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\title[Equivariant $\gamma$-positivity]{Equivariant \texorpdfstring{$\gamma$}{gamma}-positivity of Chow rings and augmented Chow rings of matroids}

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\author[H-C. Liao]{\texorpdfstring{Hsin-Chieh Liao \thanks{\href{mailto:liaoh@wustl.edu}{liaoh@wustl.edu}. %Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.
}
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}{Hsin-Chieh Liao 
 %and you
}}

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}

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\abstract{
We prove that the Chow ring and augmented Chow ring of a matroid are equivariant $\gamma$-positive under the action of any group of automorphisms of the matroid. This verifies a conjecture of Angarone, Nathanson, and Reiner. Our method gives an explicit interpretation for the coefficients of the equivariant $\gamma$-expansion, and extends the author's previous results regarding the positivity of the equivariant Charney--Davis quantity of matroids. Applying the theorems to uniform matroids, we obtain interpretations that extend Shareshian and Wachs' Schur-$\gamma$-positivity of the Eulerian and binomial Eulerian quasisymmetric functions, or equivalently, of the cohomologies of the permutahedral and the stellahedral varieties. 
}

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\keywords{matroids, Chow ring, $\gamma$-positivity, equivariant, uniform matroids, symmetric functions}

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\section{Introduction}
Given a finite sequence, there are many features that one can look for in the sequence. A sequence $\{a_i\}_{i=0}^d$ or a polynomial $f(t)=\sum_{i=0}^d a_i t^i$ of degree $d$ is said to be \emph{palindromic} if $a_j=a_{d-j}$ for all $j=0,1,\ldots, d$. It is said to be \emph{unimodal} if there is some $0\le j\le d$ such that $a_0\le a_1\le\cdots\le a_j\ge\cdots\ge a_{d-1}\ge a_d$. A palindromic polynomial $f(t)$ can be uniquely expressed as 
\[
    f(t)=\sum_{k=0}^{\lfloor\frac{d}{2}\rfloor}\gamma_{k}t^k(1+t)^{d-2k}.
\]
We call the right-hand-side the $\gamma$-expansion of $f(t)$, and say that $f(t)$ is \emph{$\gamma$-positive} (or \emph{$\gamma$-nonnegative}) if $\gamma_k\ge 0$ for all $k$. It is not hard to see that a polynomial being $\gamma$-positive implies that it is both palindromic and unimodal. Although the $\gamma$-positivity is interesting to combinatorialists in its own right, this property has a connection to Gal's conjecture in discrete geometry, and hence has drawn much attention from researchers in combinatorics, discrete geometry, and commutative algebra. For more details about $\gamma$-positivity, we refer the readers to the survey \cite{AthanasiadisGamma} by Athanasiadis.

Let $M$ be a loopless matroid on the ground set $E$. The \emph{Chow ring of the matroid $M$} encodes information about the lattice of flats $\L(M)$ and is defined as 
\[
    A(M)\coloneqq \mathbb{R}[x_{F}: F\in\L(M)\setminus\{\emptyset\}]/(I+J)
\]
where $I=(x_Fx_{F'}:F,F' \text{ not comparable})$ and $J=(\sum_{F:i\in F}x_F: i\in E)$. The \emph{augmented Chow ring of $M$} is an extension of $A(M)$ encoding both information from $\L(M)$ and the independence complex $\I(M)$, and is defined as
\[
    \widetilde{A}(M)\coloneqq \mathbb{R}[\{x_F:F\in\L(M)\setminus\{[n]\}\}\cup\{y_i:i\in E\}]/(\widetilde{I}+\widetilde{J})
\]   
where $\widetilde{I}=(x_Fx_{F'}:F,F' \text{ not comparable})+(y_ix_F: i\notin F)$ and $\widetilde{J}=(y_i-\sum_{F:i\notin F}x_F: i\in E)$.
Despite the seemingly complicated presentation, the Chow ring $A(M)$ has a very nice $\mathbb{R}$-basis called the \emph{Feichtner--Yuzvinsky basis}
\begin{equation}\label{eq:ChowFY}
    FY(M)\coloneqq\left\{x_{F_1}^{a_1}x_{F_2}^{a_2}\ldots x_{F_\ell}^{a_\ell}:\substack{~\emptyset=F_0\subsetneq F_1\subsetneq F_2\subsetneq\ldots\subsetneq F_\ell \text{ for } 0\le \ell\le n \\
    1\le a_i\le \rk_M(F_i)-\rk_M(F_{i-1})-1}\right\},
\end{equation}
which is induced by the Gr\"{o}bner basis of the ideal $I+J$ found by Feichtner and Yuzvinsky \cite{FeichtnerYuzvinsky2004}. There is a similar Feichtner--Yuzvinsky basis for the augmented Chow ring $\widetilde{A}(M)$, 
\begin{equation} \label{eq:AugFY}
	\widetilde{FY}(M)\coloneqq\left\{x_{F_1}^{a_1}x_{F_2}^{a_2}\ldots x_{F_\ell}^{a_\ell}: \substack{\emptyset\subsetneq F_1\subsetneq F_2\subsetneq\ldots\subsetneq F_\ell \text{ for }0\le\ell\le n\\
    1\le a_1\le\rk_M(F_1),\\ 
    1\le a_i\le\rk_M(F_i)-\rk_M(F_{i-1})-1 \text{ for }i\ge 2}
    \right\},
\end{equation}
found by the author \cite{Liao2023FPSAC,Liao2023One} and independently by Eur, Huh, and Larson \cite{EHL2022stellahedral}. The Chow rings and augmented Chow rings of matroids play important roles in settling the long-standing Rota--Welsh conjecture and Dowling--Wilson top-heavy conjecture, respectively. They are shown to satisfy the so-called K\"{a}hler package (see \cite{AHK2018} for Chow rings and \cite{BHMPW2020+} for augmented Chow rings). In this package, the Poincar\'{e} duality implies that the Hilbert series of the two rings are palindromic, and the hard Lefschetz theorem implies that the Hilbert series are unimodal. The $\gamma$-positivity of the Hilbert series was shown by Ferroni, Matherne, Stevens, and Vecchi \cite{Ferroni2024hilbert}, and independently by Wang \cite[p.33]{Ferroni2024hilbert}, using the semismall decomposition introduced in \cite{BHMPW2020}. However, no interpretation of the coefficients $\gamma_k$ was known. 

