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    cells={nodes={scale=#1}}}}
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\newtheorem{thm}{Theorem}
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\newcommand{\xx}{\mathbf{x}}

\newcommand{\R}{\mathbb{R}} 
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\kk}{\mathbb{F}}

\newcommand{\M}{\mathcal{M}} %matroid
\newcommand{\FF}{\mathbf{F}} %set of all flats
\newcommand{\symm}{\mathfrak{S}} %For Tor()
\newcommand{\F}{\mathcal{F}} %flag
\newcommand{\DD}{\mathfrak{D}} %dots and dashes

\newcommand{\Hom}{\mathrm{Hom}} %For Hom()
\newcommand{\Tor}{\mathrm{Tor}} %For Tor()

%\newcommand{\A}{A^\bullet} %in case we choose to change it to CH
\newcommand{\E}{E} %ground set, in case we want to change it to [n]
\newcommand{\m}{\mathbf{m}} %for monomial

\newcommand{\supp}{\mathrm{supp}} %support of a monomial
\newcommand{\esp}{\overline{\mathrm{supp}}} %essential support of a monomial
\newcommand{\fy}{\mathrm{FY}}
\newcommand{\FY}{\mathrm{FY}}
\newcommand{\spn}{\mathrm{span}}
\newcommand{\init}{\mathrm{in}}
\newcommand{\into}{\hookrightarrow}
\newcommand{\bij}{\leftrightarrow}
\newcommand{\blank}{\phantom{\bullet}}

\renewcommand{\phi}{\varphi}
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\newcommand{\verteq}[0]{\begin{turn}{90} $=$\end{turn}}


\DeclareMathOperator\rk{\mathrm{rk}}
\DeclareMathOperator\cork{\mathrm{cork}}
\DeclareMathOperator\Aut{\mathrm{Aut}}
\DeclareMathOperator\stab{\mathrm{Stab}}





%% define your title in the usual way
\title{Chow rings of matroids as permutation representations}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Robert Angarone\thanks{\href{angar017@umn.edu}{angar017@umn.edu}}\addressmark{1}, Anastasia Nathanson\thanks{\href{natha129@umn.edu}{natha129@umn.edu}}\addressmark{1}, \and Victor Reiner\thanks{\href{reiner@umn.edu}{reiner@umn.edu}}\addressmark{1}}

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\address{\addressmark{1}University of Minnesota - Twin Cities, Minneapolis MN 55455}

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\received{\today}

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%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{Given a matroid and a group of its matroid automorphisms, we study the induced group action on the Chow ring of the matroid. This turns out to always be a permutation action. Work of
Adiprasito, Huh and Katz showed that the Chow ring satisfies Poincar\'e duality and
the Hard Lefschetz theorem.  We lift these to statements about this permutation action,
and suggest further conjectures in this vein.}

%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{matroid, Chow ring, Koszul, log-concave, unimodal, Kahler package, Burnside ring, equivariant, Polya freqency, real-rooted}

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\usepackage[backend=bibtex]{biblatex}
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\begin{document}

\maketitle
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\section{Introduction}
A {\it matroid} $\M$ is a combinatorial abstraction of lists of vectors $v_1,v_2,\ldots,v_n$ in a vector space, recording only the information about which subsets of the vectors
are linearly independent or dependent, forgetting their coordinates.
In groundbreaking work, Adiprasito, Huh and Katz \cite{AHK} affirmed long-standing conjectures of Rota--Heron--Welsh and Mason about vectors and matroids via a new methodology.  Their work employed
a certain graded $\Z$-algebra
$A(\M)$
called the {\it Chow ring} of $\M$, introduced by Feichtner and Yuzvinsky \cite{FY} as a generalization of the Chow ring of DeConcini and Procesi's {\it wonderful compactifications} for hyperplane arrangement complements.
A remarkable integral Gr\"obner basis result proven by Feichtner and Yuzvinsky \cite[Thm. 2]{FY} shows that for a matroid $\M$ of rank $r+1$ with Chow ring $A(\M)=\bigoplus_{k=0}^r A^k$, each homogeneous component is free abelian:  $A^k \cong \Z^{a_k}$
for some Hilbert function $(a_0,a_1,\ldots,a_r)$.
A key step in the work of Adiprasito, Huh and Katz shows
not only {\it symmetry} and {\it unimodality} for the Hilbert function
\begin{align}
\label{symmetry}
&a_k = a_{r-k}\text{ for } r \leq k/2\\
\label{unimodality} 
&a_0 \leq a_1 \leq \cdots \leq a_{\lfloor \frac{r}{2} \rfloor} =
a_{\lceil \frac{r}{2} \rceil} \geq \cdots \geq a_{r-1} \geq a_r,
\end{align}
but in fact proves that $A(\M)$ enjoys a trio of properties referred to as the {\it K\"ahler package}, reviewed in Section~\ref{AHK-section} below.  The first of these properties is
{\it Poincar\'e duality}, proving \eqref{symmetry} via a natural
$\Z$-module isomorphism $A^{r-k} \cong \Hom_\Z(A^k,\Z)$.  The second property, called the {\it Hard Lefschetz Theorem}, shows that after tensoring over $\Z$ with $\R$ to obtain $A(\M)_\R=\bigoplus_{k=0}A^k_\R$, one can find {\it Lefschetz elements} $\omega$ in $A^1_\R$ such that multiplication by $\omega^{r-2k}$ give $\R$-linear isomorphisms $A^k_\R \rightarrow A^{r-k}_\R$ for $k \leq \frac{r}{2}$.  In particular, multiplication by $\omega$ mapping $A^k_\R \rightarrow A_\R^{k+1}$ is {\it injective} for $k < \frac{r}{2}$, strengthening the unimodality \eqref{unimodality}.


