We study the natural representations of $\S_n$ and $GL_n(\mathbb{F}_q)$ on the (augmented) Chow rings of uniform matroids and $q$-uniform matroids. The Frobenius series for uniform matroids and their $q$-analogs are computed. As a byproduct, we recover Hameister, Rao, and Simpson's formula of Hilbert series of Chow rings of $q$-uniform matroids in terms of permutations and further obtain their augmented counterpart in terms of decorated permutations.

We also show that equivariant Charney-Davis quantities of (augmented) Chow rings of general matroids are nonnegative (i.e. genuine representations of the automorphism group of the matroid). When the matroid is a uniform matroid, the representations either vanish or are Specht modules of some skew hook shapes. When descending to the usual Charney-Davis quantities, we obtain an elegant combinatorial interpretation of Hameister, Rao, and Simpson's formula for Chow rings of $q$-uniform matroids and its augmented counterpart.

§ INTRODUCTION

Matroids are combinatorial objects that unify various notions of independence in different subjects. Because of its generality, matroids appear in diverse contexts and are applicable throughout mathematics. First, and perhaps the most commonly seen example comes from linear algebra as an abstraction of linear independence of a collection of vectors; this is known as a linear or a realizable matroid. They also arise as algebraic independence of equations (algebraic matroid), forests of a graph (graphical matroid), partial matchings of a bipartite graph (transversal matroid), etc.

Recently, the interactions between matroid theory and algebraic geometry have led to the resolution of several long-standing conjectures about unimodality, log-concavity, and top-heaviness of other combinatorial sequences coming from matroids. In particular, Adiparsito, Huh, and Katz [1] proved the long-standing Heron-Rota-Welsh conjecture, which asserts the log-concavity of the Whitney numbers of the first kind for a matroid. Following the seminal paper [1], Braden, Huh, Matherne, Proudfoot, and Wang [5] resolved the Dowling-Wilson conjecture, which states that the Whitney numbers of the second kind of a matroid are top-heavy. In the proofs of the two conjectures, Chow ring $A(M)$ and augmented Chow ring $\widetilde{A}(M)$ of a matroid $M$ play important roles, respectively. They were shown to satisfy the so-called “Kähler package", which includes Poincaré duality, the Hard Lefschetz property, and the Hodge-Riemann relations. Since then, there have been extensive studies of Chow rings of matroids and their extensions, for examples  [6, 3, 15, 20, 11, 9, 2].

Among these developments, Hameister, Rao, and Simpson [15] studied the Hilbert series of Chow rings of uniform matroids $U_{r,n}$ and their $q$-analogs $U_{r,n}(q)$, and their $(-1)$-evaluations, or equivalently, the Charney-Davis quantities (see Sections <ref> and <ref> for definitions).
They showed that the Hilbert series of the Chow ring of the $q$-uniform matroid $U_{r,n}(q)$, for $1\le r\le n$, can be expressed combinatorially as

\begin{equation}\label{HRS:ChowQUniform}
\Hilb(A(U_{r,n}(q)),t)=\sum_{\sigma\in\S_n}q^{\maj(\sigma)-\exc(\sigma)}t^{\exc(\sigma)}-\sum_{j=r}^{n-1}\sum_{\substack{\sigma\in\S_n \\ \fix(\sigma)\ge n-j}}q^{\maj(\sigma)-\exc(\sigma)}t^{j-\exc(\sigma)}
\end{equation}

where $\fix(\sigma)$ is the number of fixed points of $\sigma\in\S_n$. In particular, when the rank $r$ is $n$ and $n-1$, the matroid $U_{n,n}(q)$ is exactly the $q$-Boolean matroid $\mathsf{B}_n(q)$ and the Hilbert series of the Chow rings of $U_{n,n}(q)$ and $U_{n-1,n}(q)$ are

\begin{equation}\label{HRS:qBoolean}
\Hilb(A(U_{n,n}(q)),t)=\sum_{\sigma\in\S_n}q^{\maj(\sigma)-\exc(\sigma)}t^{\exc(\sigma)}=A_n(q,t)
\end{equation}

\begin{equation} \label{HRS:ChowDerangement}
\Hilb(A(U_{n-1,n}(q)),t)=\sum_{\sigma\in D_n}q^{\maj(\sigma)-\exc(\sigma)}t^{\exc(\sigma)-1}=t^{-1}d_n(q,t),
\end{equation}

where both $A_n(q,t)$ and $d_n(q,t)$ are $q$-analogs of Eulerian polynomials for $\S_n$ and derangements $D_n$ studied by Shareshian and Wachs [25].

They obtained formulas for the evaluation $\Hilb(A(U_{r,n}(q)),-1)$, which is related to the Charney-Davis quantity (see Section <ref>) of $A(U_{r,n}(q))$ :

\begin{equation} \label{HRS:sign}
\Hilb(A(U_{r,n}(q)),-1)=\begin{cases}
0, & \mbox{ if $r$ is even;}\\
\displaystyle{\sum_{k=0}^{\frac{r-1}{2}}(-1)^k\qbin{n}{2k}E_{2k}(q)}, & \mbox{ if $r$ is odd.}
\end{cases}
\end{equation}

where $E_{2k}(q)=\sum_{\sigma\in\RAlt_{2k}}q^{\inv(\sigma)}$ is the $q$-analog of the $k$-th secant number and $\RAlt_{2k}=\{\sigma\in\S_{2k}\,:\, \sigma_1<\sigma_2>\ldots>\sigma_{2k-1}<\sigma_{2k}\}$ the set of reverse alternating permutations[In this paper, we follow the terminology in Stanley's survey [30] calling such permutations reverse alternating and the permutations with $\sigma_1>\sigma_2<\sigma_3\ldots$ alternating. In <cit.>, they do it reversely and the $q^{\exc(\sigma)}$ in the expression of $E_{2k}(q)$ is a typo.]. They also offered a different expression of the alternating sum in the case of odd $r$ as an alternating sum of determinants:

\begin{equation} \label{HRS:determinant}
\left|\begin{array}{ccccc}
\frac{1}{[2]_q!} &         1        &   0   & \ldots & 0 \\
\frac{1}{[4]_q!} & \frac{1}{[2]_q!} &   1   & \ldots & 0 \\
\vdots           & \vdots           & \vdots& \ddots & \vdots\\
\frac{1}{[2k-2]_q!} & \frac{1}{[2k-4]_q!} & \frac{1}{[2k-6]_q!} & \ldots & 1\\
\frac{1}{[2k]_q!} & \frac{1}{[2k-2]_q!}& \frac{1}{[2k-4]_q!} & \ldots & \frac{1}{[2]_q!}
\end{array}\right|.
\end{equation}

In this paper, we continue Hameister, Rao, and Simpson's study of Chow rings of uniform matroids and $q$-uniform matroids. Using the Feichtner-Yuzvinsky type basis for $\widetilde{A}(M)$ found by the author <cit.>, and independently by Eur, Huh, and Larson [9], we build the parallel story for the augmented Chow rings of $U_{r,n}$ and $U_{r,n}(q)$. Besides their Hilbert series and its $(-1)$-evaluation, we further explore the structures of $\S_n$-representations and the $GL_n(\mathbb{F}_q)$-representations on these rings. The augmented counterpart involves the objects appear in the combinatorics of the binomial Eulerian story in [19], e.g., binomial Eulerian quasisymmetric functions $\widetilde{Q}_n(\x,t)$, binomial Eulerian polynomials $\widetilde{A}_n(\x,t)$, decorated permutations $\widetilde{\S}_n$, etc.

We start by establishing the parallelity between the $\S_n$-representations on $A(U_{r,n})$, $\widetilde{A}(U_{r,n})$ and the $GL_n(\mathbb{F}_q)$-representations on $A(U_{r,n}(q))$, $\widetilde{A}(U_{r,n}(q))$ via Steinberg's theory of unipotent representations of $GL_n(\mathbb{F}_q)$. This will enable us to obtain the Hilbert series related to $q$-uniform matroids directly through the Frobenius series related to uniform matroids. We obtain two formulas in different flavors for the Frobenius series of the Chow ring $A(U_{r,n})$ and the $q$-Frobenius series (see Sec. <ref>) of the Chow ring $A(U_{r,n}(q))$ (Theorem <ref>). We also derive the analogous formula in the case of the augmented Chow ring $\widetilde{A}(U_{r,n})$ and $\widetilde{A}(U_{r,n}(q))$ (Theorem <ref>).
From these symmetric function formulas, the corresponding Hilbert series of Chow rings and augmented Chow rings of $U_{r,n}(q)$ can be obtained directly (Theorem <ref>, Theorem <ref>). In the non-augmented case, we not only recover Hameister, Rao, and Simpson's formula (<ref>), (<ref>), (<ref>), but also obtain a different expression for the Hilbert series of $A(U_{r,n}(q))$.

