t^j$ $\bB_n$-orbits of elements
$x$ of rank $k$ in this poset with second coordinate
of rank $j$ in $T_{t,n-1}$ and for each one of these,
the open interval $(\hat{0}, x)$ is isomorphic to $I_j(B_k)$
and the stabilizer of $x$ is isomorphic to $\fS_k
\times \bB_{n-k}$. We conclude that
%
\begin{multline*}
\widetilde{H}_{n-1} ((sB_n \sm \{\varnothing\})
\ast T_{t,n-1}; \CC)
\\
\cong_{\bB_n} \
\bigoplus_{k=0}^n \, (-1)^{n-k} \,
\bigoplus_{j=0}^{k-1} \, t^j \left(
\widetilde{H}_{k-2} (I_j(B_k); \CC) \otimes
1_{\bB_{n-k}} \right)
\uparrow^{\bB_n}_{\fS_k \times \bB_{n-k}}.
\end{multline*}
Applying the map $\ch_\bB$ and
using the transitivity $\uparrow^{\bB_n}_{\fS_k
\times \bB_{n-k}} \, \cong_{\bB_n} \, \uparrow^{\bB_k
\times \bB_{n-k}}_{\fS_k \times \bB_{n-k}} \
\uparrow^{\bB_n}_{\bB_k \times \bB_{n-k}}$ of
induction and properties of $\ch_\bB$ discussed in
Section~\ref{sec:pre}, the right-hand side becomes
%
\[ \sum_{k=0}^n \, (-1)^{n-k} \,
\sum_{j=0}^{k-1} \, t^j \, \ch \left(
\widetilde{H}_{k-2} (I_j(B_k); \CC) \right)
(\bx, \by) \cdot h_{n-k}(\bx).
\]
Thus, the left-hand side of Equation~\eqref{eq:prop2}
is equal to
%
\[ H(\bx; -z) \cdot \left( 1 + \sum_{n \ge 1}
\, z^n \, \sum_{j=0}^{n-1} \, t^j \, \ch \left(
\widetilde{H}_{n-2} (I_j(B_n); \CC) \right)
(\bx, \by) \right)
\]
and the result follows from Equation~\eqref{eq:Ij(Bn)}
and the identities $E(\bx; z) H(\bx; -z) = 1$ and
$E(\bx, \by; z) = E(\bx; z) E(\by; z)$.
\end{proof}
Recall the definition of the sets $\Des(w)$ and
$\Des(\pP)$ for a signed permutation $w \in \bB_n$
and standard Young bitableau $\pP$ of size $n$,
respectively, from Section~\ref{sec:pre}. Following
\cite[Section~6]{Sta82}, we define the \emph{type
$B$ descent set} of $\pP = (\pP^+, \pP^{-})$ as
$\Des_B(\pP) = \Des(\pP) \cup \{n\}$, if $n$ appears
in $\pP^+$, and $\Des_B(\pP) = \Des(\pP)$ otherwise.
The complement of $\Des_B(\pP)$ in the set $[n]$ is
called the \emph{type $B$ ascent set} of $\pP$ and
is denoted by $\Asc_B(\pP)$. Similarly, we define
the \emph{type $B$ descent set} of $w \in \bB_n$
as $\Des_B(w) = \Des(w) \cup \{n\}$, if $w(n)$ is
positive, and $\Des_B(w) = \Des(w)$ otherwise. The
complement of $\Des_B(w)$ in the set $[n]$ is
called the \emph{type $B$ ascent set} of $w$ and
is denoted by $\Asc_B(w)$. The sets $\Des_B(w)$ and
$\Des_B(\pP)$ depend only on the signed descent sets
$\sDes_B(w)$ and $\sDes_B(\pP)$, respectively, and
\cite[Proposition~5.1]{AAER17}, mentioned at the
end of Section~\ref{sec:pre}, implies that $\Des_B(w)
= \Des_B(Q^B(w))$ and $\Des_B(w^{-1}) = \Des_B
(\pP^B(w))$ for every $w \in \bB_n$.
\begin{coro} \label{cor:Ath2}
Equations~\eqref{eq:Ath2+} and~\eqref{eq:Ath2-} are
valid for the functions
\begin{equation} \label{eq:corAth2+}
\xi^+_{n,k} (\bx, \by) =
\sum_{(\lambda, \mu) \vdash n}
c^+_{(\lambda,\mu),k} \cdot s_\lambda (\bx) s_\mu
(\by) = \sum_{w} F_{\sDes(w)}(\bx, \by)
\end{equation}
and
\begin{equation} \label{eq:corAth2-}
\xi^{-}_{n,k} (\bx, \by) =
\sum_{(\lambda, \mu) \vdash n}
c^{-}_{(\lambda,\mu),k} \cdot s_\lambda (\bx) s_\mu
(\by) = \sum_{w} F_{\sDes(w)}(\bx, \by),
\end{equation}
where $c^+_{(\lambda,\mu),k}$ \textup{(}respectively,
$c^{-}_{(\lambda,\mu),k}$\textup{)} stands for the number
of bitableaux $\pP \in \SYT(\lambda,\mu)$ for which
$\Asc_B(\pP) \in \stab([2,n])$ has $k$ elements and
contains \textup{(}respectively, does not
contain\textup{)} $n$ and where, similarly, $w \in
\bB_n$ runs through all signed permutations for which
$\Asc_B(w^{-1}) \in \stab([2,n])$ has $k$
elements and contains \textup{(}respectively, does not
contain\textup{)} $n$.