We are interested in how these {\it Poincar\'e duality} and {\it Hard Lefschetz} properties interact with the group $G:=\Aut(\M)$
of symmetries of the matroid $\M$.  It is not hard to check that $G$ acts via graded $\Z$-algebra automorphisms
on $A(\M)$, giving $\Z G$-module structures on each $A^k$, and
$\R G$-module structures on each $A^k_\R$.  One can also check
(see the proof of Corollary~\ref{integral-equivariant-PD-cor} below) that $A^r \cong \Z$ with trivial $G$-action.  From
this, the Poincar\'e duality pairing immediately gives 
rise to a $\Z G$-module isomorphism
\begin{equation}
\label{integral-rep-PD-isomorphism}
A^{r-k} \cong \Hom_\Z(A^k,\Z)
\end{equation}
where $g$ in $G$ acts on $\varphi$ in $\Hom_\Z(A^k ,\Z)$ via $\varphi \mapsto \varphi \circ g^{-1}$; similarly
$A^{r-k} \cong \Hom_\R(A^k,\R)$ as $\R G$-modules.  
Furthermore,
it is not hard to check (see Corollary~\ref{AHK-equivariant-Hard-Lefschetz} below) that one can pick an explicit Lefschetz element $\omega$ as in \cite{AHK} which is $G$-fixed, giving $\R G$-module isomorphisms and injections
\begin{align}
    \label{real-Lefschetz-PD-isomorphism-injection}
    A_\R^{k} &\overset{\sim}{\longrightarrow} A_\R^{r-k} 
     \quad \text{ for }r \leq \frac{k}{2} \nonumber 
     & 
      A_\R^{k} &\hookrightarrow A_\R^{k+1} \quad \text{ for }r < \frac{k}{2} \nonumber 
     \\
    a &\longmapsto  a \cdot \omega^{r-2k}  &
     a & \longmapsto  a \cdot \omega. 
\end{align}

Our goal is to use  Feichtner and Yuzvinsky's Gr\"obner basis result to prove a combinatorial strengthening of the isomorphisms and injections \eqref{integral-rep-PD-isomorphism}, \eqref{real-Lefschetz-PD-isomorphism-injection}.  To this end, recall that the matroid $\M$ can be specified by
its family $\FF$ of {\it flats};
then the {\it Chow ring}
$A(\M)$ is presented as a quotient of the
polynomial ring $S:=\Z[x_F]$ having one variable $x_F$ for each nonempty flat $F$ in $\FF \setminus \{\varnothing\}$.  The presentation takes the form 
$
A(\M) := S / (I + J)
$
where $I ,J$ are certain ideals of $S$ defined more precisely in Definition~\ref{Chow-ring-definition} below.

Feichtner and Yuzvinsky exhibited a Gr\"obner basis for $I + J$ that leads to the following standard monomial $\Z$-basis for $A(\M)$,
which we call the {\it FY-monomials} of $\M$:
$$
\fy:=\{  x_{F_1}^{m_1} x_{F_2}^{m_2} \cdots x_{F_\ell}^{m_\ell}: 
(\varnothing =:F_0) \subsetneq F_1 \subsetneq F_2 \subsetneq \cdots 
\subsetneq F_\ell, \text{ and }
m_i \leq \rk(F_i) - \rk(F_{i-1})-1\}.
$$
Here $\rk(F)$ denotes the matroid rank of the flat $F$.
The subset $\FY^k$ of FY-monomials $x_{F_1}^{m_1} \cdots x_{F_\ell}^{m_\ell}$ of total degree $m_1+\cdots+m_\ell=k$ then gives a $\Z$-basis for $A^k$.
One can readily check
(see Corollary~\ref{Chow-ring-carries-perm-reps-cor} below) that the group $G=\Aut(\M)$ permutes the $\Z$-basis $\FY^k$ for $A^k$, 
endowing $A^k$ with the structure of
a {\it permutation representation}, or {\it $G$-set}.  
Our main result is this strengthening of the isomorphisms and injections seen in \eqref{integral-rep-PD-isomorphism}, \eqref{real-Lefschetz-PD-isomorphism-injection}.

\begin{thm}
\label{main-theorem}
For every matroid $\M$ of rank $r+1$, there exist
\begin{itemize}
\item[(i)]
$G$-equivariant  bijections 
$
\pi: \FY^k  \overset{\sim}{\longrightarrow}  \FY^{r-k}
$ for $k \leq \frac{r}{2}$, and
\item[(ii)]
$G$-equivariant injections
$
\lambda: \FY^k  \hookrightarrow  \FY^{k+1}
$
for $k < \frac{r}{2}$.
\end{itemize}
\end{thm}










\section{Background}
%%%%%%%%%%%%%%%%%%%%%

%For matroid basics and background, good references are Oxley \cite{oxley} and Ardila \cite{icm_matroids}. 
Among the many definitions of a matroid $\M$ on ground set $E$, the most useful here specifies its collection of {\it flats} $\FF\subsetneq 2^E$, satisfying certain axioms. 
When ordered by inclusion the collection of flats $(\FF,\subseteq)$ forms a geometric lattice; in an abuse of notation, we will use $\FF$ to refer to both the lattice and the set.  This lattice is ranked,
with rank function denoted $\rk(F)$.
The {\it rank} of the matroid $\M$ itself is defined to be $\rk(E)$, and we assume throughout that $\rk(E)=r+1$. 
An {\it automorphism} of the matroid $\M$ is any permutation $g: E \to E$ of the ground set $E$ that carries flats to flats:  for all $F$ in $\FF$ one has $g(F)$ in $\FF$.  Let $G=\Aut(\M)$ denote the group
of all automorphisms of $\M$.  Since such automorphisms respect the
partial order via inclusion on $\FF$, they also preserve the
rank function: $\rk(g(F))=\rk(F)$ for all $g$ in $G$ and $F$ in $\FF$.