For the $(-1)$-evaluation results and the Charney-Davis quantities, we even obtain more general formulas (Theorems <ref> and <ref>). We show that for any matroid $M$ and subgroup $G$ of the automorphism group $Aut(M)$ of $M$, the $(-1)$-evaluation of the $G$-equivariant Hilbert series of the (augmented) Chow ring of $M$ is related to the $G$-representation on the poset homology of the rank-selected subposet of the lattice of flats $\L(M)$. As a result, the $G$-equivariant Charney-Davis quantities of $A(M)$ and $\widetilde{A}(M)$ are always genuine representations of $G$ (see Section <ref>). The cases that $M$ is $U_{r,n}$ and $U_{r,n}(q)$ are corollaries that follow right away from the general theorem. In particular, we observe that Hameister, Rao, and Simpson's result (<ref>) of the $(-1)$-evaluation of the Hilbert series of $A(U_{r,n}(q))$ can be simplified to a more elegant combinatorial interpretation (Corollary <ref>). We also obtain the parallel result in the case of the augmented Chow ring (Corollary <ref>).

The paper is organized as follows. We set up some preliminary definitions and backgrounds in Section <ref>. In Section <ref>, we describe the parallelity between representations of $\S_n$ and the unipotent representations of $GL_n(\mathbb{F}_q)$ and how it works on the representations on Chow rings and augmented Chow rings of uniform matroids and $q$-uniform matroids. As a result, it reduces most of the proofs to only the case of the Frobenius series for uniform matroids. In Section <ref>, we summarize our main results for the Frobenius series, $q$-Frobenius series, Hilbert series related to uniform matroids and $q$-uniform matroids, the $(-1)$-evaluations and Charney-Davis quantities in logical order. In Section <ref>, we prove our results of the Frobenius series of Chow rings of uniform matroids. In Section <ref>, we prove the results of the Frobenius series of augmented Chow rings of uniform matroids. Section <ref> deals with the proof of the $(-1)$-evaluation of the $G$-equivariant Hilbert series of general matroids.

§ BACKGROUND

§.§ Matroids

We first briefly review some basics of matroids. For a more thorough introduction to matroids, we refer the readers to Oxley [21]. Some familiarity with terminologies on partially ordered sets (posets) is assumed. See Stanley <cit.> for a good reference. There are many ways to define a matroid, and they are cryptomorphic to each others (equivalent, but not obvious). Due to our purpose, here we define a matroid in terms of its flats.

A matroid $M=(E,\L(M))$ consists of a finite ground set $E$ and a collection of subsets $\L(M)\subseteq 2^{E}$ satisfy the following axioms:

($F_1$) $E\in \L(M)$.

($F_2$) If $F$, $F'\in\L(M)$, then $F\cap F'\in\L(M)$.

($F_3$) For any $F\in\L(M)$ and any $i\in E\setminus F$, there exists a unique $F'$ containing $i$ and covers $F$, in the sense that $F\subsetneq F'$ and no other $F''\in M$ such that $F\subsetneq F''\subsetneq F'$.

The subsets in $\L(M)$ are called the flats of $M$. These axioms are the abstraction of the model where $E$ is a collection of vectors $v_1,\ldots,v_n$ in a vector space over some field $\mathbb{F}$, and $\L(M)$ is the set of subspaces $\{\mathrm{span}_{\mathbb{F}}\{v_i\}_{i\in S}: S\subseteq E\}$ in which each subspace $W$ is represented by the maximal subcollection of $v_1,\ldots,v_n$ that spans $W$.

The axioms $(F_1)$, $(F_2)$ imply that $\L(M)$ with the inclusion order forms a lattice with meet $F\wedge F'=F\cap F'$ and join $F\vee F'=\bigcap_{F''\supseteq F,F'}F''$ for any $F$, $F'\in\L(M)$. The axiom $(F_3)$ further implies that $\L(M)$ is a geometric lattice. We call $\L(M)$ the lattice of flats of $M$. It is well-known <cit.> that any finite geometric lattice is a lattice of flats of some matroid.

Denote $\rk_M$ the rank function of $\L(M)$. We say the rank of the matroid $M$ is $r$ if $\rk_M(E)=r$. For any subset $S\subseteq E$, the closure $\overline{S}$ of $S$ is the minimal flat containing $S$, in other words, $\overline{S}=\bigcap_{F\supseteq S, F\in\L(M)}F$. This way one can extend the rank function $\rk_M$ to $2^{E}\rightarrow \mathbb{N}$ by defining $\rk_M(S)=\overline{S}$ for any $S\subseteq E$. We call an element $i\in E$ a loop if $i\in \overline{\emptyset}$.

A permutation $\sigma$ on $E$ is said to be an automorphism of the matroid $M$ if $\rk_M(\sigma(X))=\rk_M(X)$ for all subsets $X\subseteq E$. Denote $\Aut(M)$ the set of automorphisms of $M$. In particular, an automorphism $\sigma\in\Aut(M)$ maps a flat $F\in\L(M)$ to a flat  $\sigma(F)\in\L(M)$ and $\rk_M(\sigma(F))=\rk_M(F)$ for all $F\in\L(M)$. Hence, the automorphism $\sigma$ permutes the chains in $\L(M)$.

In this paper, most of the time we will focus on the uniform matroids and their $q$-analogs. The uniform matroid $U_{r,n}$ ($r\le n$) is a matroid on $E=[n]$ of rank $r$ whose collection of flats are

\[
\{S\subseteq [n]: |S|\le r-1\}\cup\{[n]\}.
\]

The lattice of flats $\L(U_{r,n})$ is a truncated Boolean lattice. In particular, when $r=n$, the lattice $\L(U_{n,n})$ is the Boolean lattice $B_n=(2^{[n]},\subseteq)$, hence $U_{n,n}$ is also called a Boolean matroid.

The lattices of flats of uniform matroids $U_{r,n}$ when $n=3$ are the truncated Boolean lattices as shown in the  figure <ref>.

Boolean lattice $\L(U_{3,3})$

Let $q>1$ be a prime power. The $q$-uniform matroid, which we refer to as $U_{r,n}(q)$, is a natural $q-$analog of $U_{r,n}$.
It is a matroid whose ground set is the vector space $\mathbb{F}_q^n$ over the finite field $\mathbb{F}_q$.
Its lattice of flats $\L(U_{r,n}(q))$ consists of subspaces $V\le\mathbb{F}_q^n$ of $\dim V\le r-1$ and the maximal subspace $\mathbb{F}_q^n$.
It will be convenient to define $U_{r,n}(1)\coloneqq U_{r,n}$.
In particular, the lattice of flats of $U_{n,n}(q)$ is the subspace lattice of $\mathbb{F}_q^n$, also known as the $q$-Boolean lattice $B_n(q)$, which is a natural $q$-analog of the Boolean lattice $B_n$.

§.§ Chow rings and augmented Chow rings of matroids

In the following, we assume $M$ to be a loopless matroid.

Let $S$ be the  polynomial ring $\mathbb{Q}[x_F:F\in\L(M)\setminus\{\emptyset\}]$. The Chow ring $A(M)$ of a matroid $M$ is a quotient $\mathbb{Q}$-algebra

\[
A(M)\coloneqq S/(I+J)
\]

where $I$ and $J$ are the following ideals of $S$:

\[
I=\left\langle x_F x_{F'}: F, F'\text{ are incomparable in }\L(M)\right\rangle \quad \text{and} \quad     J=\left\langle \sum_{F:F\ni i}x_F: i\in E\right\rangle.
\]

This Chow ring was introduced and studied by Feichtner and Yuzvinsky [10] in a more general setting, for any atomic lattice $\L$ and a certain subset $\G$ of $\L$ called a building set. In particular, they proved $A(M)$ is finite-dimensional and found a Gröbner basis of the ideal $I+J$. It leads to a monomial basis for $A(M)$, which we refer to as the Feichter-Yuzvinsky basis (FY-basis).