\end{coro}
\begin{proof}
We apply the second part of Theorem~\ref{thm:main}
for $P^- = sB_n$, thought of as a $\bB_n$-poset.
The representations $\beta_P (S)$ for $S \subseteq
[n]$ were computed in this case
in~\cite[Theorem~6.4]{Sta82}, which implies that
\[
\ch_\bB \left( \beta_{B_n} (S) \right) =
\sum_{(\lambda,\mu) \vdash n} c_{(\lambda,\mu),S}
\cdot s_\lambda (\by) s_\mu(\bx)
\]
for $S \subseteq [n]$, where $c_{(\lambda,\mu),S}$
is the number of standard Young bitableaux $\pP$ of
shape $(\lambda,\mu)$ such that $\Des_B(\pP) = S$.
Switching the roles of $\bx$ and $\by$ and
combining this result with the second part of
Theorem~\ref{thm:main} and
Proposition~\ref{prop:Ath2} we get
\begin{multline} %\nonumber
{\displaystyle \frac{(1-t) E(\bx; z)}
{E(\bx; tz) E(\by; tz) - tE(\bx; z) E(\by; z)}}
=
\sum_{n \ge 0} z^n \,
\sum_{k=0}^{\lfloor n/2 \rfloor} \
\xi^+_{n,k}(\bx, \by) \, t^k (1 + t)^{n-2k}
\\ +
\sum_{n \ge 1} z^n
\sum_{k=0}^{\lfloor (n-1)/2 \rfloor}
\xi^{-}_{n,k}(\bx, \by) \, t^k (1 + t)^{n-1-2k},
\label{eq:sum}
\end{multline}
where the $\xi^{\pm}_{n,k}(\bx, \by)$ are given
by the first equalities in~\eqref{eq:corAth2+}
and~\eqref{eq:corAth2-}. We now note that the
left-hand side of Equation~\eqref{eq:sum} is equal
to the sum of the left-hand sides, say
$\Xi^+(\bx, \by, t; z)$ and $\Xi^{-}(\bx, \by, t;
z)$, of Equations~\eqref{eq:Ath2+}
and~\eqref{eq:Ath2-}. Since, as one can readily
verify, $\Xi^+(\bx, \by, t; z)$ is left invariant
under replacing $t$ with $1/t$ and $z$ with $tz$,
while $\Xi^{-}(\bx, \by, t; z)$ is multiplied by
$t$ after these substitutions, the coefficient of
$z^n$ in $\Xi^+(\bx, \by, t; z)$ (respectively,
$\Xi^{-}(\bx, \by, t; z)$) is a symmetric
polynomial in $t$ with center of symmetry $n/2$
(respectively, $(n-1)/2$) for every $n \in \NN$.
Since the corresponding properties are clear for
the coefficient of $z^n$ in the two summands in
the right-hand side of Equation~\eqref{eq:sum}
and because of the uniqueness of the decomposition
of a polynomial $f(t)$ as a sum of two symmetric
polynomials with centers of symmetry $n/2$ and
$(n-1)/2$ (see~\cite[Section~5.1]{Ath17}), we
conclude that~\eqref{eq:Ath2+}
and~\eqref{eq:Ath2-} follow from the single
equation~\eqref{eq:sum}.
The second equalities in~\eqref{eq:corAth2+}
and~\eqref{eq:corAth2-} follow by expanding
$s_\lambda(\bx) s_\mu(\by)$ according to
Equation~\eqref{eq:sxyFexpan} and then using the
Robinson--Schensted correspondence of type $B$ and
its properties $\sDes(w) = \sDes(Q^B(w))$ and
$\Des_B(w^{-1}) = \Des_B(\pP^B(w))$, exactly as
in the proof of Corollary~\ref{cor:Gessel}.
\end{proof}
\begin{exam}
The coefficient of $z^2$ in the left-hand side
of Equations~\eqref{eq:Ath2+} and~\eqref{eq:Ath2-}
equals
\begin{itemize}
\item %[$\bullet$]
$e_1(\bx) e_1(\by) \, t + e_2(\by) \, t = s_{(1)}
(\bx) s_{(1)}(\by) \, t + s_{(1,1)}(\by) \, t$, and
\item %[$\bullet$]
$e_2(\bx) \, (1 + t) = s_{(1,1)}(\bx) \, (1 + t)$,
\end{itemize}
respectively. Hence, $\xi^+_{2,0} (\bx, \by) =
0$, $\xi^+_{2,1} (\bx, \by) = s_{(1)}(\bx) s_{(1)}(\by)
+ s_{(1,1)}(\by)$ and $\xi^{-}_{2,0} (\bx, \by) =
s_{(1,1)}(\bx)$, in agreement with
Corollary~\ref{cor:Ath2}.