%%%%%%%%%%%%%%%%%%%%%
\subsection{Chow Rings}

As defined in the Introduction, Feichtner and Yuzvinsky \cite{FY}
introduced the Chow ring $A(\M)$ of a matroid $\M$.  


\begin{defn} \rm
\label{Chow-ring-definition}
    The \emph{Chow ring} $A(\M)$ of a matroid $\M$ is the quotient $Z$-algebra 
    $$
    A(\M) := S/(I+J)
    $$
    where $S=\Z[x_F]$ is a polynomial ring having one variable $x_F$ for
    each nonempty flat $F \in \FF \setminus \{ \varnothing\}$, and where $I,J$ are the following ideals of $S$:
    \begin{itemize}
      \item $I$ is generated by products $x_F x_{F'}$ where $F, F'$ are incomparable flats,
    \item $J$ is generated by the linear elements
$\displaystyle\sum_{a \in F \in \FF} x_F$ for each atom $a$ in the lattice $\FF$.
    \end{itemize}
\end{defn}

\noindent The presentation of the Chow ring $A(\M)$ only uses
the information about the partial order on the lattice of flats $\FF$ has some consequences.
For one, the Chow ring depends
only upon the associated {\it simple matroid} of $\M$ (one without loops and parallel edges); hence, we assume all matroids to be such.
Another consequence is that  any element $g$ in $G=\Aut(\M)$ will send the generators of the ideals $I, J$ to other such generators.  Thus $I+J$ is a $G$-stable ideal, and $G$ acts on $A(\M)$.


Note if one considers $S=\Z[x_F]$ as a graded $\Z$-algebra,
 then the ideals $I, J$ are generated by homogeneous elements. 
 Hence the quotient $A(\M)=S/(I+J)$ inherits
the structure of a graded $\Z$-algebra $A(\M)=\bigoplus_{k=0}^\infty A^k$.
Since the action of $G=\Aut(\M)$ on the Chow ring preserves rank and hence degree, both $A(\M)$ and each homogeneous component $A^k$ become $\Z G$-modules.





The following crucial result appears as \cite[Thm. 2]{FY}.  To state it, define an {\it FY-monomial order} on $S=\Z[x_F]_{\varnothing \neq F \in \FF}$ to be any monomial order
based on a linear
order of the variables with $x_F > x_{F'}$ if $F\subsetneq F'$.

\begin{thm}
\label{FY-GB-theorem} 
Given a matroid $\M$ and any FY-monomial order on $S=\Z[x_F]_{\varnothing \neq F \in \FF}$,
the ideal $I+J$ presenting  $A(\M)=S/(I+J)$
has a monic Gr\"obner basis $\{ g_{F,F'} \}$ indexed by $F \neq F'$ in $\FF$, with $g_{F,F'}$ and their initial terms $\init_\prec(g_{F,F'})$ as shown here:
\begin{center}
\begin{tabular}{|c|c|c|}\hline
condition on $F \neq F'$ in $\FF$ & $g_{F,F'}$ & $\init_\prec(g_{F,F'})$ \\ \hline \hline 
& & \\
$F, F'$ non-nested & $x_F x_{F'}$ & $x_F x_{F'}$ \\ 
 & & \\
 \hline
 & & \\
$\varnothing \neq F \subsetneq F'$ & $\displaystyle x_F \left( \sum_{\substack{F'' \in \FF: \\ F'' \supseteq F'}} x_{F''} \right)^{\rk(F')-\rk(F)}$ & $\displaystyle x_F \cdot  x_{F'}^{\rk(F')-\rk(F)}$ \\ 
& & \\
\hline
& & \\
$\varnothing = F \subsetneq F'$ & $\displaystyle  \left( \sum_{\substack{F'' \in \FF:\\ F'' \supseteq F'}} x_{F''}\right)^{\rk(F')}$ & $\displaystyle x_{F'}^{\rk(F')}$ \\ 
& & \\ 
\hline
\end{tabular}
\end{center}
\end{thm}



\begin{cor} (\cite[Cor. 1]{FY}) \label{cor: mon_basis}
For a matroid $\M$ of rank $r+1$, the
Chow ring $A(\M)$ has these properties:
\begin{itemize}
    \item[(i)] $A(\M)$ is free as a $\Z$-module, with $\Z$-basis given by the set of what we call \emph{FY-monomials}
\begin{equation}
\label{FY-monomials-definition}
\fy:=\{  x_{F_1}^{m_1} x_{F_2}^{m_2} \cdots x_{F_\ell}^{m_\ell} 
\colon F_1 \subsetneq \cdots 
\subsetneq F_\ell \in \FF, \text{ and }
m_i \leq \rk(F_i) - \rk(F_{i-1})-1\}.
\end{equation}
\item[(ii)] $A(\M)$ vanishes in degrees strictly above $r$,
that is, $A(\M)=\bigoplus_{k=0}^r A^k$.
\item[(iii)] $A^r$ has $\Z$-basis $\{x_E^r\}$, and so a $\Z$-module isomorphism
$\deg: A^r \longrightarrow \Z$ sending $x_E^r \longmapsto 1$.
\end{itemize}
\end{cor}

Assertions (ii) and (iii) follow immediately from the first. To see this, note that the typical FY-monomial $x_{F_1}^{m_1} x_{F_2}^{m_2} \cdots x_{F_\ell}^{m_\ell}$,
has total degree
\[
\sum_{i=1}^\ell m_i
\leq \sum_{i=1}^\ell (\rk(F_i)-\rk(F_{i-1}) -1)
= \rk(F_\ell) - \ell
\leq (r+1) - 1 = r+1.
\]
Equality here occurs only if $\ell=1$ and $F_\ell=E$, in
which case the FY-monomial is $x_E^r$.