In \cite{ANR2023PermRepchow}, Angarone, Nathanson, and Reiner further conjectured that the matroid Chow rings, under the action of groups of automorphisms of the corresponding matroid, are equivariant $\gamma$-positive (see \Cref{sec:EqGamma} for the definition). In this extended abstract, we present a proof of this conjecture using only the \emph{Feichtner--Yuzvinsky bases} and a result in Stanley's classical paper \cite{Stanley1982GroupPoset} (see \Cref{thm:BetaGenRep}). Our method is motivated by the author's proof of the positivity of the equivariant Charney--Davis quantity of matroids in \cite{Liao2023Two}, and it gives an explicit interpretation of the $\gamma$-coefficients, which is stated in \Cref{sec:EqGamma}. We present the proof in \Cref{sec:Proof}. In \Cref{sec:Uniform}, we apply our results to uniform matroids, in which case the Hilbert series of the (augmented) Chow rings have been studied in Hameister, Rao, and Simpson \cite{HameisterRaoSimpson2021}, Liao \cite{Liao2023Two}, and Hoster \cite{Hoster2024ChowUniform}. Our results extend Shareshian and Wachs' Schur-$\gamma$-positivity of the (binomial) Eulerian quasisymmetric functions, and equivalently the equivariant $\gamma$-positivities on the cohomologies of the permutahedral and the stellohedral varieties in \cite{ShareshianWachs2010,ShareshianWachs2020}. 

In a recent preprint, Stump \cite{StumpChow2024} also independently gives an interpretation for the coefficients of the (non-equivariant) $\gamma$-expansion for Chow rings and augmented Chow rings of matroids using a certain evaluation of the \emph{Poincar\'{e} extended $\mathbf{ab}$-index} introduced in \cite{DorpalenMaglioneStump2023ab}. He interprets the $\gamma$-coefficient as the number of maximal chains in $\L(M)$ with fixed descent set under some $R$-labeling. This agrees with the result in this paper, following from Stanley \cite[Theorem 3.1]{StanleyRlabel1974}. %In future work, we would like to study the relationship between the Feichtner--Yuzvinsky basis and the Poincar\'{e} extended $\mathbf{ab}$-index, and this extended $\mathbf{ab}$-index in the equivariant setting. 

In \cite{Ferroni2024hilbert}, the Hilbert series of Chow rings and augmented Chow rings of matroids are called Chow and augmented Chow polynomial of matroids. In an upcoming paper \cite{FMVChowPoly}, Ferroni, Matherne, and Vecchi generalize Chow and augmented Chow polynomials to arbitrary graded posets. They show that Chow and augmented Chow polynomials are $\gamma$-positive for Cohen--Macaulay posets using a method similar to this work.

\section{Group actions on posets}
For non-negative integers $a\le b$, let $[a,b]\coloneqq\{a,a+1,\ldots, b\}$; and in particular, if $a=1$, we write $[b]$ instead of $[1,b]$.

Let $P$ be a finite poset with a unique minimal element $\hat{0}$ and a unique maximal element $\hat{1}$. If $P$ is graded of rank $n$ with rank function $\rk_P: P\longrightarrow [0,n]$, then for $S\subseteq [n-1]$, the \emph{rank-selected subposet} of $P$ is
\[
    P_S\coloneqq\{x\in P~:~\rk_P(x)\in S\}\cup\{\hat{0},\hat{1}\}.
\]
Consider a group $G$ of automorphisms of $P$ acting on the poset $P$. The action of $G$ preserves the rank of the elements in $P$; hence for any $S\subseteq [n-1]$, the group $G$ permutes the maximal chains in $P_S$. Let us denote $\alpha_P(S)$ the permutation representation of $G$ generated by the maximal chains in $P_S$. Consider the virtual representation
\[
    \beta_P(S)\coloneqq\sum_{T\subseteq S}(-1)^{|S|-|T|}\alpha_P(T).
\]
By M\"{o}bius inversion, we also have $\alpha_P(S)=\sum_{T\subseteq S}\beta_P(T)$.


\begin{thm}[Stanley \cite{Stanley1982GroupPoset}] \label{thm:BetaGenRep}
If $P$ is Cohen--Macaulay, then $\beta_P(S)$ is a genuine representation of $G$ and 
\[
    \beta_P(S)\cong_G \tilde{H}_{|S|-1}(P_S).
\]
\end{thm}
\begin{rem}
The dimension (rank) of $\beta_P(S)$ is given by the M\"{o}bius function 
{\parfillskip=0pt \par}\noindent
$(-1)^{|S|-1}\mu(P_S)$ and is the number of maximal chains in $P$ whose Jordan--H\"{o}lder sequences have descent set $S$. See Stanley \cite[Theorem 3.1]{StanleyRlabel1974} or Bj\"{o}rner \cite[Theorem 2.7]{BjornerCMposet1980}.
\end{rem}

Throughout this paper, the poset $P$ we care about is always the lattice of flats of a matroid. It is well known that the lattice of flats of a matroid is always Cohen--Macaulay.