The following set of monomials forms a basis for $A(M)$

\begin{equation} \label{eq:FYChowBasis}
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 any } \ell \\
1\le a_i\le \rk_M(F_i)-\rk_M(F_{i-1})-1}\right\}.
\end{equation}

In order to proved Dowling-Wilson's top-heavy conjecture, Braden, Huh, Matherne, Proudfoot, and Wang [6, 5] introduced an extension of the Chow ring of a matroid called the augmented Chow ring.
[[6, 5]]
Let $\widetilde{S}$ be the polynomial ring $\mathbb{Q}\left[\{x_F\}_{F\in\L(M) \setminus\{\emptyset\}\setminus \{E\}}\cup\{y_i\}_{i\in E}\right]$. The augmented Chow ring of $M$ is

\[
\widetilde{A}(M)\coloneqq\widetilde{S}/(\widetilde{I}_1+\widetilde{I}_2+\widetilde{J}).
\]

where $\widetilde{I}_1$, $\widetilde{I}_2$, and $\widetilde{J}$ are the following ideals of $\widetilde{S}$:

* $\widetilde{I}_1$ is generated by $x_F x_{F'}$ for any proper flats $F$ and $F'$.

* $\widetilde{I}_2$ is generated by $y_i x_F$ for any $i\in E$ and any proper flats $F$ such that $i\notin F$.

* $\widetilde{J}$ is generated by the linear elements $y_i-\sum_{F:i\notin F}x_F$ for all $i\in E$.

In the author's previous work [19], and independently by Eur, Huh, and Larson [9], they showed the augmented Chow ring $\widetilde{A}(M)$ can be realized as Feichtner and Yuzvinsky's Chow ring of lattice of flats of $M\times e$ (free coextension of $M$) and certain building set; thereby, they obtained a Feichtner-Yuzivinsky type basis for the augmented Chow ring.

The following set of monomials forms a basis for $\widetilde{A}(M)$

\begin{equation} \label{eq:AugChowBasis}
\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 any }\ell \\
1\le a_1\le\rk_M(F_1),~ a_i\le\rk_M(F_i)-\rk_M(F_{i-1})-1 \text{ for }i\ge 2}
\right\}.
\end{equation}

When $M$ is the uniform matroid $U_{r,n}$, we can consider $\S_n$ acting on the polynomial ring $S$ and $\widetilde{S}$ by permuting the variables $x_F$ which are indexed by subsets $F$ of $[n]$ and $y_i$ for $i\in [n]$. Both of the ideals $I+J$ of $S$ and $\widetilde{I}_1+\widetilde{I}_2+\widetilde{J}$ of $\widetilde{S}$ are generated by homogeneous polynomials. One can easily check that both ideals are $\S_n$-invariant under this case. Hence the Chow rings $A(U_{r,n})$ and $\widetilde{A}(U_{r,n})$ and each of their degree components form $\S_n$-modules. Furthermore, the basis $FY(U_{r,n})$ in (<ref>) and $\widetilde{FY}(U_{r,n})$ are permutation bases for the $\S_n$-modules  $A(U_{r,n})$ and $\widetilde{A}(U_{r,n})$ in <ref>), respectively. Similarly, when $M$ is the $q$-uniform matroid $U_{r,n}(q)$, the rings $A(U_{r,n}(q))$, $\widetilde{A}(U_{r,n}(q))$ are $GL_n(\mathbb{F}_q)$-modules and $FY(U_{r,n}(q))$, $\widetilde{FY}(U_{r,n}(q))$ are permutation bases for them respectively.

§.§ Charney-Davis quantity

Let $\mathbb{F}$ be any field and $A=\bigoplus_{j\in\mathbb{N}}A^j$ be an $\mathbb{N}$-graded $\mathbb{F}$-algebra such that each degree-$j$ homogeneous component $A^j$ ($j\in\mathbb{N}$) satisfies $\dim_{\mathbb{F}}A^j<\infty$.
Then the Hilbert series of $A$ is defined to be

\[
\Hilb(A,t)\coloneqq \sum_{j\in\mathbb{N}}\dim_{\mathbb{F}}(A^j)t^j.
\]

We say that the Hilbert series of $A$ is palindromic, if there is some $n\ge 0$ such that $\dim_{\mathbb{F}}(A^N)=0$ for $N>n$ and $\Hilb(A,t)=\sum_{j=0}^n a_jt^j$ with $a_j=a_{n-j}$ for $j\le n/2$. When the Hilbert series of $A$ is a polynomial of degree $n$, we call the number

\[
\begin{cases}
(-1)^{\frac{n}{2}}\Hilb(A,-1) & \text{ if $n$ is even}\\
\Hilb(A,-1) & \text{ if $n$ is odd}
\end{cases}
\]

the Charney-Davis quantity of $A$. In particular, when $A$ has a palindromic Hilbert series of odd degree, then $CD(A)=0$.

This quantity was first introduced by Charney and Davis in [7] in connection with a conjecture in geometry.
The conjecture has an equivalent form stating that the $h$-polynomial $h_{\Delta}(t)$ of a simplicial $(2n-1)$-sphere $\Delta$ satisfies $(-1)^nh_{\Delta}(-1)\ge 0$ if $\Delta$ has some geometric property called flagness.
This nonnegative quantity $(-1)^nh_{\Delta}(-1)$ is now referred to as the Charney-Davis quantity of $\Delta$ and the conjecture is referred to as the Charney-Davis conjecture. The reader interested in more detail related to this conjecture can see [35], <cit.>.

In this paper, we consider the Charney-Davis quantities of the Chow ring $A(M)$ and the augmented Chow ring $\widetilde{A}(M)$ of a matroid $M$ of rank $r$. The Hilbert series of $A(M)$ and $\widetilde{A}(M)$ are palindromic polynomials of degree $r-1$ and $r$ respectively (see remark <ref>). If $r$ is even, the Charney-Davis quantity $CD(A(M))$ of Chow ring $A(M)$ vanishes; if $r$ is odd, then $CD(A(M))$ has another interpretation in terms of the signatures of the quadratic form on $A^{\frac{r-1}{2}}(M)$. Similarly, for the augmented Chow ring $\widetilde{A}(M)$, if $r$ is odd, then $CD(\widetilde{A}(M)$ vanishes; while if $r$ is even, the quantity $CD(\widetilde{A}(M))$ has a similar interpretation as in the non-augmented case (see remark <ref> for more details).

Hameister, Rao, Simpson [15] use the FY-basis in (<ref>) to obtain formulas for the Hilbert series of Chow ring of uniform matroids $U_{r,n}$ and their $q$-analogs $U_{r,n}(q)$ and the corresponding Charney-Davis quantity $CD(A(U_{r,n}))$ and $CD(A(U_{r,n}(q))$. Following the discovery of the Feichtner-Yuzvinsky type basis for the augmented Chow ring $\widetilde{A}(M)$, one of the original motivations of this paper is to establish the parallel formulas for the augmented Chow ring of uniform matroids and their $q$-analogs.

It is shown by Adiprasito, Huh, and Katz [1] and by Bradon, Huh, Matherne, Proudfoodt, and Wang [5] that $A(M)$ and $\widetilde{A}(M)$ respectively satisfy  the so-called Kähler package, which includes Poincaré duality, Hard Lefschetz theorem, and Hodge-Riemann relation. The Poincaré duality for $A(M)$ states that the multiplication maps

\[
A^i(M)\times A^{r-1-i}(M)\longrightarrow A^{r-1}(M)
\]

for $0\le i\le r-1$ induce isomorphisms

\[
A^{r-1-i}(M)\longrightarrow \Hom_{\mathbb{Q}}(A^i(M),A^{r-1}(M)).
\]

Combining with the fact that the top degree component satisfies $A^{r-1}(M)\cong\mathbb{Q}$ gives the isomprohisms $A^{r-1-i}(M)\cong\Hom_{\mathbb{Q}}(A^i(M),\mathbb{Q})$ which implies that $\dim_{\mathbb{Q}}(A^i(M))=\dim_{\mathbb{Q}}(A^{r-1-i}(M))$ for $0\le i\le r-1$. Therefore, the Hilbert series of $A(M)$ is palindromic. Similarly, the Poincaré duality for $\widetilde{A}(M)$ and the fact that $\widetilde{A}^{r}(M)\cong\mathbb{Q}$ together imply the palindromicity of the Hilbert series of $\widetilde{A}(M)$.