For any matroid automorphism $g$,
the fact that $\rk(g(F))=\rk(F)$
for every flat $F$ in $\FF$ implies that $g$ sends any FY-monomial to another FY-monomial:
$$
x_{F_1}^{m_1} x_{F_2}^{m_2} \cdots x_{F_\ell}^{m_\ell}
\,\,
\overset{g}{\longmapsto}
\,\,
x_{g(F_1)}^{m_1} x_{g(F_2)}^{m_2} \cdots x_{g(F_\ell)}^{m_\ell}.
$$
This has a corollary, 
inspired by work of H.-C. Liao on Boolean matroids \cite[Thm. 2.5]{Liao}.

\begin{cor}
\label{Chow-ring-carries-perm-reps-cor}
For any matroid $\M$, the group $G=\Aut(\M)$ permutes the set $\FY$,
as well as its subset of degree $k$ monomials $\FY^k \subset \FY$.  Consequently, the $\Z G$-modules on the Chow ring $A(\M)$ and each of its homogeneous components $A^k$ lift to $G$-permutation representations on $\FY$ and each $\FY^k$.  
\end{cor}





\begin{example} \rm 
        Let $\M=U_{4,5}$ be the uniform matroid of rank $4$ on $E=\{1,2,3,4,5\}$, associated to a list of $5$ {\it generic} vectors $v_1,v_2,v_3,v_4,v_5$ in a $4$-dimensional vector space, so that any quadruple $v_i,v_j,v_k,v_\ell$ is linearly independent.  One has these flats of various ranks:

\begin{center}
\begin{tabular}{|c|c|}\hline
rank & flats $F \in \FF$ \\ \hline\hline
 $0$ & $\varnothing$ \\ \hline
 $1$ & $1,2,3,4,5$ \\ \hline
 $2$ & $12,13,14,15,23,24,25,34,35,45$ \\ \hline
 $3$ & $123,124,125,134,135,145,234,235,245,345$ \\ \hline
 $4$ & $E=12345$ \\ \hline
\end{tabular}
\end{center}
The Chow ring $A(\M)=S/(I+J)$, where 
$
S= \Z[ x_i, x_{jk}, x_{\ell m n}, x_E]
$
with
$\{i\}, \{j,k\}, \{\ell, m, n\}$ running through all one, two and three-element subsets of 
$E=\{1,2,3,4,5\}$,
and 
$$
I=\Big( x_F x_{F'} \Big)_{F \not\subset F', F' \not \subset F},
\qquad
J=\bigg( x_i 
+ \sum_{\substack{1 \leq j < k \leq 5\\i \in \{j,k\}}} x_{jk}
+ \sum_{\substack{1 \leq \ell  < m < n \leq 5\\i \in \{\ell,m,n\} }} x_{\ell m n}
\,\,\, + x_E \bigg)_{i=1,2,3,4,5}.
$$
The FY-monomial bases for $A^0,A^1,A^2,A^3$ are shown here,
together with the $G$-equivariant maps $\lambda$:
$$
\begin{array}{cccccccc}
\bf{FY^0} & & \bf{FY^1}& & \bf{FY^2} & & &\bf{FY^3} \\
% & & & & & & & \\
1 &\overset{\lambda}{\longmapsto}& x_E &\overset{\lambda}{\longmapsto} & x_E^2 &  & &x_E^3 \\
 & & & & & & &\\
  & & x_{ijk} &\overset{\lambda}{\longmapsto} & x_{ijk}^2& & &\\
       & & & & & & &\\
  & &x_{ij} &\overset{\lambda}{\longmapsto} &x_{ij} \cdot x_E& & &
\end{array} 
$$

Thus in this case, the ranks of the free $\Z$-modules $(A^0,A^1,A^2,A^3)$ form the symmetric, unimodal sequence $(a_0,a_1,a_2,a_3)=(1,21,21,1)$.   Here the bijection $\pi: \fy^0 \rightarrow \fy^3$ 
necessarily maps $1 \longmapsto x_E^3$, and 
the bijection $\pi: \fy^1 \rightarrow \fy^2$  coincides with
the map $\lambda: \fy^1 \rightarrow \fy^2$ above.  
\end{example}










%%%%%%%%%%
\subsection{The K\"ahler package}
\label{AHK-section}
The following theorem on the K\"ahler package for $A(\M)$ compiles
some of the main results of the work of
Adiprasito, Huh and Katz \cite{AHK}.

\begin{thm}
\label{AHK-theorem}
For a matroid $\M$ of rank $r+1$, the Chow ring $A(\M)$ satisfies the 
K\"ahler package:
\begin{itemize}
\item (Poincar\'e duality)
For every $k \leq \frac{r}{2}$, 
one has a perfect $\Z$-bilinear pairing
\begin{align*}
    A^k \times A^{r-k} &\longrightarrow \Z \\ 
    (a,b) &\longmapsto \deg(a \cdot b)
\end{align*}
\item (Hard Lefschetz)
Tensoring over $\Z$ with $\R$, the (real) Chow ring $A_\R(\M)=\sum_{k=0}^r A^k_\R$ contains {\textbf{Lefschetz elements}} $\omega$ in $A^1_\R$, meaning
that $a \mapsto a \cdot \omega^{r-2k}$ is an $\R$-linear isomorphism $A^k_\R \rightarrow A^{r-k}_\R$ for $k \leq \frac{r}{2}$.
In particular, multiplication by $\omega$ is an injection $A^k_\R \rightarrow A_\R^{k+1}$ for $k < \frac{r}{2}$.
\item (Hodge-Riemann-Minkowski inequalities)
The Lefschetz elements $\omega$ define quadratic forms 
$a \longmapsto (-1)^k \deg(a \cdot \omega^{r-2k} \cdot a)$
on $A^k_\R$ that become positive definite 
upon restriction to the kernel of the map
$A^k_\R \longrightarrow A^{r-k+1}_\R$ that sends 
$a \longmapsto  a \cdot \omega^{r-2k+1}$.
\end{itemize}
\end{thm}