\section{Main results on equivariant \texorpdfstring{$\gamma$}{gamma}-positivity}\label{sec:EqGamma}

Let $G$ be a finite group. For a $\mathbb{C}G$-module $M$, let $[M]$ be the isomorphism class of $\mathbb{C}G$-modules containing $M$. 

\begin{defi}
The \emph{Grothendieck ring} $R_{\mathbb{C}}(G)$ of $\mathbb{C}G$-modules is a free abelian group having the transversal of isomorphism classes of simple $\mathbb{C}G$-modules $\{[S_i],\ldots,[S_{cc(G)}]\}$ as a $\mathbb{Z}$-basis, where $cc(G)$ is the number of conjugacy classes of $G$, with the addition and multiplication relations given by 
\[
    [S_i]+[S_j]\coloneqq [S_i\oplus S_j] \quad  \text{ and } \quad  [S_i]\cdot[S_j]\coloneqq[S_i\otimes_{\mathbb{C}} S_j]
\]
for $1\le i,j\le cc(G)$ and extended linearly over $\mathbb{C}$. Now every element $A$ in $R_{\mathbb{C}}(G)$ has a unique expression as $a=\sum_{i=1}^{cc(G)}a_i[S_i]$. We say $A$ is a \emph{genuine representation} of $G$ if $a_i\ge 0$ for all $i$, denoted by $A\ge_{R_{\mathbb{C}}(G)}0$.
\end{defi}

Given a graded $\mathbb{C}G$-module $V=\bigoplus_{i} V_i$, define the \emph{equivariant Hilbert series} of $V$ to be the formal power series 
\[
    \Hilb_{G}(V,t)=\sum_{i}[V_i] t^i\in R_{\mathbb{C}}(G)[[t]].
\]
 
\begin{defi}
For a finitely dimensional graded $\mathbb{C}G$-module $V=\bigoplus_{i=0}^d V_i$, we say that $V$ is ($G$-~)\emph{equivarant $\gamma$-positive} if its equivariant Hilbert series can be expressed as 
\[
    \Hilb_G(V,t)=\sum_{i=0}^d[V_i]t^i=\sum_{k=0}^{\lfloor\frac{d}{2}\rfloor}\gamma_k t^k(1+t)^{d-2k}
\]
and the uniquely defined coefficient $\gamma_k\in R_{\mathbb{C}}(G)$ is a class of a genuine representation of $G$ over $\mathbb{C}$ for all $k$, i.e. $\gamma_k\ge_{R_{\mathbb{C}}(G)}0$ for all $k$.    
\end{defi}

Let $M$ be a loopless matroid of rank $r$ on the ground set $E$ with the lattice of flats $\L(M)$. Write $A(M)_\mathbb{C}\coloneqq A(M)\otimes_{\mathbb{R}}\mathbb{C}$ and $\widetilde{A}(M)_\mathbb{C}\coloneqq \widetilde{A}(M)\otimes_{\mathbb{R}}\mathbb{C}$. 

From now on, we let $G$ be a group of automorphisms of matroid $M$.
Consider the induced action of $G$ on the Chow ring $A(M)_{\mathbb{C}}=\bigoplus_{i=0}^{r-1}A_{\mathbb{C}}^i$ which gives each graded piece $A_{\mathbb{C}}^i$ a $\mathbb{C}G$-module structure. The following conjecture was proposed by Angarone, Nathanson, and Reiner \cite{ANR2023PermRepchow}.

\begin{conj}[{\cite[Conjecture 5.2]{ANR2023PermRepchow}}] \label{conj:Equiv}
For any group $G$ of automorphisms of $M$, the matroid Chow ring $A(M)_{\mathbb{C}}=\bigoplus_{i=0}^{r-1}A^i_{\mathbb{C}}$ is ($G$-)equivariant $\gamma$-positive. 
\end{conj}

We prove \Cref{conj:Equiv} and the equivariant $\gamma$-positivity of the augmented Chow ring $\widetilde{A}(M)$ and obtain explicit $G$-representations as the coefficients in the equivariant $\gamma$-expansions. Our method uses only the Feichtner--Yuzvinsky bases \eqref{eq:ChowFY} and \eqref{eq:AugFY} and \Cref{thm:BetaGenRep}. For $S\subseteq [n]$, let $\Stab(S)$ denote the collection of subsets of $S$ containing no consecutive integers.

\begin{thm} \label{thm:GammaChow}
The Chow ring $A(M)$ is equivariant $\gamma$-positive with the following $\gamma$-expansion
\begin{align*}
    \Hilb_{G}(A(M)_{\mathbb{C}},t) 
    &=\sum_{S\in\Stab([2,r-1])}[\tilde{H}_{|S|-1}(\L(M)_S)] t^{|S|}(1+t)^{r-1-2|S|}\\
    &=\sum_{k=0}^{\floor{\frac{r-1}{2}}}\left(\sum_{\substack{S\in\Stab([2,r-1])\\ |S|=k}}[\tilde{H}_{|S|-1}(\L(M)_S)]\right)t^k(1+t)^{r-1-2k}.
\end{align*}
\end{thm}

\begin{thm} \label{thm:GammaAug}
The augmented Chow ring $\widetilde{A}(M)_{\mathbb{C}}$ is equivariant $\gamma$-positve with    
\begin{align*}
    \Hilb_{G}(\widetilde{A}(M)_{\mathbb{C}},t) 
    &=\sum_{S\in\Stab([r-1])}[\tilde{H}_{|S|-1}(\L(M)_S)] t^{|S|}(1+t)^{r-2|S|}\\
    &=\sum_{k=0}^{\floor{\frac{r}{2}}}\left(\sum_{\substack{S\in\Stab([r-1])\\ |S|=k}}[\tilde{H}_{|S|-1}(\L(M)_S)]\right)t^k(1+t)^{r-2k}.
\end{align*}

\end{thm}

\begin{rem}
\Cref{thm:GammaChow} and \Cref{thm:GammaAug} generalize the author's previous results in \cite{Liao2023Two}, Theorem 4.9 and Theorem 4.12 respectively,  regarding the equivariant Charney-Davis quantities of Chow and augmented Chow rings of matroids.   
\end{rem}

\section{Proof of \texorpdfstring{\Cref{thm:GammaChow}}{Theorem 3.4}} \label{sec:Proof}

In this section, we present our proof of \Cref{thm:GammaChow}, the equivariant $\gamma$-positivity of Chow rings of matroids. The augmented case, \Cref{thm:GammaAug}, can be proved in a similar way.