In the non-augmented case, if $r$ is odd, the multiplication map

\[
Q:A^{\frac{r-1}{2}}(M)\times A^{\frac{r-1}{2}}(M)\longrightarrow A^{r-1}(M)\cong\mathbb{Q}
\]

with $Q(x)=x^2$ defines a quadratic form
on $A^{\frac{r-1}{2}}(M)$.
Analogous to the proof of Theorem 1.1 in [16], the Hodge-Riemann relation for $A(M)$ implies that the signature of the quadratic form $Q$ is equal to the Charney-Davis quantity of $A(M)$.
In the augmented case, if $r$ is even, a similar quadratic form on $\widetilde{A}^{\frac{r}{2}}(M)$ can be defined. Hence the Charney-Davis quantity of the augmented Chow ring $\widetilde{A}(M)$ agrees with the signature of this quadratic form by the Hodge-Reimann relation for $\widetilde{A}(M)$.

§.§ Symmetric functions and representations of symmetric groups

We assume familiarity with symmetric functions and the representation theory of symmetric group $\S_n$. We will review the connection between the ring of symmetric functions $\Lambda$ and the representations of symmetric groups; see Sagan [24] and Stanley [29] for references.

Let $\Lambda_{\mathbb{Z}}$ be the ring of symmetric functions of bounded degrees with infinite variables $\x=(x_1,x_2,x_3,\ldots)$. It can be viewed as the polynomial ring $\mathbb{Z}[e_1,e_2,e_3,\ldots]=\mathbb{Z}[h_1,h_2,h_3,\ldots]$ where $e_n$ and $h_n$ are the elemenatary symmetric function and the complete homogeneous symmetric function of degree $n$ respectively. The ring $\Lambda_{\mathbb{Z}}=\bigoplus_{i\ge 0}\Lambda_i$, where $\Lambda_i$ is the space of symmetric functions homogeneous of degree $i$, forms a graded $\mathbb{Z}$-algebra via the usual multiplication.

For any positive integer $n$, let $\C(\S_n)$ be the space of virtual characters of $\S_n$, i.e. the free $\mathbb{Z}$-module generated by the irreducible characters $\chi^{\lambda}$ of $\S_n$ for any partition $\lambda$ of $n$, denoted by $\lambda\vdash n$. The irreducible representation corresponds to the character $\chi^\lambda$ is called the Specht module $S^\lambda$ for any partition $\lambda$. Then the direct sum

\[
\C(\S)\coloneqq\bigoplus_{n\ge 0}\C(\S_n)
\]

forms a graded $\mathbb{Z}$-algebra under the induction product defined by

\begin{align*}
*:\C(\S_{n_1})\times \C(\S_{n_2}) &\longrightarrow \C(\S_{n_1+n_2})\\
(f, g) &\longmapsto f*g\coloneqq(f\otimes g)\uparrow_{\S_{n_1}\times\S_{n_2}}^{\S_{n_1+n_2}}
\end{align*}

where for subgroup $H$ of $G$ the operation $(-)\uparrow_{H}^{G}$ is the usual induction from the class functions of $H$ to the class functions of $G$.

There is a $\mathbb{Z}$-algebra isomorphism between $\C(\S)$ and $\Lambda_{\mathbb{Z}}$ called the Frobenius characteristic map $\ch$ such that

\begin{align*}
\ch: \C(\S_n) &\longrightarrow \Lambda_{\mathbb{Z}}\\
\mathbf{1}_{\S_n} & \longmapsto h_n\\
{\rm sgn}_{\S_n}  & \longmapsto e_n\\
{\chi}^{\lambda} & \longmapsto s_{\lambda}
\end{align*}

where $\mathbf{1}_{\S_n}$, ${\rm sgn}_{\S_n}$ are the trivial and signed characters of $\S_n$ respectively, and $s_{\lambda}$ is the Schur function. To simplify the notations, we sometimes denote the representation by the corresponding character. For a composition $\nu=(\nu_1,\nu_2,\ldots,\nu_{\ell})$ of $n$, let $\S_{\nu}$ be the corresponding Young subgroup. Then the $\S_n$-permutation module corresponding to $\nu$ is isomorphic to the induced representation $\mathbf{1}\uparrow_{\S_{\nu}}^{\S_n}$ and has the Frobenius characteristic

\[
\ch(\mathbf{1}\uparrow_{\S_{\nu}}^{\S_n})=\ch(1_{\S_{\nu_1}}*1_{\S_{\nu_2}}*\ldots*1_{\S_{\nu_{\ell}}})=h_{\nu_{1}}h_{\nu_{2}}\ldots h_{\nu_{\ell}}\eqqcolon h_{\nu}.
\]

Furthermore, the dimension of the permutation module is given by

\[
\dim(\mathbf{1}_{\S_{\nu}}\uparrow_{\S_{\nu}}^{\S_n})=\mathbb{C}\S_n\mathbf{1}_{\S_{\nu}}=|\S_n:\S_{\nu}|=\frac{n!}{\nu_1!\ldots\nu_{\ell}!}=\binom{n}{\nu_1,\ldots,\nu_{\ell}}.
\]

Let $V=\bigoplus_i V^i$ be a graded $\mathbb{C}[\S_n]$-module. The graded Frobenius series of $V$ is defined to be $\grFrob(V,t)\coloneqq \sum_{i}\ch(V^i)t^i$. In this way, one can encode representations of $\S_n$ in terms of symmetric functions via $\ch$ and this will be the method we apply to study the representations of $\S_n$ in this paper.

§.§ Eulerian polynomials for derangements, permutations, and decorated permutations and their extensions

Denote by $\S_n$ the set of permutations on $[n]\coloneqq\{1,2,\ldots,n\}$, also called the symmetric group, and by $D_n$ the subset of derangements in $\S_n$ i.e., permutations without fixed points.

For any permutation $\sigma=\sigma_1\sigma_2\ldots\sigma_n$ in $\S_n$, one can consider the following statistics:

* Excedance set $\EXC(\sigma)\coloneqq\{i\in[n-1]: \sigma_i>i\}$ and excedance number $\exc(\sigma)\coloneqq |\EXC(\sigma)|$.

* Descent set $\DES(\sigma)\coloneqq\{i\in[n-1]: \sigma_i> \sigma_{i+1}\}$ and descent number $\des(\sigma)\coloneqq |\DES(\sigma)|$.

* Inversion number $\inv(\sigma)\coloneqq |\{(i,j)\in[n]\times[n]: i<j,~\sigma_i> \sigma_j\}|$.

* Major index $\maj(\sigma)\coloneqq \sum_{i\in\DES(\sigma)}i$.

It is well known that $\exc$ and $\des$ are equidistributed statistics on $\S_n$ and belong to the class of Eulerian statistics. The generating function of their distribution

\begin{equation} \label{eq:DefEulPoly}
A_n(t)\coloneqq \sum_{\sigma\in\S_n}t^{\des(\sigma)}=\sum_{\sigma\in\S_n} t^{\exc(\sigma)}
\end{equation}

is known as the Eulerian polynomial whose coefficients are called the Eulerian numbers. Similarly, we can consider the analogous Eulerian polynomial for derangements

\[
d_n(t)\coloneqq \sum_{\sigma\in D_n}t^{\exc(\sigma)}.
\]

Led by their study on Rees product of posets, Shareshian and Wachs [25] studied certain $q$-analogs and symmetric function analogs of the Eulerian polynomials $A_n(t)$ and $d_n(t)$, which we will introduce in the following.

[Shareshain, Wachs[25]]
Let $[\overline{n}]\coloneqq\{\overline{1},\overline{2},\ldots,\overline{n}\}$ and assign to $[n]\cup[\overline{n}]$ the total order $\overline{1}<\ldots<\overline{n}<1<\ldots<n$. Let $\overline{\sigma}=\overline{\sigma}_1\ldots\overline{\sigma}_n$ be a word over $[n]\cup[\overline{n}]$ obtained from $\sigma$ by letting $\overline{\sigma}_i=\overline{j}$ if $\sigma_i=j$ and $i\in\EXC(\sigma)$, letting $\overline{\sigma}_i=\sigma_i$ if $i\notin\EXC(\sigma)$. Define the set statistic

\[
\DEX(\sigma)\coloneqq\DES(\overline{\sigma}).
\]

The statistic $\DEX$ is a variant of descent set $\DES$ and has amazing properties that connect the classical statistics $\maj$ and $\exc$.