\noindent
In fact, they show that one obtains a Lefschetz element $\omega$ whenever
$\omega=\sum_{\varnothing \neq F \in \FF} c_F x_F$ has coefficients $c_F$
coming from restricting to $\FF$ any function $A \mapsto c_A$ that maps $2^E \rightarrow \R$ 
and satisfies these two properties: 
\begin{itemize}
    \item[(1)] the {\it strict submodular inequality}
$
c_{A} + c_{B} > c_{A \cap B} + c_{A \cup B}
$
for all $A \neq B$, and
\item[(2)] $c_{\varnothing}=c_E=0$.
\end{itemize}

This has consequences for $G$ acting
on $A(\M)$ and each $A^k$.  Properties (1) and (2) above are refined by Theorem~\ref{main-theorem}'s parts (i) and (ii) respectively.


\begin{cor}
\label{integral-equivariant-PD-cor} \label{AHK-equivariant-Hard-Lefschetz} 
For any matroid $\M$, one has an isomorphism of $\Z G$-modules 
$
A^{r-k} \rightarrow A^k
$
for each $k \leq \frac{r}{2}$
and $\R G$-module maps
$A^k_\R \rightarrow A_\R^{k+1}$ which are
 injective for $k < \frac{r}{2}$.
\end{cor}



\section{Results}
We recall the statement of the theorem, involving 
the FY-monomial $\Z$-basis for $A(\M)$ in Corollary~\ref{cor: mon_basis}:
$$
\FY:=\{  x_{F_1}^{m_1} x_{F_2}^{m_2} \cdots x_{F_\ell}^{m_\ell} \colon \subsetneq F_1 \subsetneq F_2 \subsetneq \cdots 
\subsetneq F_\ell \text{ in }\FF, \text{ and }
m_i \leq \rk(F_i) - \rk(F_{i-1})-1\}
$$
This also means that the FY-monomials $\FY^k$ 
of degree $k$ form a $\Z$-basis for $A^k$ for each $k=0,1,2,\ldots,r$.

\vskip.1in
\noindent
{\bf Theorem~\ref{main-theorem}}
{\it 
For every matroid $\M$ of rank $r+1$, there exist
\begin{itemize}
\item[(i)]
$G$-equivariant  bijections 
$
\pi: \FY^k  \overset{\sim}{\longrightarrow}  \FY^{r-k}
$ for $k \leq \frac{r}{2}$, and
\item[(ii)]
$G$-equivariant injections
$
\lambda: \FY^k  \hookrightarrow  \FY^{k+1}
$
for $k < \frac{r}{2}$.
\end{itemize}
}

The prove this, we organize monomials according to the fibers of the following map.


\begin{defn}\rm Define the {\it extended support} $\supp_+(a) \subset \FF$ of an FY-monomial $a=x_{F_1}^{m_1} \cdots x_{F_\ell}^{m_\ell}
$ by
\vspace{-0.1in}
$$
\supp_+(a):=\{F_1,\ldots,F_\ell\} \cup \{E\}
=\begin{cases}
    \{F_1,\ldots,F_\ell\} \cup \{E\} & \text{ if }F_\ell \subsetneq E,\\
    \{F_1,\ldots,F_\ell\} &\text{ if }F_\ell=E.
\end{cases}
$$
Define a partial order $<_+$ on the FY-monomials in which $a <_+ b$ if $a$ divides $b$ and $\supp_+(a)=\supp_+(b)$.
\end{defn}


For integers $p < q$, let $[p,q]$ denote
the integer linear order inclusively from $p$ to $q$.  Given a sequence of
such pairs $p_i < q_i$ for $i=1,2,\ldots,m$, let
\begin{equation}
\label{typical-product-of-chains}
\prod_{i=1}^n [p_i,q_i]
=[p_1,q_1] \times [p_2,q_2] \times \cdots \times [p_m,q_m]
\end{equation}
denote their Cartesian product, partially ordered componentwise.

\begin{prop}
\label{iso-to-product-of-chains}
    For any nested flag 
    $\{F_1 \subsetneq \cdots \subsetneq F_\ell \subsetneq E\}$ in $\FF$
    containing $E$, with conventions $F_0:=\varnothing$ and $F_{\ell+1}:=E$,
    the fiber 
    $
    \supp_+^{-1}\{F_1,\ldots,F_\ell,E\}
    $
    is the set of monomials 
    $\{
    x_{F_1}^{m_1} x_{F_2}^{m_2} \cdots x_{F_\ell}^{m_\ell} x_E^{m_{\ell+1}}
    \}$
    satisfying these inequalities:  
    \begin{align*}1 &\leq m_{i} \leq \rk(F_i)-\rk(F_{i-1})-1 \text{ for } i\leq \ell, \\ 0 &\leq m_{\ell+1} \leq \rk(E)-\rk(F_{\ell})-1 =r-\rk(F_\ell).\end{align*}

\noindent Consequently, the minimum and maximum degree of
monomials in
$
\supp_+^{-1}\{F_1,\ldots,F_\ell,E\}
$
are $\ell$ and $r-\ell$, and one has a poset isomorphism
$$
\begin{array}{rcl}
(\supp_+^{-1}\{F_1,\ldots,F_\ell,E\},<_+)
&\longrightarrow &
\displaystyle \prod_{i=1}^\ell [1,\rk(F_i)-\rk(F_{i-1})-1]  \,\, \times \,\,  [0,r-\rk(F_{\ell})]\\
& & \\
x_{F_1}^{m_1} x_{F_2}^{m_2} \cdots x_{F_\ell}^{m_\ell} x_E^{m_{\ell+1}}
&\longmapsto &
(m_1,m_2,\ldots,m_\ell,m_{\ell+1}).
\end{array}
$$
\end{prop}