It is not hard to check that under the action of $G$, the Feichtner--Yuzvinsky bases $FY(M)$ and $\widetilde{FY}(M)$ are permutation bases for $A(M)_{\mathbb{C}}$ and $\widetilde{A}(M)_{\mathbb{C}}$, respectively.

\begin{proof}[Proof of \Cref{thm:GammaChow}]
Recall that the permutation basis $FY(M)$ consisting of monomials $x_{F_1}^{a_1}x_{F_2}^{a_2}\ldots x_{F_{\ell}}^{a_{\ell}}$ for any chains (including empty chain)
\[
    \emptyset\neq F_1\subsetneq F_2\subsetneq\ldots\subsetneq F_{\ell}\subseteq{E}
\]
such that $1\le a_j\le \rk(F_j)-\rk(F_{j-1})-1$ for all $j$.
For a subset $S=\{s_1<s_2<\ldots<s_{\ell}\}\subseteq [r-1]$, the permutation $\mathbb{C}G$-module generated by chains in $\L(M)$ whose rank set is $S$ is
\begin{align*}
    \alpha_{\L(M)}(S)
    &=\mathbb{C}G\{F_1\subsetneq \ldots\subsetneq F_{\ell}: \rk(F_i)=s_i~\forall i\}.\\
    &\cong \mathbb{C}G\{F_1\subsetneq \ldots\subsetneq F_{\ell}\subsetneq E: \rk(F_i)=s_i~\forall i\}
\end{align*}
Therefore, $A(M)$ as a $\mathbb{C}G$-module is the direct sum of $\alpha_{\L(M)}(S)$ with some multiplicities for $S\subseteq [r-1]$. Hence we have 
\begin{equation}\label{eq:Hilb}
    \Hilb_G(A(M)_{\mathbb{C}},t)=\sum_{S\subseteq [r-1]}\phi_{S,r}(t)[\alpha_{\L(M)}(S)]
\end{equation}
for some polynomials $\phi_{S,r}(t)$. To see what the polynomial $\phi_{S,t}(t)$ is, we consider a map $f:FY(M)\longrightarrow 2^{[r-1]}$ defined by $f(x_{F_1}^{a_1}x_{F_2}^{a_2}\ldots x_{F_{\ell}}^{a_{\ell}}x_E^i)=\{\rk(F_1),\rk(F_2),\ldots,\rk(F_{\ell})\}$ where all flats $F_i$ are distinct and are not $E$. Since for any $S=\{s_1<\cdots<s_{\ell}\}\subseteq [r-1]$, 
\[
    f^{-1}(S)=\left\{x_{F_1}^{a_1}x_{F_2}^{a_2}\ldots x_{F_{\ell}}^{a_{\ell}}x_E^i : \substack{\emptyset=F_0\subsetneq F_1\subsetneq\cdots\subsetneq F_{\ell}\subsetneq E \text{ with }\rk(F_j)=s_j\\
    1\le a_j\le s_j-s_{j-1}-1  \text{ for }j=1,2\ldots,\ell\\
    0\le i\le r-s_{\ell}-1
    }\right\},
\]
the polynomials that count the multiplicities are
\begin{align*}
    \phi_{S,r}(t)
    &=(t+t^2+\cdots+t^{s_1-1})\ldots(t+t^2+\cdots+t^{s_{\ell}-s_{\ell-1}-1})(1+t+\cdots+t^{r-s_{\ell}-1})\\
    &=t^{|S|}[s_1-1]_t[s_2-s_1-1]_t\ldots[s_{\ell}-s_{\ell-1}-1]_t[r-s_{\ell}]_t,
\end{align*}
where $[n]_t\coloneqq 1+t+\ldots+t^{n-1}$ for $n\ge 1$ and $[0]_t\coloneqq 0$.  
Note that $\phi_{S,r}(t)=0$ if $s_j-s_{j-1}=1$ for some $1\le j\le \ell$, and $[r-s_{\ell}]_t=1$ if $r-1\in S$. Therefore, $\phi_{S,r}(t)$ takes nonzero value when $S\in\Stab([2,r-1])$. The identity \eqref{eq:Hilb} can be rewritten as follows:
\begin{align*}
    \Hilb_G(A(M)_{\mathbb{C}},t)
    &=\sum_{S\in \Stab([2,r-1])}\phi_{S,r}(t)[\alpha_{\L(M)}(S)]
    =\sum_{S\in \Stab([2,r-1])}\phi_{S,r}(t)\sum_{T:T\subseteq S}[\beta_{\L(M)}(T)]\\
    &=\sum_{T\in\Stab([2,r-1])}\left(\sum_{T\subseteq S\subseteq [r-1]}\phi_{S,r}(t)\right)[\beta_{\L(M)}(T)].
\end{align*}
Surprisingly, it turns out that  
\[
    \sum_{T\subseteq S\subseteq [r-1]}\phi_{S,r}(t)=t^{|T|}(1+t)^{r-1-2|T|}.
\]
This fact and its proof will be presented later in \Cref{lem:ChowIdentity}. Combining with \Cref{thm:BetaGenRep}, which states that $\beta_{\L(M)}(T)$ is isomorphic to $\tilde{H}_{|T|-1}(\L(M)_T)$ as $\mathbb{C}G$-modules, we obtain
\begin{align*}
    \Hilb_G(A(M)_{\mathbb{C}},t)
    &=\sum_{T\in\Stab([2,r-1])}[\beta_{\L(M)}(T)]t^{|T|}(1+t)^{r-1-2|T|}\\
    &=\sum_{T\in\Stab([2,r-1])}[\tilde{H}_{|T|-1}(\L(M)_T)]t^{|T|}(1+t)^{r-1-2|T|}.
\end{align*}
\end{proof}