\end{exam}
\begin{prop} \label{prop:Ath1}
For the $\bB_n$-poset $sB_n$ we have
\begin{equation} \label{eq:prop1}
1 + \sum_{n \ge 1} \, \ch_\bB
\left( \widetilde{H}_{n-1}
((sB_n \ast T_{t,n})_{-}; \CC) \right) z^n =
\frac{(1-t) E(\by; z) E(\bx; tz) E(\by; tz)}
{E(\bx; tz) E(\by; tz) - tE(\bx; z) E(\by; z)}.
\end{equation}
\end{prop}
\begin{proof}
Following the reasoning in the proof
of~\cite[Equation~(3.3)]{SW09}, we set
\[
L_n (\bx, \by; t) \ \coloneqq \ \ch_\bB \left(
L((sB_n \ast T_{t,n})_{-}; \bB_n) \right),
\]
where $L(P; G)$ denotes the Lefschetz character of
the $G$-poset $P$ over $\CC$ (see
Section~\ref{sec:pre}). Since $(sB_n \ast
T_{t,n})_{-}$ is Cohen--Macaulay over $\CC$ of rank
$n-1$, we have
\[
\ch_\bB \left( \widetilde{H}_{n-1} ((sB_n \ast
T_{t,n})_{-}; \CC) \right) = (-1)^{n-1} \,
L_n(\bx, \by; t).
\]
Thus, the left-hand side of~\eqref{eq:prop1} is
equal to $-\sum_{n \ge 0} L_n(\bx, \by; t) (-z)^n$.
The sequence of posets $(sB_0, sB_1,\ldots,sB_n)$ can
easily be verified to be $(\bB_0 \times \fS_n, \bB_1
\times \fS_{n-1},\ldots,\bB_n \times \fS_0)$-uniform
(see Section~\ref{sec:pre}). Moreover, there
is a single $\bB_n$-orbit of elements of $sB_n$ of
rank $k$ for each $k \in \{0, 1,\ldots,n\}$.
Thus, applying~\eqref{eq:uniform} to this
sequence gives
%
\[ 1_{\bB_n} \oplus \, \bigoplus_{k=0}^n \, [k + 1]_t
\, L((sB_{n-k} \ast T_{t,n-k})_{-}; \bB_{n-k}
\times \fS_k) \uparrow^{\bB_n}_{\bB_{n-k} \times
\fS_k} \ \, \cong_{\bB_n} \ 0.
\]
Applying the characteristic map $\ch_\bB$, as in
the proof of Proposition~\ref{prop:Ath2}, gives
%
\[ \sum_{k=0}^n \, [k + 1]_t \, h_k(\bx, \by) \,
L_{n-k} (\bx, \by; t) = - h_n (\bx).
\]
Standard manipulation with generating functions,
as in the proof of~\cite[Equation~(3.3)]{SW09},
results in the formula
%
\[ \sum_{n \ge 0} L_n(\bx, \by; t) z^n = - \
\frac{H(\bx; z)}{\sum_{n \ge 0} \, [n+1]_t \,
h_n(\bx, \by) z^n} = - \ \frac{(1-t) H(\bx; z)}
{H(\bx, \by; z) - t H(\bx, \by; tz)}.
\]
The proof now follows by switching $z$ to $-z$ and
using the identities $E(\bx; z) H(\bx; -z) = 1$ and
$E(\bx, \by; z) = E(\bx; z) E(\by; z)$.
\end{proof}
\begin{coro} \label{cor:Ath1}
Equations~\eqref{eq:Ath1+} and~\eqref{eq:Ath1-}
are valid for the functions
\begin{equation} \label{eq:corAth1+}
\gamma^+_{n,k} (\bx, \by) =
\sum_{(\lambda, \mu) \vdash n}
d^+_{(\lambda,\mu),k} \cdot s_\lambda (\bx) s_\mu
(\by) = \sum_{w} F_{\sDes(w)}(\bx, \by)
\end{equation}
and
\begin{equation} \label{eq:corAth1-}
\gamma^{-}_{n,k} (\bx, \by) =
\sum_{(\lambda, \mu) \vdash n}
d^{-}_{(\lambda,\mu),k} \cdot s_\lambda (\bx) s_\mu
(\by) = \sum_{w} F_{\sDes(w)}(\bx, \by),
\end{equation}
where $d^+_{(\lambda,\mu),k}$ \textup{(}respectively,
$d^{-}_{(\lambda,\mu),k}$\textup{)} is the
number of bitableaux $\pP \in \SYT(\lambda,\mu)$ for
which $\Asc_B(\pP) \in \stab([n])$ has $k$
elements and does not contain \textup{(}respectively,
contains\textup{)} $n$ and, similarly, $w \in \bB_n$
runs through all signed permutations for which
$\Asc_B(w^{-1}) \in \stab([n])$ has $k$ elements
and does not contain \textup{(}respectively,
contains\textup{)}~$n$.