For all $\sigma \in \S_n$,

\[
\sum_{i\in\DEX(\sigma)}i=\maj(\sigma)-\exc(\sigma)
\]

\[
\des(\sigma)  & \text{ if }\sigma(1)=1\\
\des(\sigma)-1 &  \text{ if }\sigma(1)\neq 1
\end{cases}.
\]

For integer $n\ge 1$ and $S\subseteq[n-1]$, define Gessel's fundamental quasisymmetric function

\[
F_{S,n}(\x)\coloneqq \sum_{\substack{i_1\ge i_2\ge \ldots\ge i_n\\ j\in S\Rightarrow i_j>i_{j+1}}}x_{i_1}x_{i_2}\ldots x_{i_n}
\]

and $F_{\phi,0}\coloneqq 1$. In particular, when $S=\emptyset$, $F_{\emptyset, n}=h_n$; when $S=[n-1]$, $F_{[n-1],n}=e_n$.

The Eulerian quasisymmetric functions for .
For integer $n\ge 1$, define

\[
\]

\[
Q_n^0(\x,t)=\sum_{\sigma\in D_n}F_{\DEX(\sigma),n}(\x)t^{\exc(\sigma)}.
\]

It is useful to set $Q_0(\x,t)\coloneqq 1$. For $0\le j\le n-1$, we denote $Q_{n,j}(\x)$ the quasisymmetric function coefficient of $t^j$ in $Q_n(\x,t)$.

It is shown by Shareshian and Wachs that both $Q_n(\x,t)$ and $Q_n^0(\x,t)$ are polynomials with symmetric function coefficients. In particular, the coefficient

\begin{equation} \label{F-expansionQ}
Q_{n,j}(\x)=\sum_{\substack{\sigma\in\S_n\\ \exc(\sigma)=j}}F_{\DEX(\sigma),n}(\x)
\end{equation}

of $t^j$ is symmetric for all $j$. Hence one can view $Q_{n,j}$ as a symmetric function analog of the Eulerian number and was referred to as the Eulerian quasisymmetric function in [25]. The polynomial $Q_n$ and $Q_n^0$ are symmetric function analogs of the Eulerian polynomials $A_n(t)$ and $d_n(t)$. In fact, in [25], they show that the sum is still symmetric even if we further refine it over fixed $\exc$ and fixed cycle type of permutations in $\S_n$.

The symmetric function $Q_{n,j}(\x)$ is $h$-positive (has positive coefficients when expanded as a linear combination of $\{h_{\lambda}\}_{\lambda\vdash n}$) and is the Frobenius characteristic of the $\S_n$-modules appearing in several different contexts, including the cohomology of the permutahedral variety [33], the Chow ring of the Boolean matroid [18, 19], and the poset homology of the Rees product of the Boolean lattice and a chain [25].

The stable principal specialization of a (quasi)symmetric function $f(\x)$ is defined by $\ps(f)\coloneqq f(1,q,q^2,\ldots)$.

For all $n\ge 1$, $S\subseteq [n-1]$, we have

\[
\ps(F_{S,n})=\frac{q^{\sum_{i\in S}i}}{(1-q)(1-q^2)\ldots(1-q^n)}
\]

By Lemma <ref>, Theorem <ref> implies the following corollary.

For all $\sigma\in\S_n$,

\[
\ps(F_{\DEX(\sigma),n})=\frac{q^{\maj(\sigma)-\exc(\sigma)}}{(1-q)(1-q^2)\ldots(1-q^n)}.
\]

By Corollary <ref>, the stable principal specialization of $Q_n(\x,t)$ and $Q_n^0(\x,t)$ gives the following interesting $q$-analogs of the Eulerian polynomials $A_n(t)$ and $d_n(t)$.

For all $n\ge 1$,

(a) $\prod_{i=1}^n\left(1-q^i\right)\ps(Q_n(\mathbf{x},t))=\sum_{\sigma\in\S_n}q^{\maj(\sigma)-\exc(\sigma)}t^{\exc(\sigma)}$.

(b) $\prod_{i=1}^n\left(1-q^i\right)\ps(Q_n^0(\mathbf{x},t))=\sum_{\sigma\in D_n}q^{\maj(\sigma)-\exc(\sigma)}t^{\exc(\sigma)}$

Throughout this paper, we denote

\begin{equation}\label{qEulerian}
\end{equation}

\[
d_n(q,t)\coloneqq\sum_{\sigma\in D_n}q^{\maj(\sigma)-\exc(\sigma)}t^{\exc(\sigma)}.
\]

Another polynomial that shares many interesting properties with the Eulerian polynomial $A_n(t)$ is the binomial Eulerian polynomial

\[
\widetilde{A}_n(t)\coloneqq 1+t\sum_{k=1}^n\binom{n}{k}A_k(t),
\]

named by Shareshian and Wachs [26]. Same as the Eulerian polynomial, it is palindromic and the palindromicity is equivalent to an identity studied by Chung, Graham, and Knuth [8].
This connection also motivated Shareshian and Wachs to consider $\widetilde{A}_n(t)$.
The polynomial $\widetilde{A}_n(t)$ was first appeared in Postnikov, Reiner, and Williams' paper [23] as the $h$-polyonomial of certain simple polytope called the stellohedron.

Shareshian and Wachs [26] introduced a natural $q$-analog and a symmetric function analog of $\widetilde{A}_n(t)$

\begin{equation}
\widetilde{A}_n(q,t)\coloneqq 1+t\sum_{k=1}^n\qbin{n}{k}A_k(q,t)
\end{equation}

\begin{equation} \label{defBinomQ}
\widetilde{Q}_n(\x,t)\coloneqq h_n(\x)+t\sum_{k=1}^n h_{n-k}(\x)Q_k(\x,t).
\end{equation}

Similar to $Q_n(\x,t)$ being the graded Frobenius series of the natural $\S_n$-representation on the cohomology ring of the permutahedral variety, Shareshian and Wachs showed that $\widetilde{Q}_n(\x,t)$ is the graded Frobenius series of the cohomology of the stellohedral variety. They further showed that its stable principal specialization gives exactly the following

\begin{equation}  \label{eq:PSBinomialQ}
\prod_{i=1}^n(1-q^i)\ps(\widetilde{Q}_n(\x,t))=\widetilde{A}_n(q,t).
\end{equation}

As an analog of the set of permutations $\mathfrak{S}_n$ in the Eulerian story, Liao [19] showed that the Postnikov's decorated permutations gives a nice combinatorial model to interpret the binomial Eulerian story. A decorated permutation of $[n]$ can be viewed as a permutation on a subset of $[n]$. Denote $\widetilde{\mathfrak{S}}_n$ the set of decorated permutations of $[n]$.

For example, $215\in\widetilde{\mathfrak{S}_5}$ is a permutation on $\{1,2,5\}\subset[5]$ that maps $1$ to $2$, $2$ to $1$, $5$ to $5$. We can express it in two-line notation and in one-line notation respectively as the following

\[
1 & 2 & 3 & 4 & 5\\
2 & 1 & 0 & 0 & 5
\end{array}\right)=21005.
\]

In particular, denote $\theta$ the permutation on $\emptyset$ in $\widetilde{\mathfrak{S}_5}$. It has the following two-line notation and one-line notation

\[
\theta\coloneqq\left(\begin{array}{ccccc}
1 & 2 & 3 & 4 & 5\\
0 & 0 & 0 & 0 & 0
\end{array}\right)=00000.
\]

Our definition of decorated permutations is different from Postinkov's in [22] where he introduced them as permutations with two kinds of fixed points. An easy bijection is given by replacing the $0$s in our one-line notation with the second kind of fixed points.

Extend the set statistic $\DEX$ over $\S_n$ to $\widetilde{\S}_n$ in the following way.