\begin{lem} \label{lem:ChowIdentity}
For $n\ge 2$ and any subset $T\in\Stab([2,n-1])$, 
\begin{equation} \label{eq:ChowIdentity}
    \sum_{T\subseteq S\subseteq [n-1]}\phi_{S,n}(t)=t^{|T|}(1+t)^{n-1-2|T|},
\end{equation}
where 
\[
    \phi_{S,n}(t)=t^{|S|}[s_1-1]_t[s_2-s_1-1]_t\ldots[s_\ell-s_{\ell-1}-1]_t[n-s_\ell]_t.
\]
Note that $\phi_{S,n}(t)=0$ if $S\notin\Stab([2,n-1])$.

\end{lem}


To prove \Cref{lem:ChowIdentity}, we will use a combinatorial model motivated by Angarone, Nathanson, and Reiner's combinatorial proof of the permutation representation lift of the Poincar\'{e} duality and the Hard Lefschetz theorem in \cite[Theorem 1.1]{ANR2023PermRepchow}.

\begin{defi} \label{def:ChowModel}
For $n\ge 2$, consider sequences $\mathbf{w}=w_1 w_2\ldots w_{n-1}$ in three symbols $\bullet$, $\times$, and a blank space , defined as follows:
\begin{itemize}
    \item[(W1)] If $w_i=\bullet$, then $w_{i-1}=\times$ (hence $w_1\neq\bullet$ and there is no consecutive $\bullet$ in $\mathbf{w}$).
\end{itemize}
Say that $S=\{s_1<s_2<\ldots<s_\ell\}\subseteq \Stab([2,n-1])$ is the set of indices $s$ with $w_s=\bullet$.
\begin{itemize}
    \item[(W2)] If there is $s_i<j<s_{i+1}$ such that $w_j=\times$, then $w_j=w_{j+1}=\ldots=w_{s_{i+1}-1}=\times$.
    \item[(W3)] If there is $s_{\ell}<j<n$ such that $w_j=\times$, then $w_j=w_{j+1}=\ldots=w_{n-1}=\times$.
\end{itemize}
Denote by $\mathcal{W}_n$ the set of such sequences of length $n-1$. For $\mathbf{w}\in\mathcal{W}_n$, let $\Dot(\mathbf{w})=\{i\in [n-1]: w_i=\bullet\}$ and let $\cro(\mathbf{w})$ be the number of $\times$ in $\mathbf{w}$.
\end{defi}

\begin{proof}[Proof of \Cref{lem:ChowIdentity}]
We show that \eqref{eq:ChowIdentity} counts in two ways the sum of $t^{\cro(\mathbf{w})}$ for sequences $\mathbf{w}$ in $\mathcal{W}_n$ satisfying $\Dot(\mathbf{w})\supseteq T$.

First, for fixed $S\supseteq T$, $S=\{s_1<\ldots<s_\ell\}\in\Stab([2,n-1])$, a sequence $\mathbf{w}=w_1\ldots w_{n-1}\in \mathcal{W}_n$ with $\Dot(\mathbf{w})=S$ has $w_{s_{j}}=\bullet$ for all $j$. Between $w_{s_{j-1}}$ and $w_{s_{j}}$, we have $w_{s_{j}-1}=\times$ and we can choose to place from $0$ to $s_j-s_{j-1}-1$ consecutive $\times$s on the left of $\times=w_{s_{j}-1}$. Hence the section between $w_{s_{j-1}}$ and $w_{s_j}$ contributes $t[s_j-s_{j-1}-1]_t$. In total, we have
\begin{align*}
    \sum_{\substack{\mathbf{w}\in\mathcal{W}_n\\ \Dot(\mathbf{w})=S}}t^{\cro(\mathbf{w})}
    &=t^{|S|}[s_1-1]_t[s_2-s_1-1]_t\ldots[s_\ell-s_{\ell-1}-1]_t[n-s_\ell]_t
    % \begin{cases}
    % t^{|S|}[s_1-1]_t[s_2-s_1-1]_t\ldots[n-s_{\ell-1}-2]_t & \text{ if }n-1\in S\\
    % t^{|S|}[s_1-1]_t[s_2-s_1-1]_t\ldots[s_\ell-s_{\ell-1}-1]_t[n-s_\ell]_t & \text{ if }n-1\notin S
    % \end{cases}\\
    =\phi_{S,n}(t).
\end{align*}
Therefore, 
$
    \sum_{\substack{\mathbf{w}\in\mathcal{W}_n\\ \Dot(\mathbf{w})\supseteq T}}t^{\cro(\mathbf{w})}=\sum_{T\subseteq S\subseteq [n-1]}\phi_{S,n}(t).
$

On the other hand, for any sequence $\mathbf{w}=w_1\ldots w_{n-1}\in\mathcal{W}_n$ with $\Dot(\mathbf{w})\supseteq T$, since $w_i=\bullet$ for $i\in T$, by the definition of the sequence we have $w_{i-1}=\times$ for $i\in T$. These $\times$ contribute the factor $t^{|T|}$ on the right hand side of \eqref{eq:ChowIdentity}. In the remaining $n-1-2|T|$ positions, we can choose to place a $\times$ or not, which gives the factor $(1+t)^{n-1-2|T|}$ on the right-hand side of \eqref{eq:ChowIdentity}. After this process, there is a unique way to add $\bullet$ so that the sequence is in $\mathcal{W}_n$. This implies that
\[
    \sum_{\substack{\mathbf{w}\in\mathcal{W}_n\\ \Dot(\mathbf{w})\supseteq T}}t^{\cro(\mathbf{w})}=t^{|T|}(1+t)^{n-1-2|T|}.
\]    
\end{proof}

\begin{exa}
Here is an example of the procedure in the proof of \Cref{lem:ChowIdentity}. Let $n=12$, $T=\{3,9\}\subseteq\Stab([2,11])$. 