\end{coro}
\begin{proof}
This statement follows by the same reasoning as in
the proof of Corollary~\ref{cor:Ath2}, provided one
appeals to the first part of Theorem~\ref{thm:main}
and Proposition~\ref{prop:Ath1} instead.
\end{proof}
\goodbreak
\begin{exam}
The coefficient of $z^2$ in the left-hand side
of Equations~\eqref{eq:Ath1+} and~\eqref{eq:Ath1-}
equals
\begin{itemize}
%\itemsep=3pt
\item %[$\bullet$]
$e_2(\bx) \, (1 + t + t^2) + e_1(\bx)^2 \, t +
e_1(\bx) e_1(\by) \, t = s_{(1,1)}(\bx) \, (1 + t)^2
+ s_{(2)} (\bx) \, t + s_{(1)}(\bx) s_{(1)}(\by) \,
t$,
\item %[$\bullet$]
$e_1(\bx) e_1(\by) \, (t + t^2) + e_2(\by) \,
(t + t^2) = s_{(1)} (\bx) s_{(1)}(\by) \, t(1+t) +
s_{(1,1)}(\by) \, t(1+t)$,
\end{itemize}
respectively. Hence, we have $\gamma^+_{2,0} (\bx,
\by) = s_{(1,1)}(\bx)$, $\gamma^+_{2,1} (\bx, \by) =
s_{(2)}(\bx) + s_{(1)}(\bx) s_{(1)}(\by)$ and
$\gamma^{-}_{2,1} (\bx, \by) = s_{(1)}(\bx) s_{(1)}
(\by) + s_{(2)}(\by)$, in agreement with
Corollary~\ref{cor:Ath1}.
\end{exam}
\section{An instance of the local equivariant Gal
phenomenon} \label{sec:face}
This section uses Equation~\eqref{eq:Ath2+} to
verify an equivariant analogue of Gal's
conjecture~\cite{Ga05} for the local face module
of a certain triangulation of the simplex with
interesting combinatorial properties. Background
and any undefined terminology on simplicial
complexes can be found in~\cite{StaCCA}.
To explain the setup, let $V_n = \{ \varepsilon_1,
\varepsilon_2,\ldots,\varepsilon_n \}$ be the set
of unit coordinate vectors in $\RR^n$ and
$\Sigma_n$ be the geometric simplex on the vertex
set $V_n$. Consider a triangulation $\Gamma$ of
$\Sigma_n$ (meaning, a geometric simplicial
complex which subdivides $\Sigma_n$) with
vertex set $V_\Gamma$ and the polynomial ring
$S = \CC[x_v: v \in V_\Gamma]$ in indeterminates
which are in one-to-one correspondence with the
vertices of $\Gamma$. The \emph{face ring}
\cite[Chapter~II]{StaCCA} of $\Gamma$ is defined
as the quotient ring $\CC[\Gamma] = S/I_\Gamma$,
where $I_\Gamma$ is the ideal of $S$ generated by
the square-free monomials which correspond to the
non-faces of $\Gamma$. Following
\cite[p.~824]{Sta92}, we consider the linear forms
%
\begin{equation} \label{eq:lsop}
\theta_i = \sum_{v \in V_\Gamma} \langle
v, \varepsilon_i \rangle x_v
\end{equation}
%
for $i \in [n]$, where $\langle \ , \, \rangle$
is the standard inner product on $\RR^n$, and
denote by $\Theta$ the ideal in $\CC[\Gamma]$
generated by $\theta_1,\theta_2,\ldots,\theta_n$.
This sequence is a special linear system of
parameters for $\CC[\Gamma]$, in the sense of
\cite[Definition~4.2]{Sta92}. As a result, the
quotient ring $\CC(\Gamma) = \CC[\Gamma] / \Theta$
is a finite dimensional, graded $\CC$-vector space
and so is the \emph{local face module} $L_{V_n}
(\Gamma)$, defined
\cite[Definition~4.5]{Sta92} as the image in
$\CC(\Gamma)$ of the ideal of $\CC[\Gamma]$
generated by the square-free monomials which
correspond to the faces of $\Gamma$ lying in the
interior of $\Sigma_n$. The Hilbert polynomials
$\sum_{i=0}^n \dim_\CC (\CC(\Gamma))_i t^i$ and
$\sum_{i=0}^n \dim_\CC (L_{V_n}(\Gamma))_i t^i$
of $\CC(\Gamma)$ and $L_{V_n} (\Gamma)$ are two
important enumerative invariants of $\Gamma$,
namely the $h$-polynomial~\cite[Section~II.2]
{StaCCA} and the local $h$-polynomial
\cite[Section~2]{Sta92}~\cite[Section~III.10]
{StaCCA}, respectively.