\noindent Most assertions of the proposition are immediate
from the definition of the order on FY-monomials $<_+$ and
the map $\supp_+$. The minimum and maximum degrees of 
monomials in 
$\supp_+^{-1}
\{F_1,\ldots,F_\ell,E\})$ 
are achieved by
$$
\begin{aligned}
\deg(x_{F_1}^1 x_{F_2}^1 \cdots x_{F_\ell}^1 x_E^0)&=\ell \;\;\;\; \text{ and } \\
\deg\left(\prod_{i=1}^\ell x_{F_i}^{\rk(F_i)-\rk(F_{i})-1} \cdot
x_E^{\rk(E)-\rk(F_{\ell})-1)}\right)
&=\sum_{i=1}^{\ell+1} \left(\rk(F_i)-\rk(F_{i-1})-1\right) \\ 
&=\rk(E) - (\ell+1) \\
 &= r-\ell. \qedhere
\end{aligned}
$$




The idea behind the proof of Theorem~\ref{main-theorem} stems from the observation that all products of chains, as in \eqref{typical-product-of-chains}, have {\it symmetric chain decompositions}, which can then be pulled back to each fiber $\supp_+^{-1}\{F_1,\ldots,F_\ell,E\}.$

\begin{defn} \rm
    A {\it symmetric chain decomposition (SCD)} of a finite ranked poset $P$ of rank $r$ is a disjoint decomposition $P=\bigsqcup_{i=1}^t C_i$
    in which each $C_i$ is a totally ordered subset containing one element of each 
    rank $\{\rho_i,\rho_i+1,\ldots,r-\rho_i-1,r-\rho_i\}$ for some $\rho_i \leq \lfloor \frac{r}{2}\rfloor$.
\end{defn}




It is not hard to check that when posets $P_1, P_2$ each have an SCD, then so does their Cartesian product. In particular, all products of chains have an SCD. {\it Fix one such SCD for each product poset in \eqref{typical-product-of-chains}, once and for all,} and use the isomorphisms from Proposition~\ref{iso-to-product-of-chains} to induce an SCD on each fiber $\supp_+^{-1}\{F_1,\ldots,F_\ell,E\}.$  




\begin{example} \rm
\label{extended-support-map-example}
Assume $\M$ has $\rk(E)=10=r+1$ with $r=9$,
and one has a pair of nested flats $F \subset F'$ with $\rk(F)=3, \rk(F')=7$.  Then the
poset $\supp_+^{-1}\{F,F',E\}$ and one choice of SCD for it look as follows: 




\begin{center}
\begin{tikzpicture}
  [scale=.25,auto=left]
   \node (110) at (-6,0) {$x_F x_{F'}$};
  \node (120) at (-12,6) {$x_F x_{F'}^2$};
  \node (210) at (-6,6) {$x_F^2 x_{F'}$};
  \node (111) at (0,6) {$x_F x_{F'} x_E$};
  \node (130) at (-18,12) {$x_F x_{F'}^3$};
  \node (220) at (-12,12) {$x_F^2 x_{F'}^2$};
  \node (121) at (-6,12) {$x_F x_{F'}^2 x_E$};
  \node (211) at (0,12) {$x_F^2 x_{F'} x_E$};
  \node (112) at (6,12) {$x_F x_{F'} x_E^2$};
  \node (230) at (-18,18) {$x_F^2 x_{F'}^3$};
  \node (131) at (-12,18) {$x_F x_{F'}^3 x_E$};
  \node (221) at (-6,18) {$x_F^2 x_{F'}^2 x_E$};
  \node (122) at (0,18) {$x_F x_{F'}^2 x_E^2$};
  \node (212) at (6,18) {$x_F^2 x_{F'} x_E^2$};
  \node (231) at (-12,24) {$x_F^2 x_{F'}^3 x_E$};
  \node (132) at (-6,24) {$x_F x_{F'}^3 x_E^2$};
  \node (222) at (0,24) {$x_F^2 x_{F'}^2 x_E^2$};
   \node (232) at (-6,30) {$x_F^2 x_{F'}^3 x_E^2$};
  \foreach \from/\to in {110/120,110/210,110/111,  120/130,120/220,120/121,210/220,210/211,111/211,111/121,111/112,
130/230,130/131,220/230,220/221,121/221,121/131,211/221,211/212,112/122,112/212,121/122,
230/231,131/231,221/231,131/132,221/222,122/132,212/222,122/222,
    231/232,132/232,222/232}
    \draw (\from) -- (\to);
  [scale=.25,auto=left,every node]%/.style={circle,fill=black!20}]
   \node (110) at (27,0) {$x_F x_{F'}$};
  \node (120) at (21,6) {$x_F x_{F'}^2$};
  \node (210) at (27,6) {$x_F^2 x_{F'}$};
  \node (111) at (33,6) {$x_F x_{F'} x_E$};
  \node (130) at (15,12) {$x_F x_{F'}^3$};
  \node (220) at (21,12) {$x_F^2 x_{F'}^2$};
  \node (121) at (27,12) {$x_F x_{F'}^2 x_E$};
  \node (211) at (33,12) {$x_F^2 x_{F'} x_E$};
  \node (112) at (39,12) {$x_F x_{F'} x_E^2$};
  \node (230) at (15,18) {$x_F^2 x_{F'}^3$};
  \node (131) at (21,18) {$x_F x_{F'}^3 x_E$};
  \node (221) at (27,18) {$x_F^2 x_{F'}^2 x_E$};
  \node (122) at (33,18) {$x_F x_{F'}^2 x_E^2$};
  \node (212) at (39,18) {$x_F^2 x_{F'} x_E^2$};
  \node (231) at (21,24) {$x_F^2 x_{F'}^3 x_E$};
  \node (132) at (27,24) {$x_F x_{F'}^3 x_E^2$};
  \node (222) at (33,24) {$x_F^2 x_{F'}^2 x_E^2$};
   \node (232) at (27,30) {$x_F^2 x_{F'}^3 x_E^2$};
  \foreach \from/\to in {110/111,
  120/121,210/211,111/112,
130/230,220/221,121/131,211/212,112/122,
131/231,122/132,212/222,
132/232}
    \draw[line width=0.1mm] (\from) -- (\to);
\end{tikzpicture} 
\end{center}

\end{example}





\section{Further questions and conjectures}
\label{rep-theoreric-inequalities-section}