[Liao [19]]
Consider the set $[n]\cup[\overline{n}]\cup\{0\}$ with the total order $\overline{1}<\ldots<\overline{n}<0<1<\ldots<n$. For $\sigma\in\widetilde{\S}_n\setminus\{\theta\}$, let $\overline{\sigma}$ be a word over $[n]\cup[\overline{n}]\cup\{0\}$ obtained from $\sigma$ by replacing $\sigma_i$ with $\overline{\sigma_i}$ whenever $i\in\EXC(\sigma)\coloneqq\{i\in[n]:i<\sigma_i\}$. Define the set statistic

\[
\DEX(\sigma)\coloneqq\begin{cases}
\DES(\overline{\sigma}) & \mbox{ if }\sigma\neq \theta\\
\emptyset & \mbox{ if }\sigma=\theta
\end{cases}.
\]

Similarly, the defintion of the excedance number and the major index can be extended to $\sigma\in\widetilde{\S}_n$ as

\[
\exc(\sigma)\coloneqq\begin{cases}
|\{i\in[n]: i<\sigma_i\}| & \text{ if }\sigma\neq\theta\\
-1  & \text{ if }\sigma=\theta
\end{cases} ~\text{ and }~
\maj(\sigma)\coloneqq\begin{cases}
\sum_{i:\sigma_i<\sigma_{i+1}} i & \text{ if }\sigma\neq\theta\\
-1  & \text{ if }\sigma=\theta
\end{cases}.
\]

Then the extension of $\DEX$ to $\widetilde{\S}_n$ still preserves the nice property it has as a statistic over $\S_n$. The following lemma is analogous to Theorem <ref>.

For $\sigma\in\widetilde{\S}_n$,

\[
\sum_{i\in\DEX(\sigma)}i=\maj(\sigma)-\exc(\sigma).
\]

And for $\sigma\in\widetilde{\S}_n\setminus\{\theta\}$,

\[
\des(\sigma) & \mbox{ if }\sigma(1)=0 \text{ or }1\\
\des(\sigma)-1 & \mbox{ if }\sigma(1)>1
\end{cases}.
\]

Liao used the extension of $\DEX$ to $\widetilde{\S}_n$ to obtain the $F$-expansion of $\widetilde{Q}_n(\x,t)$ analogous to Definition <ref>.

For any integer $n\ge 1$,

\[
\widetilde{Q}_n(\x,t)=\sum_{\sigma\in\widetilde{\mathfrak{S}}_n}F_{\DEX(\sigma),n}(\x)t^{\exc(\sigma)+1}.
\]

Or equivalently, for all $0\le j\le n$,

\[
\widetilde{Q}_{n,j}(\x)=\sum_{\substack{\sigma\in\widetilde{\S}_n\\ \exc(\sigma)=j-1}} F_{\DEX(\sigma),n}(\x).
\]

Combining Lemma <ref>, (<ref>) and Proposition <ref> gives the following Theorem analogous to Theorem <ref>.

For all integer $n\ge 1$,

\[
\prod_{i=1}^n(1-q^i)\ps(\widetilde{Q}_n(\x,t))=\sum_{\sigma\in\widetilde{\mathfrak{S}}_n}q^{\maj(\sigma)-\exc(\sigma)}t^{\exc(\sigma)+1}.
\]

We summarize the relations between the combinatorial models and their enumerators lifted to different extensions discuss in this subsection in table <ref>.

\[
\begin{array}{c|ccccc}
& \substack{\text{enumerator}\\ \exc} &  & \substack{q-\text{enumerator}\\ (\maj-\exc,\exc)} &  &  \substack{\text{equivariant enumerator}\\ (\DEX,\exc)}\\
\hline
D_n   &   d_n(t)     &   &   d_n(q,t)  &  &   Q_n^0(\x,t)\\
\cap \\
\S_n  &   A_n(t)     & \xleftarrow{q=1}   &   A_n(q,t)  & \xleftarrow{\ps} &   Q_n(\x,t)  \\
\cap \\
\widetilde{\S}_n  & \widetilde{A}_n(t) &  & \widetilde{A}_n(q,t)  &  & \widetilde{Q}_n(\x,t)
\end{array}
\]

§.§ Group actions on rank selected posets

In this section, we recall some terminology and facts about rank selected posets from [28] and [34]. The readers who are interested in knowing more detail can read these two papers.

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 $\rk_P: P\longrightarrow\{0,1,\ldots,n\}$, then for $S\subseteq [n-1]$, we can define the rank selected subposet of $P$ by

\[
P_S\coloneqq\{x\in P~:~\rk_P(x)\in S\}\cup\{\hat{0},\hat{1}\}.
\]

Let $G$ be a group of automorphisms of $P$. 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 Möbius inversion, we also have

\[
\alpha_P(S)=\sum_{T\subseteq S}\beta_P(T).
\]

If $P$ is Cohen-Macaulay, then $\beta_P(S)$ is a genuine representation of $G$ and

\[
\beta_P(S)\cong_G \tilde{H}_{|S|-1}(P_S).
\]

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.

When the poset $P$ is the Boolean lattice $B_n$ and $G$ is the symmetric group $\S_n$, the representation $\beta_{B_n}(S)$ is known for any selected rank set $S\subseteq [n-1]$. Here we follow the notations from <cit.>.

Let $\lambda$, $\mu$ be two partitions such that $\mu\subseteq\lambda$ (i.e. $\mu_i\le \lambda_i$ for all $i$). The skew shape $\lambda/\mu$ is said to be a skew hook (or a ribbon or a border strip) if its corresponding Ferrer diagram does not contain any $2\times 2$ subdiagram. A skew hook can be described in terms of its number of cells and its descent set. Given a skew hook $H$ of $n$ cells, we label the cells of $H$ with numbers $1$ through $n$, starting from the southwestern end cell, through the adjacent cell, ending at the northeastern end cell. We say a cell $i$ is a descent of $H$ if cell $i+1$ is directly above cell $i$. The descent set of $H$ is the set of all descents of $H$. If the descent set of $H$ is $R\subseteq [n-1]$, then we denote $H$ by $H_{R,n}$.

Let $n=7$. Then the diagrams of $H_{\{2,4\},7}$ and $H_{\{1,3\},7}$ are

\[
\begin{array}{ccccc}
H_{\{2,4\},7} &   &    &    & H_{\{1,3\},7} \\
& &      &  &            \\
\begin{ytableau}
\none & \none & 5 & 6 & 7\\
\none & 3 & *(green)4\\
1     & *(green)2
\end{ytableau}
&   &     &     &
\begin{ytableau}
\none & 4 & 5 & 6 & 7\\
2     & *(green)3 \\
*(green) 1
\end{ytableau}
\end{array}.
\]

Let $S^{\lambda/\mu}$ be the Specht module of skew shape $\lambda/\mu$. If $\lambda/\mu$ is a skew hook, the corresponding Specht module is known as a Foulkes representation.

Let $R\subseteq [n-1]$. Then

\[
\tilde{H}_{|R|-1}({(B_n)}_R)\cong_{\S_n} S^{H_{R,n}}.
\]

§ PARALLELITY BETWEEN REPRESENTATIONS OF $\S_N$ AND UNIPOTENT REPRESENTATIONS OF $GL_N(\MATHBB{F}_Q)$

Our goal is to study the representations of $\S_n$ on $A(U_{r,n})$ and $\widetilde{A}(U_{r,n})$ and the representations of $GL_n(q)$ on $A(U_{r,n}(q))$ and $\widetilde{A}(U_{r,n}(q))$ and obtain the augmented counterpart of Hameister, Rao, and Simpson's formula of the Hilbert series of $A(U_{r,n}(q))$. In this section, we will introduce Steinberg's results on unipotent representations of $GL_n(q)$ in [32] and apply it on $q$-uniform matroids. It enables us to unify the results on Frobenius series of (augmented) Chow rings of $U_{r,n}$ and the Hilbert series of (augmented) Chow rings of $U_{r,n}(q)$.

For a matroids $M$, we write $A(M)_{\mathbb{C}}\coloneqq\mathbb{C}\otimes_{\mathbb{Q}}A(M)$ and  $\widetilde{A}(M)_{\mathbb{C}}\coloneqq\mathbb{C}\otimes_{\mathbb{Q}}\widetilde{A}(M)$.
Recall from the last paragraph of Section <ref> that the Chow ring  $A(U_{r,n})$ and the augmented Chow ring $\widetilde{A}(U_{r,n})$ are $\S_n$-permutation modules having $FY(U_{r,n})$ and $\widetilde{FY}(U_{r,n})$ as permutation bases respectively for $1\le r\le n$. Hence, so are $A(U_{r,n})_{\mathbb{C}}$ and $\widetilde{A}(U_{r,n})_{\mathbb{C}}$.