Suppose that $G$ is a subgroup of the
automorphism group $\fS_n$ of $\Sigma_n$ which
acts simplicially on $\Gamma$. Then, $G$ acts on
the polynomial ring $S$ and (as discussed on
\cite[p.~250]{Ste94}) leaves the $\CC$-linear
span of $\theta_1, \theta_2,\ldots,\theta_n$
invariant. Therefore, $G$ acts on the
graded $\CC$-vector spaces $\CC(\Gamma)$
and $L_{V_n} (\Gamma)$ as well and the
polynomials $\sum_{i=0}^n (\CC(\Gamma))_i t^i$
and $\sum_{i=0}^n (L_{V_n}(\Gamma))_i t^i$, whose
coefficients lie in the representation ring of
$G$, are equivariant generalizations of the
$h$-polynomial and local $h$-polynomial of
$\Gamma$, respectively. The pair $(\Gamma, G)$
is said (see also~\cite[Section~5.2]{Ath17}) to
satisfy the \emph{local equivariant Gal
phenomenon} if
%
\begin{equation} \label{eq:Gal}
\sum_{i=0}^n \, (L_{V_n}(\Gamma))_i \, t^i =
\sum_{k=0}^{\lfloor n/2 \rfloor} M_k \, t^k
(1+t)^{n-2k}
\end{equation}
%
for some non-virtual $G$-representations $M_k$.
This is an analogue for local face modules of
the equivariant Gal phenomenon, formulated by
Shareshian and Wachs~\cite[Section~5]{SW17} for
group actions on (flag) triangulations of spheres
as an equivariant version of Gal's conjecture
\cite[Conjecture~2.1.7]{Ga05}. For trivial
actions on flag triangulations of simplices,
the validity of the local equivariant Gal
phenomenon was conjectured in~\cite{Ath12} and
has been verified in many special cases; see
\cite[Section~4]{Ath16a}~\cite[Section~3.2]{Ath17}
and references therein.
Although it would be too optimistic to expect
that the local equivariant Gal phenomenon holds
for all group actions on flag triangulations of
$\Sigma_n$, the case $G = \fS_n$ deserves special
attention. We then use the notation
%
\begin{align*}
\ch \left( \CC(\Gamma), t \right) & \coloneqq
\sum_{i=0}^n \, \ch \left( \CC(\Gamma) \right)_i
t^i, \\
\ch \left( L_{V_n}(\Gamma), t \right) & \coloneqq
\sum_{i=0}^n \, \ch \left( L_{V_n}(\Gamma) \right)_i
t^i.
\end{align*}
%
For the (first) barycentric subdivision of
$\Sigma_n$ we have the following result of Stanley.
%
\newbox\toto
\setbox\toto=\hbox{\cite[Proposition~4.20]{Sta92}}
\begin{prop}[\box\toto] \label{prop:sdn}
%{\rm ()}
For the $\fS_n$-action on the barycentric
subdivision $\Gamma_n$ of the simplex $\Sigma_n$,
we have
%
\begin{equation} \label{eq:Lsdn}
1 + \sum_{n \ge 1} \, \ch
\left( L_{V_n}(\Gamma_n), t \right) z^n =
\frac{1 - t}{H(\bx; tz) - tH(\bx; z)}.
\end{equation}
%
\end{prop}
Combining this result with Gessel's identity
\eqref{eq:Ge2} gives
%
\[ \ch \left( L_{V_n}(\Gamma_n), t \right) =
\sum_{k=0}^{\lfloor (n-2)/2 \rfloor} \omega \,
\xi_{n,k}(\bx) \, t^{k+1} (1 + t)^{n-2k-2}, \]
%
where $\omega$ is the standard involution on
symmetric functions exchanging $e_\lambda(\bx)$
and $h_\lambda(\bx)$ for every $\lambda$, whence
it follows that $(\Gamma_n, \fS_n)$ satisfies the
local equivariant Gal phenomenon for every $n$.
The combinatorics of the barycentric subdivision
$\Gamma_n$ is related to the symmetric group $\fS_n$.
We now consider a triangulation $K_n$ of the simplex
$\Sigma_n$, studied in~\cite[Chapter~3]{Sav13} (see
also~\cite[Remark~4.5]{Ath16a}
\cite[Section~3.3]{Ath17}) and shown on the right of
Figure~\ref{fig:K3} for $n=3$, the combinatorics of
which is related to the hyperoctahedral group $\bB_n$.