So far, we have mentioned that the unimodality statement \eqref{unimodality}, asserting
for $k < \frac{r}{2}$ that one has
$
a_k \leq a_{k+1},
$
is weaker than the statement in Corollary~\ref{AHK-equivariant-Hard-Lefschetz} asserting that there are injective $\R G$-module maps
$
A^k_\R \rightarrow A_\R^{k+1},
$
which is weaker than 
Theorem~\ref{main-theorem}(ii) asserting that there are
injective $G$-equivariant maps of the $G$-sets
$
\FY^k \hookrightarrow \FY^{k+1}.
$
Here, we wish to consider 
not only unimodality for $(a_0,a_1,\ldots,a_r)$, but other properties
like {\it log-concavity}, {\it the P\'olya frequency property}, and how to similarly lift them to statements regarding
$\R G$-modules and $G$-permutation representations.  In phrasing this, it helps to consider the character and Burnside rings.

\begin{defn} \rm For a finite group $G$, its {\it virtual (complex) character ring} $R_\C(G)$ is the
free $\Z$-submodule of the ring of (conjugacy) class functions $\{f: G \rightarrow \C\}$, having as a $\Z$-basis the irreducible complex characters of $G$.  
If a character $\chi$ can be written as a positive linear combination of irreducible characters of $G$, we say that $\chi$ is a {\it genuine character}, and write $\chi \geq_{R_\C(G)} 0$.


Similarly,  one can define its {\it Burnside ring} $B(G)$ by now having as basis the isomorphism classes $[X]$ of finite $G$-sets $X$. Then $B(G)$ is the $\Z$-module that mods out by the span of all elements $[X \sqcup Y] - ([X]+[Y])$ and if $b\in B(G)$ can be written as a positive linear combination of isomorphism classes, then $b$ is a {\it genuine permutation representation}, and $b \geq_{B(G)} 0$.
\end{defn}


\subsection{PF sequences and log-concavity}
For a sequence of {\it positive} real numbers $(a_0,a_1,\ldots,a_r)$, the property of {\it unimodality} lies at the bottom of a hierarchy of concepts
\begin{equation}
\label{PF-hierarchy}
\begin{array}{rccccccccccccccc}
\text{unimodal}
&\Leftarrow&
PF_2
&\Leftarrow&
&PF_3&
&\Leftarrow&
\cdots&
\Leftarrow&
&PF_\infty\\
&  & \Vert & & & &
& & & & 
&\Vert\\
&  & \text{(strongly) log-concave} & & & &
& & & &   
&PF\\
\end{array}
\end{equation}
which we next review, along with their equivariant and Burnside ring extensions.  


\begin{defn}\rm
\label{numerical-inequality-conditions}
    Say a sequence of positive reals $(a_0,a_1,\ldots,a_r)$ is {\it strongly
    log-concave}
(or {\it $PF_2$}) if  $0 \leq i\leq j \leq k \leq \ell\leq r$ and $i+\ell=j+k$ implies
    $$
    a_i a_\ell \leq a_j a_k, \text{ or equivalently,  }
    \det\left[ \begin{matrix} a_j & a_\ell \\ a_i & a_k \end{matrix}\right] \geq 0.
    $$
    
    For $\ell=2,3,4,\ldots$, say that the sequence is $PF_\ell$ if the associated (infinite) {\it Toeplitz matrix} 
    $$
    T(a_0,\ldots,a_r):=
    \left[
    \begin{matrix}
        a_0 & a_1& a_2& \cdots & a_{r-1} & a_r     & 0   & 0 & \cdots  \\
        0   & a_0 & a_1 & \cdots & a_{r-2} & a_{r-1} & a_{r} & 0 &\cdots \\
        0   & 0   & a_0 & \cdots & a_{r-3} & a_{r-2} & a_{r-1} & a_{r}& \cdots \\
        \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots 
    \end{matrix}
    \right]
    $$
    has all {\it nonnegative} square minor subdeterminants of size 
    $m \times m$ for $1\leq m  \leq \ell$.
    Say that the sequence is a {\it P\'olya frequency sequence} (or {\it $PF_\infty$}, or just {\it $PF$}) if it is $PF_\ell$ for all $\ell=2,3,\ldots$.
\end{defn}




\begin{defn} \rm
For a finite group $G$ and (genuine, nonzero) $\C G$-modules $(A^0,A^1,\ldots,A^r)$, define the analogous
notions of {\it equivariant unimodality}, {\it equivariant strong log-concavity}, {\it equivariant $PF_r$ or $PF_\infty$} by replacing the numerical inequalities in 
Definition~\ref{numerical-inequality-conditions} by
inequalities in $\R_\C(G)$, or, similarly, one can define all these concepts to be Burnside if these inequalities are in the Burnside ring $B(G)$. 
\end{defn}

We've seen for Chow rings $A(\M)$ of rank $r+1$ matroids $\M$, and $G=\Aut(\M)$, 
the sequence $(a_0,a_1,\ldots,a_r)$ with $a_k:=\rk_\Z A_k$ is {\it unimodal}; after tensoring with $\C$, the sequence of $\C G$-modules $(A^0_\C,A^1_\C,\ldots,A^r_\C)$ is {\it equivariantly unimodal}; and the sequence of $G$-sets $(\FY^0,\FY^1,\ldots,\FY^r)$
is {\it Burnside unimodal}.

\begin{conj}
\label{log-concavity-conjectures}
In the Chow ring of a rank $r+1$ matroid $\M$, one has that
    \begin{itemize}
    \item[(i)] (Ferroni-Schr\"oter \cite[Conj. 10.19]{FerroniSchroter}) $(a_0,\ldots,a_r)$ is $PF_\infty$.
\item[(ii)] $(A^0_\C,\ldots,A^r_\C)$ is equivariantly $PF_\infty$.
\item[(iii)] $(\FY^0,\ldots,\FY^r)$
is Burnside $PF_2$ (Burnside log-concave).
\end{itemize}
\end{conj}
Of course, in Conjecture~\ref{log-concavity-conjectures}, 
assertion (ii) implies assertion (i).
However the same is not true of assertion (iii): it would only imply the weaker $PF_2$ part of 
the conjectural assertion (ii), and only imply the $PF_2$ part of Ferroni and Schr\"oter's assertion (i), but not their $PF_\infty$ assertions. We have some evidence for the following two further conjectures.