Since the basis elements are monomials indexed by chains in the Boolean lattice $B_n$, recalling our notation from Section <ref>, both $A(U_{r,n})_{\mathbb{C}}$ and $\widetilde{A}(U_{r,n})_{\mathbb{C}}$ are isomorphic to the direct sum of $\S_n$-permutation modules $\alpha_{B_n}(S)$, which is generated by the maximal chains of the rank selected poset $(B_n)_S$ for $S\subseteq [n-1]$. If $S=\{s_1<s_2<\ldots<s_\ell\}$, then

\begin{equation} \label{eq:PermRepBooleanChain}
\alpha_{B_n}(S)\cong_{\S_n} \mathbb{C}[\S_n/\S_{\nu}]\cong_{\S_n} \mathbf{1}\uparrow_{\S_{\nu}}^{\S_n},
\end{equation}

where $\nu$ is the comoposition $\nu(S)=(s_1,s_2-s_1,\ldots,s_\ell-s_{\ell-1},n-s_{\ell})$ and $\S_{\nu}$ is the Young subgroup for the composition $\alpha$. A well-known fact in the representation theory of $\S_n$ states that the induced representation (<ref>) can be decomposed into irreducible $\S_n$-modules $\{S^{\lambda}\}_{\lambda\vdash n}$ as

\begin{equation} \label{eq:MDecomposionSn}
\mathbf{1}_{\S_{\nu}}\uparrow_{\S_{\nu}}^{\S_n}\cong_{\S_n} \bigoplus_{\lambda\vdash n}K_{\lambda,\nu}S^{\lambda}
\end{equation}

where $K_{\lambda,\nu}$ is the well-known Kostka number (see Propostion 5.3.6 in [24]).

Similarly, since the general linear group $GL_n(q)\coloneqq GL_n(\mathbb{F}_q)$ acts on the $q$-Boolean lattice $B_n(q)$, i.e., the subspace lattice of $\left(\mathbb{F}_q\right)^n$, in an obvious way, both $A(U_{r,n}(q))_{\mathbb{C}}$ and $\widetilde{A}(U_{r,n}(q))_{\mathbb{C}}$ are permutation representations of $GL_n(q)$ with permutation bases $FY(U_{r,n}(q))$ and $\widetilde{FY}(U_{r,n}(q))$, respectively, for $1\le r\le n$. Since the basis elements are monomials indexed by chains in the $q$-Boolean lattice $B_n(q)$, both $A(U_{r,n}(q))_{\mathbb{C}}$ and $\widetilde{A}(U_{r,n}(q))_{\mathbb{C}}$ are isomorphic to the direct sum of $GL_n(q)$-permutation modules $\alpha_{B_n(q)}(S)$, which is generated by the maximal chains of the rank selected poset $(B_n(q))_S$ for $S\subseteq [n-1]$. If $S=\{s_1<s_2<\ldots<s_\ell\}$, then

\[
\alpha_{B_n(q)}(S)\cong_{GL_n(q)} \mathbb{C}[GL_n(q)/P_{\nu}]\cong_{GL_n(q)}  \mathbf{1}_{P_{\nu}}\uparrow_{P_{\nu}}^{GL_n(q)},
\]

where $\nu=(s_1,s_2-s_1,\ldots,s_\ell-s_{\ell-1},n-s_{\ell})$ and $P_{\nu}$ is the maximal parabolic subgroup of $GL_n(q)$ that stabilizes the flag

\[
\langle e_1,\ldots,e_{s_1}\rangle\subsetneq \langle e_1,\ldots, e_{s_1},\ldots,e_{s_2}\rangle\subsetneq\ldots\subsetneq \langle e_1,\ldots,e_{s_1},\ldots,e_{s_2},\ldots,e_{s_\ell}\rangle,
\]

i.e., the subgroup of matrices having main diagonal block matrices of size $s_1$,$s_2-s_1$,$\ldots$,$n-s_{\ell}$ and $0$ for entries below the main diagonal block matrices.

There is a parallelity between the representations of $\S_n$ and some representations of $GL_n(q)$. In the following, we will briefly describe this story mostly following <cit.> and the lecture notes <cit.>. The reader interested in knowing more details can read these references.

Steinberg [32] showed that the decomposition of $\mathbf{1_{P_{\nu}}}\uparrow_{P_{\nu}}^{GL_n(q)}$ into irreducible representations of $GL_n(q)$ is parallel to the decomposition of $\mathbf{1_{\S_{\nu}}}\uparrow_{\S_{\nu}}^{\S_n}$ given in (<ref>). More precisely, there exists a subset $\{S^{\lambda}_q\}$ of the set of irreducible representations of $GL_n(q)$ indexed by partitions $\lambda$ of $n$ such that for any composition $\alpha$, we have

\begin{equation} \label{eq:parallel}
\mathbf{1}\uparrow_{P_{\nu}}^{GL_n(q)}\cong_{GL_n(q)} \bigoplus_{\lambda\vdash n}K_{\lambda,\nu}S^{\lambda}_q.
\end{equation}

These irreducible representations $\{S^{\lambda}_q\}_{\lambda\vdash n}$ studied by Steinberg are now called the unipotent representations of $GL_n(q)$. Let $\chi_q^\lambda$ be the character of the unipotent representation $S_q^\lambda$ for any partition $\lambda$. Let $\C(GL_n(q))$ be the free $\mathbb{Z}$-module spanned by the characters of unipotent reprentations of $GL_n(q)$ and $\C(GL(q))\coloneqq \bigoplus_{n=0}^\infty \C(GL_n(q))$. One can make $\C(GL(q))$ a $\mathbb{Z}$-graded algebra via the parabolic induction product:

\begin{align*}
\C(GL_{n_1}(q))\times \C(GL_{n_2}(q)) &\longrightarrow \C(GL_{n_1+n_2}(q))\\
(f_1, f_2) & \longmapsto f_1*f_2\coloneqq \left(f_1\otimes f_2 \Uparrow_{GL_{n_1}(q)\times GL_{n_2}(q)}^{P_{(n_1,n_2)}}\right)\uparrow_{P_{(n_1,n_2)}}^{GL_{n_1+n_2}(q)}
\end{align*}

where $(-)\Uparrow_{GL_{n_1}(q)\times GL_{n_2}(q)}^{P_{n_1,n_2}}$ is the inflation operation that precomposes a representation of $GL_{n_1}(q)\times GL_{n_2}(q)$ with the surjective map $P_{n_1,n_2}\twoheadrightarrow GL_{n_1}(q)\times GL_{n_2}(q)$ given by

\[
\left[\begin{array}{cc}
A  &  C \\
0  &  B
\end{array}\right]\mapsto
\left[\begin{array}{cc}
A  &  0 \\
0  &  B
\end{array}\right].
\]

Because of the parallelity in (<ref>), there is a $q$-analog of the Frobenius characteristic map, which is also a $\mathbb{Z}$-graded isomorphism between $\C(GL(q))$ and the ring of symmetric functions $\Lambda_\mathbb{Z}$ over $\mathbb{Z}$,

\[
\ch_q: \C(GL(q))\longrightarrow \Lambda_{\mathbb{Z}}
\]

sending $\ch_q(\chi^{\lambda}_q)=s_{\lambda}$ and $\ch_q(\mathbf{1}_{GL_n(q)})=h_n$. Since the one-to-one correspondence between the representation of the characters and the isomorphism classes of representations, we sometimes let the Frobenius map $\ch$ and $q$-Frobenius map $\ch_q$ take representations instead of characters as inputs.