The triangulation $K_n$ can be defined as the
barycentric subdivision of the standard cubical
subdivision of $\Sigma_n$, shown on the left of
Figure~\ref{fig:K3} for $n=3$, whose faces are in
inclusion-preserving bijection with the nonempty
closed intervals in the truncated Boolean lattice
$B_n \sm \{ \varnothing \}$. Thus, the faces of
$K_n$ correspond bijectively to chains of nonempty
closed intervals in $B_n \sm \{ \varnothing \}$
and $\fS_n$ acts simplicially on $K_n$ in the
obvious way. As a simplicial complex, $K_n$ can be
thought of as a ``half Coxeter complex'' for $\bB_n$.
\begin{figure}[htb]\centering
\includegraphics[width=\textwidth]{206_Figure/K3}
\caption{The triangulation $K_3$} \label{fig:K3}
\end{figure}
\begin{prop} \label{prop:Kn}
For the $\fS_n$-action on $K_n$ we have
%
\begin{equation} \label{eq:Kn}
1 + \sum_{n \ge 1} \, \ch
\left( \CC(K_n), t \right) z^n = \frac{H(\bx; z)
\left( H(\bx; tz) - tH(\bx; z) \right)}
{H(\bx; tz)^2 - tH(\bx; z)^2}
\end{equation}
%
and
%
\begin{equation} \label{eq:localKn}
1 + \sum_{n \ge 1} \, \ch
\left( L_{V_n}(K_n), t \right) z^n =
\frac{H(\bx; tz) - tH(\bx; z)}
{H(\bx; tz)^2 - tH(\bx; z)^2}.
\end{equation}
%
\looseness-1
Moreover,$\mk$ the$\mk$ pair$\mk$ $(K_n,\mk \fS_n)\mk$ satisfies$\mk$ the$\mk$
local$\mk$ equivariant$\mk$ Gal$\mk$ phenomenon$\mk$ for$\mk$ every$\mk$ $n$.
\end{prop}
The proof relies on methods developed by
Stembridge~\cite{Ste94} to study representations of
Weyl groups on the cohomology of the toric varieties
associated to Coxeter complexes. To prepare for it,
we recall that the \emph{$h$-polynomial} of a
simplicial complex $\Delta$ of dimension $n-1$
is defined as
%
\[ h(\Delta, t) = \sum_{i=0}^n f_{i-1}
(\Delta) \, t^i (1-t)^{n-i}, \]
%
where $f_i(\Delta)$ stands for the number of
$i$-dimensional faces of $\Delta$. Consider
a pair $(\Gamma, G)$, consisting of a
triangulation $\Gamma$ of $\Sigma_n$ and a
subgroup $G$ of $\fS_n$ acting on $\Gamma$, as
discussed earlier. Following~\cite[Section~1]
{Ste94}, we call the action of $G$
on $\Gamma$ \emph{proper} if $w$ fixes all
vertices of every face $F \in \Delta$ which is
fixed by $w$, for every $w \in G$. Note that
group actions, such as the $\fS_n$-actions on
$\Gamma_n$ and $K_n$, on the order complex
(simplicial complex of chains) of a poset which
are induced by an action on the poset itself,
are always proper. Under this assumption, the
set $\Gamma^w$ of faces of $\Gamma$ which are
fixed by $w$ forms an induced subcomplex of
$\Gamma$, for every $w \in G$.
Although Stembridge~\cite{Ste94} deals with
triangulations of spheres, rather than simplices,
his methods apply to our setting and his Theorem~1.4,
combined with the considerations of Section~6 in
\cite{Ste94}, imply that
%
\begin{equation} \label{eq:ste}
\ch \left( \CC(\Gamma), t \right) =
\frac{1}{n!} \sum_{w \in \fS_n}
\frac{h(\Gamma^w, t)}{(1 - t)^{1+\dim(\Gamma^w)}}
\, \prod_{i \ge 1} \, (1 - t^{\lambda_i(w)}) \,
p_{\lambda_i(w)} (\bx)
\end{equation}
%
for every proper $\fS_n$-action on $\Gamma$, where
$\lambda_1(w) \ge \lambda_2(w) \ge \cdots$ are the
sizes of the cycles of $w \in \fS_n$ and $p_k(\bx)$
is a power sum symmetric function.
\begin{proof}[Proof of Proposition~\ref{prop:Kn}] To prove
Equation~\eqref{eq:Kn}, we follow the analogous
computation in~\cite[Section~6]{Ste94} for the
barycentric subdivision of the boundary complex of
the simplex. We first note that
$(K_n)^w$ is combinatorially isomorphic to
$K_{c(w)}$ for every $w \in \fS_n$, where $c(w)$ is
the number of cycles of $w$. Furthermore, it was
shown in~\cite[Section~3.6]{Sav13} that, in the
notation of Section~\ref{sec:proof}, $h(K_n, t)$
is the ``half $\bB_n$-Eulerian polynomial''
%
\begin{equation}
B^+_n(t) = \sum_{w \in \bB^+_n}
t^{|\Des_B(w)|},
\end{equation}
%
where $\bB^+_n$ consists of the signed permutations
$w \in \bB_n$ with \emph{negative} first coordinate.