\begin{conj}
\label{boolean-h-log-concavity-conj}
For a Boolean matroid $\M$ of rank $n$
and $i \leq j \leq k \leq \ell$ with  $i+\ell=j+k$, not only is the element
$
[\FY^j][\FY^k]-[\FY^i][\FY^\ell] \geq_{B(\symm_n)} 0,
$
so that it is a genuine permutation representation,
but furthermore one whose 
orbit-stabilizers are all Young subgroups
$\symm_\lambda$.
%:=\symm_{\lambda_1} \times \symm_{\lambda_2} \times \cdots \times \symm_{\lambda_\ell}$.
\end{conj}

\begin{conj}
\label{Burnside-Koszul-nonnegativity}
For a matroid $\M$ of rank $r+1$ with
Chow ring $A(\M)$, and any composition $\alpha=(\alpha_1,\ldots,\alpha_\ell)$ with $m:=\sum_i \alpha \leq r$, the analogous Toeplitz minors of $G$-sets have
$$
\det\left[
\begin{matrix}
[\FY^{\alpha_1}] & [\FY^{\alpha_1+\alpha_2}]  & [\FY^{\alpha_1+\alpha_2+\alpha_3}]  & \cdots &[\FY^m]  \\
[\FY^0] & [\FY^{\alpha_2}] & [\FY^{\alpha_2+\alpha_3}]  &\cdots  &[\FY^{m-\alpha_1}]    \\
0 & [\FY^0] & [\FY^{\alpha_3}] & \cdots & [\FY^{m-(\alpha_1+\alpha_2)}]   \\
0 & 0 &   &   & \vdots\\
\vdots & \vdots & & & [\FY^{\alpha_{\ell-1}+\alpha_\ell}]\\
0 & 0   &   \cdots     & [\FY^0] &[\FY^{\alpha_\ell}]
\end{matrix}
\right] \geq_{B(G)} 0.
$$
\end{conj}




\subsection{Further Questions}
So far, we have focused on the Chow ring of a matroid $\M$ using its {\it maximal} building set. 

One relevant example of such a building set is the {\it minimal} building set, which is stable under the full automorphism group $\Aut(\M)$, and which arises, for example, in the study of the moduli space $\overline{M}_{0,n}$ of genus $0$ curves with $n$ marked points; see, e.g., Dotsenko \cite{dotsenko}, Gibney and Maclagan \cite{GibneyMaclagan}.


\begin{question} 
    Does the analogue of Theorem~\ref{main-theorem} hold for the Chow ring of a matroid $\M$ with respect to {\it any} $G$-stable building set? 
In particular, what about the {\it minimal} building set?
\end{question}

 
In \cite[Lem. 3.1]{Stembridge}, Stembridge provides a generating function for the symmetric group representations on each graded component of the Chow ring for all Boolean matroids; see also Liao \cite{Liao}.  Furthermore, Stembridge's expression exhibits them as {\it permutation representations}, whose orbit-stabilizers are all {\it Young subgroups} in the symmetric group.
\begin{question}
Can one provide such explicit expressions as permutation representations for other families of
matroids with symmetry?
\end{question}



Hilbert functions $(a_0,a_1,\ldots,a_r)$ for
Chow rings of rank $r+1$ matroids are
not only symmetric and unimodal, but satisfy the stronger condition of
{\it $\gamma$-positivity}, as shown by :  one has {\it nonnegativity} for all coefficients 
$\gamma=(\gamma_0,\gamma_1,\ldots,\gamma_{\lfloor \frac{r}{2} \rfloor})$ appearing in the unique expansion
\[
\sum_{i=0}^r a_i t^i = \sum_{i=0}^{\lfloor\frac{r}{2}\rfloor} \gamma_i \,\, t^i(1+t)^{r-2i}.
\]
This has been shown, independently 
by Ferroni, Matherne, Stevens and Vecchi \cite[Thm. 3.25]{FMSV}
and by Wang (see \cite[p.~29]{FMSV}), that the $\gamma$-positivity for Hilbert series of Chow rings of matroids
follows from results of \cite{BHMPW}.

One also has the notion of {\it equivariant $\gamma$-positivity} for a sequence of $G$-representations $(A_0,A_1,\ldots,A_r)$: 
upon replacing each
$a_i$ with the element $[A_i]$ of $R_\C(G)$, one asks that the uniquely defined coefficients $\gamma_i$ in $R_\C(G)$ have $\gamma_i \geq_{R_\C(G)} 0$.


\begin{conj}
\label{equivariant-gamma-positivity-conj}
For any matroid $\M$ of rank $r+1$ and its Chow ring
$A(\M)=\bigoplus_i A^i$,
the sequence of $G$-representations
$(A^0_\C,A^1_\C,\ldots,A^r_\C)$ is equivariantly $\gamma$-positive.
\end{conj}

\acknowledgements{The authors thank Alessio D'Ali, Chris Eur, Luis Ferroni, Matt Larson, Hsin-Chieh Liao,  Diane Maclagan, Matt Maestroni, Jacob Matherne, Connor Simpson, Lorenzo Vecchi and Peter Webb for helpful conversations and references.  They thank Trevor Karn for his wonderful {\tt Sage/Cocalc} code that checks whether a symmetric group representation is a permutation representation.  The authors received partial support from NSF grants 
DMS-1949896, DMS-1745638, DMS-2053288. Second author received partial support from PROM project no. POWR.03.03.00-00-PN13/18.}


\printbibliography
\end{document}