In the first part of the two-way counting, say $S=\{3,5,9\}\supseteq T$, so we place $\bullet$ at $3$, $9$ and $\times$ at $2$, $8$; and \textcolor{blue}{$\bullet$} at $5$, \textcolor{blue}{$\times$} at $4$ as follows:
\begin{center}
\begin{tabular}{m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm}}
    1 & 2 & 3 & 4 & 5 &6 &7 &8 & 9 & 10 &11\\
    $\underline{~~~}$ & $\times$ 
 & $\bullet$ & \textcolor{blue}{$\times$} & \textcolor{blue}{$\bullet$} & $\underline{~~~}$ & $\underline{~~~}$ & $\times$  & $\bullet$ & $\underline{~~~}$ & $\underline{~~~}$\\
 
 \multicolumn{3}{|c|}{\textcolor{red}{$t[2]_t$}} & \multicolumn{2}{|m{1 cm}|}{\textcolor{red}{$t[1]_t$}} & \multicolumn{4}{|c|}{\textcolor{red}{$t[3]_t$}} & \multicolumn{2}{|c|}{\textcolor{red}{$[3]_t$}}
\end{tabular}
\end{center}
Then for example, between $w_5=\bullet$ and $w_9=\bullet$, we can have either $w_6=w_7=$blank spaces, $w_8=\times$; or $w_6=$blank space, $w_7=w_8=\times$; or $w_6=w_7=w_8=\times$. These three cases contribute $t+t^2+t^3=t[3]_t$. Hence we have
\[
    \sum_{\substack{\mathbf{w}\in\mathcal{W}_{12}\\ \Dot(\mathbf{w})=\{3,5,9\}}}t^{\cro(\mathbf{w})}=t[2]_t\cdot t[1]_t\cdot t[3]_t\cdot [3]_t=t^3[2]_t[1]_t[3]_t[3]_t.
\]

In the second part of the two-way counting, we count sequences $\mathbf{w}$ with $w_3=w_9=\bullet$ and $w_2=w_8=\times$. We can choose each of the remaining $7$ positions to be either a $\times$ or a blank space. For each choice there is exactly one way to add $\bullet$ to complete the sequence in $\mathcal{W}_{12}$. Hence this implies 
\[
    \sum_{\substack{\mathbf{w}\in\mathcal{W}_{12}\\ \Dot(\mathbf{w})\supseteq\{3,9\}}}t^{\cro(\mathbf{w})}=t^{|\{2,8\}|}(1+t)^{|\{1,4,5,6,7,10,11\}|}=t^2(1+t)^{12-1-2\cdot 2}.
\]
For example, say we choose $w_1=w_4=w_5=w_{10}=\textcolor{blue}{\times}$, then we have 
\begin{center}
\begin{tabular}{m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm}}
    1 & 2 & 3 & 4 & 5 &6 &7 &8 & 9 & 10 &11\\
    \textcolor{blue}{$\times$} & $\times$ 
 & $\bullet$ & \textcolor{blue}{$\times$} & \textcolor{blue}{$\times$} & $\underline{~~~}$ & $\underline{~~~}$ & $\times$  & $\bullet$ & \textcolor{blue}{$\times$} & \underline{~~~}
\end{tabular}.
\end{center}
The only way to complete the above sequence into a sequence in $\mathcal{W}_{12}$ by adding $\bullet$ is as follows:
\begin{center}
\begin{tabular}{m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm} m{0.5 cm}}
    1 & 2 & 3 & 4 & 5 &6 &7 &8 & 9 & 10 &11\\
    \textcolor{blue}{$\times$} & $\times$ 
 & $\bullet$ & \textcolor{blue}{$\times$} & \textcolor{blue}{$\times$} & $\textcolor{red}{\bullet}$ & $\underline{~~~}$ & $\times$  & $\bullet$ & \textcolor{blue}{$\times$} & $\textcolor{red}{\bullet}$
\end{tabular}
\end{center}

\end{exa}

The augmented case can be proved similarly with a slightly different combinatorial model.


\section{Uniform matroids and \texorpdfstring{$q$}{q}-uniform matroids} \label{sec:Uniform}
In this section, we apply \Cref{thm:GammaChow} and \Cref{thm:GammaAug} to the uniform matroid $U_{r,n}$ of rank $r$ and the symmetric group $\S_n$. In this case, the lattice of flats $\L(U_{r,n})$ is the rank-selected subposet $(B_n)_{[r-1]}$ of the Boolean lattice $B_n=(2^{[n]},\subseteq)$. 
%The equivariant $\gamma$-coefficient $\tilde{H}_{|R|-1}((B_n)_R)$ has a well-studied $\mathbb{C}\S_n$-module structure for any $R\subseteq[n-1]$. 
For the background on representation theory of $\S_n$, standard Young tableaux, and symmetric functions in this section, we refer the readers to Wachs \cite{WachsTool2007} and Stanley \cite{StanleyEC2}. For an introduction to the permutation statistics $\inv$, $\maj$, etc., see Chapter 1 of Stanley \cite{StanleyEC1}.