These remarks and Equation~\eqref{eq:ste} imply
that
%
\[ \ch \left( \CC(K_n), t \right) z^n \ =
\sum_{\lambda = (\lambda_1, \lambda_2,\ldots)
\vdash n} m^{-1}_\lambda \,
\frac{B^+_{\ell(\lambda)}(t)}
{(1 - t)^{\ell(\lambda)}} \, \prod_{i \ge 1} \,
(1 - t^{\lambda_i}) \, p_{\lambda_i} (\bx)
z^{\lambda_i}, \]
%
where $n!/m_\lambda$ is the cardinality of the
conjugacy class of $\fS_n$ which corresponds to
$\lambda \vdash n$ and $\ell(\lambda)$ is the number
of parts of $\lambda$. The polynomials $B^+_n(t)$
are known (see, for instance,
\cite[Equation~3.7.5]{Sav13}) to satisfy
%
\begin{equation}
\frac{B^+_n(t)} {(1 - t)^n} = \sum_{k \ge 0}
\left( (2k+1)^n - (2k)^n \right) t^k
\end{equation}
%
and hence, we may rewrite the previous formula as
%
\begin{multline*}
\ch \left( \CC(K_n), t \right) z^n
\\
=
\sum_{k \ge 0} \, t^k
\sum_{\lambda = (\lambda_1, \lambda_2,\ldots)
\vdash n} m^{-1}_\lambda
\left( (2k+1)^{\ell(\lambda)} -
(2k)^{\ell(\lambda)} \right) \prod_{i \ge 1} \,
(1 - t^{\lambda_i}) \, p_{\lambda_i} (\bx)
z^{\lambda_i} .
\end{multline*}
Summing over all $n \ge 1$ and using the standard
identities
%
\[ H(\bx; z) = \sum_\lambda m^{-1}_\lambda
p_\lambda (\bx) z^{|\lambda|} = \exp \left(
\, \sum_{n \ge 1} p_n(\bx) z^n / n \right) \]
%
just as in the proof of~\cite[Theorem~6.2]{Ste94}
(one considers the $p_n$ as algebraically
independent indeterminates and replaces first each
$p_n$ with $(2k+1) (1 - t^n)p_n$, then with $(2k)
(1 - t^n)p_n$), we conclude that
%
\begin{align*}
1 + \sum_{n \ge 1} \, \ch
\left( \CC(K_n), t \right) z^n & = 1 +
\sum_{k \ge 0} \, t^k
\left( \frac{H(\bx; z)^{2k+1}}{H(\bx; tz)^{2k+1}}
- \frac{H(\bx; z)^{2k}}{H(\bx; tz)^{2k}} \right)
\\
& = 1 + \left( \frac{H(\bx; z)}{H(\bx; tz)}
- 1 \right) \left( 1 - t \, \frac{H(\bx; z)^2}
{H(\bx; tz)^2} \right)^{-1} \\
& = \frac{H(\bx; z)
\left( H(\bx; tz) - tH(\bx; z) \right)}
{H(\bx; tz)^2 - tH(\bx; z)^2}
\end{align*}
and the proof of~\eqref{eq:Kn} follows. To prove
\eqref{eq:localKn}, it suffices to observe that
%
\[ 1 + \sum_{n \ge 1} \, \ch \left(
L_{V_n}(K_n), t \right) z^n = E(\bx, -z)
\left( 1 + \sum_{n \ge 1} \, \ch \left(
\CC(K_n), t \right) z^n \right). \]
%
The latter follows exactly as the corresponding
identity for the barycentric subdivision
$\Gamma_n$, shown in the proof of
\cite[Proposition~4.20]{Sta92}. Finally, from
Equations~\eqref{eq:Ath2+} and~\eqref{eq:localKn}
we deduce that
%
\[ \ch \left( L_{V_n}(K_n), t \right) =
\sum_{k=0}^{\lfloor n/2 \rfloor} \omega \,
\xi^+_{n,k} (\bx, \bx) \, t^k (1 + t)^{n-2k}. \]
%
This expression, Corollary~\ref{cor:Ath2} and the
well known fact that $s_\lambda(\bx) s_\mu(\bx)$
is Schur-positive for all partitions $\lambda, \mu$
imply that $\ch \left( L_{V_n}(K_n), t \right)$ is
Schur $\gamma$-positive for every $n$, as claimed
in the last statement of the proposition.
\end{proof}
\section{An instance of the equivariant Gal
phenomenon} \label{sec:toric}
A very interesting group action on a simplicial
complex is that of a finite Coxeter group $W$ on
its Coxeter complex~\cite{Bj84}. When $W$
is crystallographic, this action induces a graded
$W$-representation on the (even degree) cohomology of
the associated projective toric variety which has been
studied by Procesi~\cite{Pro90},
Stanley~\cite[p.~529]{Sta89}, Dolgachev and
Lunts~\cite{DL94}, Stembridge~\cite{Ste94} and
Lehrer~\cite{Le08}, among others. The graded dimension
of this representation is equal to the $W$-Eulerian
polynomial. Its equivariant $\gamma$-positivity is
a consequence of a variant of Equation~\eqref{eq:Ge1}
in the case of the symmetric group $\fS_n$; see
\cite[Section~5]{Ath17}~\cite[Section~5]{SW17}. In the
case of the hyperoctahedral group $\bB_n$, by
\cite[Theorem~6.3]{DL94} or~\cite[Theorem~7.6]{Ste94},
the Frobenius characteristic of this graded
$\bB_n$-representation is equal to the coefficient of
$z^n$ in
%
\begin{equation} \label{eq:genrefEulerB}
\frac{(1-t) H(\bx; z) H(\bx; tz)}
{H(\bx, \by; tz) - tH(\bx, \by; z)} .