Let $\ch$ be the \emph{Frobenius characteristic map}. For a graded $\mathbb{C}\S_n$-module $V=\bigoplus_i V_i$, the \emph{graded Frobenius series} of $V$ is defined to be $\grFrob(V,t)\coloneqq \sum_{i}\ch(V_i)t^i$.

Let $\lambda$, $\mu$ be two partitions such that $\mu\subseteq\lambda$ (i.e. $\mu_i\le \lambda_i$ for all $i$). The connected skew shape $\lambda/\mu$ is said to be a \emph{ribbon} if two consecutive rows always overlap in exactly one cell. A ribbon $H_{R,n}$ can be described in terms of the number $n$ of its cells and its \emph{descent set} $R\subseteq [n-1]$. Given a ribbon of $n$ cells, we number its cells from $1$ to $n$, filling left to right in each row and filling the rows from bottommost to topmost. We call a cell $i$ a \emph{descent} of the ribbon if cell $i+1$ is directly above cell $i$. Then the collection of all its descents is called its \emph{descent set}.

\begin{exa}
Let $n=7$. Then the diagrams of $H_{\{2,4\},7}$ and $H_{\{1,3\},7}$ are 
\[
    \ytableausetup{smalltableaux}
    \begin{ytableau}
    \none & \none & 5 & 6 & 7\\
    \none & 3 & *(green)4\\ 
    1     & *(green)2 
    \end{ytableau}  
    \quad \text{ and }\quad 
    \begin{ytableau}
    \none & 4 & 5 & 6 & 7\\
    2     & *(green)3 \\
    *(green) 1
    \end{ytableau}.
\]
\end{exa}
Let $S^{H_{R,n}}$ be the \emph{Specht module} of a ribbon shape, also known as the \emph{Foulkes representation} of $\S_n$, with $\ch(S^{H_{R,n}})=s_{H_{R,n}}$ the \emph{ribbon Schur function}.
%We summarize some facts of this representation from  \cite[Theorem 4.3]{Stanley1982GroupPoset} and \cite[Theorem 3.4.4]{WachsTool2007} in the following theorem. 



% \begin{thm} \label{thm:RankHomologyBn}
% Let $R\subseteq [n-1]$. 
% \begin{itemize}
%     \item[(i)] Then
% \[
%     \tilde{H}_{|R|-1}({(B_n)}_R)\cong_{\S_n} S^{H_{R,n}}\cong_{\S_n}\bigoplus_{\lambda\vdash n}b_{\lambda}(R)S^{\lambda},
% \]    
% where $b_{\lambda}(R)$ is the number of $P\in SYT(\lambda)$ for which $\DES(P)=R$.
%     \item[(ii)] Let $\mathbb{P}$ denote the set of positive integers and $\mathbb{P}_n$ the set of words of length $n$ over $\mathbb{P}$. For $w=w_1\ldots w_n\in\mathbb{P}_n$, we say $i\in\DES(w)$ if $w_i>w_{i+1}$ and write $\x_w=x_{w_1}\ldots x_{w_n}$. Then the Frobenius characteristic is the ribbon Schur function
% \[
%     \ch(\tilde{H}_{|R|-1}({(B_n)}_R))=s_{H_{R,n}}=\sum_{\substack{w\in\mathbb{P}_n\\ \DES(w)=R}}\mathbf{x}_w,
% \]

% \end{itemize}

% % The dimension is given by
% % \[
% %   \dim \tilde{H}_{|R|-1}({(B_n)}_R)=\sum_{\lambda\vdash n}b_R(\lambda)f^{\lambda}=|\{\sigma\in\S_n: \DES(\sigma)=R\}|.
% % \]
% \end{thm}

Applying \Cref{thm:GammaChow} to the uniform matroids $U_{r,n}$ then from \cite[Theorem 4.3]{Stanley1982GroupPoset} or \cite[Theorem 3.4.4]{WachsTool2007}, one obtains the graded Frobenius series of $A(U_{r,n})_{\mathbb{C}}$. 
Applying the operator  $\prod_{i=1}^n(1-q^i)\ps(-)$, where $\ps$ is the stable principle specialization, to the graded Frobenius series gives the Hilbert series of the $q$-Uniform matroid $U_{r,n}(q)$ (See \cite[Section 3]{Liao2023Two} for the explanation). Note that the stable principle specialization on the ribbon Schur function gives
\[
    \prod_{i=1}^n(1-q^i)\ps(s_{H_{R,n}})=\sum_{\substack{\sigma\in\S_n \\ \DES(\sigma)=R}}q^{\maj(\sigma^{-1})}=\sum_{\substack{\sigma\in\S_n \\ \DES(\sigma)=R}}q^{\inv(\sigma)}
\]
(see \cite[Theorem 4.4]{ShareshianWachs2020} and its proof for a detailed discussion of the above identity). 

The following corollary generalizes Corollary 4.10 and 4.11 in the author's previous work \cite{Liao2023Two}. It also extends Shareshian and Wachs' Theorem 7.3 in \cite{ShareshianWachs2010} and Corollary 3.2, Theorem 4.4, Theorem 6.1 in \cite{ShareshianWachs2020} (when $r=n$ or $n-1$).

\begin{cor} \label{cor:ChowUniform}
For any positive integer $n$ and $1\le r\le n$,
\begin{align*}
    \grFrob(A(U_{r,n})_{\mathbb{C}},t)
    &=\sum_{R\in\Stab([2,r-1])}s_{H_{R,n}}t^{|R|}(1+t)^{r-1-2|R|}\\
    &=\sum_{k=0}^{\lfloor\frac{r-1}{2}\rfloor}\left(\sum_{\substack{R\in\Stab([2,r-1])\\ |R|=k}}s_{H_{R,n}}\right)t^k(1+t)^{r-1-2k}
\end{align*}
and 
\begin{align*}
    \Hilb(A(U_{r,n}(q)),t)
    =\sum_{k=0}^{\lfloor\frac{r-1}{2}\rfloor}\xi_{r,n,k}(q)t^k(1+t)^{r-1-2k}
\end{align*}
where 
\[
    \xi_{r,n,k}(q)=\sum_{\sigma}q^{\maj(\sigma^{-1})}=\sum_{\sigma}q^{\inv(\sigma)}
\]
and the sum runs through all $\sigma\in\S_n$ for which $\DES(\sigma)\in \Stab([2,r-1])$ has $k$ elements.
\end{cor}

% \begin{rem}
% \Cref{cor:ChowUniform} generalizes Corollary 4.10 and 4.11 in the author's previous work \cite{Liao2023Two}. \Cref{cor:ChowUniform} also extends Theorem 7.3 in \cite{ShareshianWachs2010} and Corollary 3.2, Theorem 4.4, Theorem 6.1 in \cite{ShareshianWachs2020} (when $r=n$ or $n-1$). 
% \end{rem}

Similarly, for the augmented case, the following corollary generalizes Corollary 4.13 and 4.14 in the author's previous work in \cite{Liao2023Two}. It also extends Shareshian and Wachs' Corollary 3.2, Theorem 3.4, 4.4, 4.5, and Corollary 5.4 in \cite{ShareshianWachs2020} (when $r=n$ or $n-1$).

\begin{cor} \label{cor:AugUniform}
For any positive integer $n$ and $1\le r\le n$,
\begin{align*}
    \grFrob(\widetilde{A}(U_{r,n})_{\mathbb{C}},t)
    &=\sum_{R\in\Stab([r-1])}s_{H_{R,n}}t^{|R|}(1+t)^{r-2|R|}\\
    &=\sum_{k=0}^{\lfloor\frac{r}{2}\rfloor}\left(\sum_{\substack{R\in\Stab([r-1])\\ |R|=k}}s_{H_{R,n}}\right)t^k(1+t)^{r-2k}
\end{align*}
and 
\[
    \Hilb(\widetilde{A}(U_{r,n}(q)),t)=\sum_{k=0}^{\lfloor\frac{r}{2}\rfloor}\widetilde{\xi}_{r,n,k}(q)t^k(1+t)^{r-2k},
\]
where 
\[
    \widetilde{\xi}_{r,n,k}(q)=\sum_{\sigma}q^{\maj(\sigma^{-1})}=\sum_{\sigma}q^{\inv(\sigma)}
\]
and the sum runs through all $\sigma\in\S_n$ for which $\DES(\sigma)\in\Stab([r-1])$ has $k$ elements.
\end{cor}

% \begin{rem}
% \Cref{cor:AugUniform} generalizes Corollary 4.13 and 4.14 in the author's previous work in \cite{Liao2023Two}. \Cref{cor:AugUniform} also extends Corollary 3.2, Theorem 3.4, 4.4, 4.5, and Corollary 5.4 in \cite{ShareshianWachs2020} (when $r=n$ or $n-1$).
% \end{rem}

For a partition $\lambda$ of $n$, denoted by $\lambda\vdash n$, let $SYT(\lambda)$ be the set of standard Young tableaux of shape $\lambda$. For $P\in SYT(\lambda)$, let $\DES(P)$ be the set of entries (called descents) $i\in[n-1]$ such that $i+1$ appears in a lower row than $i$ in $P$, and $\des(P)=|\DES(P)|$. Let $\grFrob(A(U_{r,n})_{\mathbb{C}},t)=\sum_{\lambda\vdash n} P_{\lambda}^r(t)s_{\lambda}$ and $\grFrob(\widetilde{A}(U_{r,n})_{\mathbb{C}},t)=\sum_{\lambda\vdash n}\widetilde{P_\lambda^r}(t)s_{\lambda}$. The following result generalizes Corollary 2.41 in Athanasiadis \cite{AthanasiadisGamma}.
\begin{cor}\label{cor:UniformSchurCoeff}
For $\lambda\vdash n$, we have
\[
    P_{\lambda}^r(t)=\sum_{\substack{P\in SYT(\lambda)\\ \DES(P)\in\Stab([2,r-1])}} t^{\des(P)}(1+t)^{r-1-2\des(P)}=\sum_{k=0}^{\lfloor\frac{r-1}{2}\rfloor}\xi_{r,\lambda,k}~ t^k(1+t)^{r-1-2k}
\]
and 
\[
    \widetilde{P}_{\lambda}^r(t)=\sum_{\substack{P\in SYT(\lambda)\\ \DES(P)\in\Stab([r-1])}} t^{\des(P)}(1+t)^{r-2\des(P)}=\sum_{k=0}^{\lfloor\frac{r}{2}\rfloor}\widetilde{\xi}_{r,\lambda,k}~ t^k(1+t)^{r-2k}
\]
where $\xi_{r,\lambda,k}$ (respectively, $\widetilde{\xi}_{r,\lambda,k}$) is the number of tableaux $P\in SYT(\lambda)$ for which $\DES(P)\in\Stab([2,r-1])$ (respectively, $\Stab([r-1])$) has $k$ elements. In particular, $P_\lambda^r(t)$ and $\widetilde{P}_{\lambda}^r(t)$ are $\gamma$-positive for every $r$ and every partition $\lambda$.
\end{cor}


\acknowledgements{
The author thanks Victor Reiner for suggesting the problem of extending the equivariant Charney-Davis quantity results in \cite{Liao2023Two} which motivates this work.
}



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