\end{equation}
The following statement (and its proof) shows that the
equivariant $\gamma$-positivity of this graded
representation is a consequence of the results of
Section~\ref{sec:app} and confirms another instance
of the equivariant Gal phenomenon of Shareshian and
Wachs. As discussed in~\cite[Section~5]{Ath17}, it is
reasonable to expect that the same holds for the
action of any finite crystallographic Coxeter group
$W$ on its Coxeter complex; that would provide a
natural equivariant analogue to the $\gamma$-positivity
of $W$-Eulerian polynomials~\cite[Section~2.1.3]{Ath17}.
\begin{prop}
The coefficient of $z^n$ in \eqref{eq:genrefEulerB}
is Schur $\gamma$-positive for every $n \in \NN$.
\end{prop}
\begin{proof}
Using Equations~~\eqref{eq:Ge2} with~\eqref{eq:Ath1+}
we find that
\[
\begin{aligned}
\frac{(1-t) E(\bx; z) E(\bx; tz)}
{E(\bx, \by; tz) \Mk - \Mk tE(\bx, \by; tz)}
& \Mk = \Mk \frac{E(\bx; z) E(\bx; tz)(E(\by; tz) - tE(\by; z))}
{E(\bx; tz)E(\by; tz) - tE(\bx; z)E(\by; tz)} \cdot
\frac{1-t}
{E(\by; tz) - tE(\by; z)} \\
& \Mk = \Mk \left( \sum_{n \ge 0} z^n
\sum_{i=0}^{\lfloor n/2 \rfloor} \gamma^{+}_{n,i}
(\bx, \by) \, t^i (1 + t)^{n-2i} \right) \\
&\Mk \leavevmode\phantom{{}={}}\Mk \cdot \left( 1 + \sum_{n \ge 2} z^n
\sum_{j=1}^{\lfloor n/2 \rfloor}
\xi_{n,j-1}(\by) \, t^j (1 + t)^{n-2j} \right)
\\ & \Mk = \Mk \sum_{n \ge 0} z^n \! \sum_{k+\ell=n}
\sum_{i, j} \, \gamma^{+}_{k,i} (\bx, \by)
\, \xi_{\ell,j-1}(\by) \, t^{i+j} (1+t)^{n-2i-2j} ,
\end{aligned}
\]
where we have set $\xi_{0, -1}(\by) \coloneqq 1$ and $\xi_{1, -1}
(\by) \coloneqq 0$. Since Schur-positivity is preserved
by sums, products and the standard involution on symmetric
functions, this computation implies that the coefficient of
$z^n$ in \eqref{eq:genrefEulerB} is Schur $\gamma$-positive
for every $n \in \NN$ and the proof follows.
\end{proof}
\begin{rema}
We have shown that
%
\begin{equation} \label{eq:gammaB(x,y)}
\frac{(1-t) H(\bx; z) H(\bx; tz)}
{H(\bx, \by; tz) - tH(\bx, \by; z)} = 1 +
\sum_{n \ge 1} z^n \sum_{i=0}^{\lfloor n/2 \rfloor}
\gamma^B_{n,i}(\bx,\by) \, t^i (1 + t)^{n-2i}
\end{equation}
%
for some Schur-positive symmetric functions $\gamma^B_{n,i}
(\bx, \by)$. It is an open problem to find an explicit
combinatorial interpretation of the coefficient
$c^B_{(\lambda, \mu), i}$ of $s_\lambda(\bx) s_\mu(\by)$ in
$\gamma^B_{n,i}(\bx, \by)$, for $(\lambda, \mu) \vdash n$.
Comparing the graded dimensions of the
$\bB_n$-representations whose Frobenius characteristic is
given by the two sides of Equation~\eqref{eq:gammaB(x,y)} we
get
%
\[ B_n(t) = \sum_{(\lambda,\mu) \vdash n}
\binom{n}{|\lambda|} f^\lambda f^\mu
\sum_{i=0}^{\lfloor n/2 \rfloor} c^B_{(\lambda,\mu), i} \,
t^i (1 + t)^{n-2i} , \]
%
where $B_n(t) \coloneqq \sum_{w \in \bB_n} t^{|\Des_B(w)|}$ is the
$\bB_n$-Eulerian polynomial and $f^\lambda$ stands for the
number of standard Young tableaux of shape $\lambda$. Thus,
a solution to this problem would provide a refinement of the
known $\gamma$-positivity of $\bB_n$-Eulerian polynomials;
see~\cite[Section~2.1.3]{Ath17}.
\end{rema}
\longthanks{The author wishes to thank Eric Katz and Kalle
Karu for some helpful e-discussions about the local
equivariant Gal phenomenon and the anonymous referees
for their useful comments.}
%\nocite{*}
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