If we consider the representations $\alpha_{B_n}(S)$ and $\alpha_{B_n(q)}(S)$ for the subset $S\subseteq [n-1]$ that corresponds to the composition $\nu=\nu(S)=(\nu_1,\ldots,\nu_{\ell+1})$, then we have

\[
\ch_q(\mathbf{1}\uparrow_{P_{\nu}}^{GL_n(q)})=\ch(\mathbf{1}\uparrow_{\S_{\nu}}^{\S_n})=h_{\nu_1}h_{\nu_2}\ldots h_{\nu_{\ell+1}}=h_{\nu}.
\]

Moreover, the dimension of the representation satisfies

\begin{equation} \label{eq:DimIsPSQ}
\dim_{\mathbb{C}}(\mathbf{1}\uparrow_{P_{\nu}}^{GL_n(q)})=|GL_n(q)/P_{\nu}|=\qbin{n}{\nu_1,\ldots,\nu_{\ell+1}}=\prod_{i=1}^n(1-q^i)
\ps(h_{\nu}).
\end{equation}

Recall the conditions that the monomials satisfy in $FY(U_{r,n})$ and $FY(U_{r,n}(q))$. Notice that for any fixed $0\le i\le r-1$, both the multiplicity of $\alpha_{B_n}(S)$ in $A^i(U_{r,n})_{\mathbb{C}}$ and  $\alpha_{B_n(q)}(S)$ in $A^i(U_{r,n}(q))_{\mathbb{C}}$ for $S=\{s_1<s_2<\ldots<s_{\ell}\}\subseteq[n-1]$ are the number of integer solutions to the equation

\[
\left\{\begin{array}{l}
1\le a_j\le s_j-s_{j-1}-1 \text{ for }j=1,\ldots,\ell \quad (\text{set }s_0\coloneqq 0)\\
0\le b\le r-s_{\ell}-1
\end{array}
\right..
\]

This implies that the $\S_n$-represetation on $A^i(U_{r,n})_{\mathbb{C}}$ and the $GL_n(q)$-representation on $A^i(U_{r,n}(q))_{\mathbb{C}}$ give the same results after applying the Frobenius map and the $q$-Frobenius map respectively. A similar thing occurs for the $\S_n$-represetations on $\widetilde{A}^i(U_{r,n})_{\mathbb{C}}$ and the $GL_n(q)$-representations on $\widetilde{A}^i(U_{r,n}(q))_{\mathbb{C}}$ as well, i.e., for all $i$,

\begin{equation} \label{eq:FrobIsQFrob}
\ch(A^i(U_{r,n})_{\mathbb{C}})=\ch_q(A^i(U_{r,n}(q)_{\mathbb{C}}) \quad \text{ and }\quad \ch(\widetilde{A}^i(U_{r,n})_{\mathbb{C}})=\ch_q(\widetilde{A}^i(U_{r,n}(q))_{\mathbb{C}}).
\end{equation}

Furthermore, the $\mathbb{Q}$-dimensions of $A^i(U_{r,n}(q))$ and $\widetilde{A}^i(U_{r,n}(q))$ can be obtained by taking the stable principal specialization of the symmetric functions $\ch(A^i(U_{r,n})_{\mathbb{C}})$ and $\ch(\widetilde{A}^i(U_{r,n})_{\mathbb{C}})$, respectively, for all $i$.

§ MAIN RESULTS

Following from Section <ref>, we know that the Frobenius characteristic on the (augmented) Chow ring of $U_{r,n}$ and the $q$-Frobenius characteristic on the (augmented) Chow ring of $U_{r,n}(q)$ gives the same symmetric function. And the Hilbert series of the (augmented) Chow ring of the $U_{r,n}(q)$ can be obtained by taking the stable principal specialization of this symmetric function. Therefore, to study these $\S_n$ and $GL_n(q)$-representations and their dimensions, it is sufficient to compute the Frobenius series of the (augmented) Chow ring of $U_{r,n}$.

In this section, we state without proof of our main results. These include formulas for the Frobenius series of the (augmented) Chow ring of $U_{n,r}$ and its $q$-analog, the Hilbert series of the (augmented) Chow ring of $U_{r,n}(q)$, and the $(-1)$-evaluation of the Hilbert series. From our previous discussion, we only need to prove the formulas for the Frobenius series of the (augmented) Chow ring of $U_{n,r}$ and the $(-1)$-evaluation of the Hilbert series. We postpone the proofs to the upcoming sections.

§.§ Frobenius series and $q$-Frobenius series

Let $V=\bigoplus_i V^i$ be a graded $\mathbb{C}[\S_n]$-module and $W=\bigoplus_{j} W^j$ be a graded $\mathbb{C}[GL_n(q)]$-module constituted by unipotent representations of $GL_n(q)$. The graded Frobenius series of $V$ is defined to be $\grFrob(V,t)\coloneqq \sum_{i}\ch(V^i)t^i$. Similarly, we can define the $q$-graded Frobenius series of $W$ by

\[
\grFrob_q(W,t)\coloneqq\sum_{j}\ch_q(W^j)t^j.
\]

We have seen in (<ref>) that

\[
\grFrob(A(U_{r,n})_{\mathbb{C}},t)=\grFrob_q(A(U_{r,n}(q))_{\mathbb{C}},t)
\]

\[
\grFrob(\widetilde{A}(U_{r,n})_{\mathbb{C}},t)=\grFrob_q(\widetilde{A}(U_{r,n}(q))_{\mathbb{C}},t)
\]

for all $1\le r\le n$.
Then we have the following formulas for Chow rings of $U_{r,n}$ and $U_{r,n}(q)$.

The series $\grFrob(A(U_{r,n})_{\mathbb{C}},t)$ and $\grFrob_q(A(U_{r,n}(q))_{\mathbb{C}},t)$ coincide and have two expressions:

(i) \(\displaystyle \sum_{j=0}^{r-1}h_{n-j}Q_j^0(\x,t)(1+t+\ldots+t^{r-j-1})\),

(ii) \(\displaystyle Q_n(\x,t)-\sum_{j=r}^{n-1}\sum_{\substack{\sigma\in\S_n\\ \fix(\sigma)\ge n-j}}F_{\DEX(\sigma),n}t^{j-\exc(\sigma)}\).

For the special case that $r=n$ and $r=n-1$, we have

(i) \(\displaystyle \grFrob(A(U_{n,n})_{\mathbb{C}},t)=\grFrob_q(A(U_{n,n}(q))_{\mathbb{C}},t) =Q_n(\x,t),\)

(ii)\(\displaystyle \grFrob(A(U_{n-1,n})_{\mathbb{C}},t)=\grFrob_q(A(U_{n-1,n}(q))_{\mathbb{C}},t) =t^{-1}Q_n^0(\x,t).\)

The fact that $\grFrob(A(U_{n,n})_{\mathbb{C}},t)=Q_n(\x,t)$ in Corollary <ref> is proved in <cit.> by constructing an $\S_n$-equivariant bijection between $FY(U_{n,n})$ and the Stembridge codes introduced by Stembridge in [33].

Recall from Theorem <ref> that $\widetilde{\S}_n$ is the set of decorated permutations of length $n$ and

\[
\widetilde{Q}_n(\x,t)=\sum_{\sigma\in\widetilde{\S}_n}F_{\DEX(\sigma),n}(\x)t^{\exc(\sigma)+1}.
\]

Then the ($q$-)Frobenius series of the augmented Chow rings of $U_{r,n}$ and $U_{r,n}(q)$ have expressions similar to Theorem <ref>.

The series $\grFrob(\widetilde{A}(U_{r,n})_{\mathbb{C}},t)$ and $\grFrob_q(\widetilde{A}(U_{r,n}(q))_{\mathbb{C}},t)$ coincide and have two expressions:

(i) \(\displaystyle h_n+t\sum_{j=0}^{r-1}h_{n-j}Q_j(\x,t)(1+t+\ldots+t^{r-j-1})\),

(ii) \(\displaystyle \widetilde{Q}_n(\x,t)-\sum_{j=r}^{n-1}\sum_{\substack{\sigma\in\widetilde{\S}_n\\ \fixTwo(\sigma)\ge n-j}}F_{\DEX(\sigma),n}t^{j-\exc(\sigma)}\),

where $\fixTwo(\sigma)$ is the number of $0$s in the one-line notation of the decorated permutation $\sigma$.

For the special case that $r=n$ and $r=n-1$, we have

(i) \(\displaystyle \grFrob(\widetilde{A}(U_{n,n})_{\mathbb{C}},t)=\grFrob_q(\widetilde{A}(U_{n,n}(q))_{\mathbb{C}},t) =\widetilde{Q}_n(\x,t).\)

(ii)\(\displaystyle \grFrob(\widetilde{A}(U_{n-1,n})_{\mathbb{C}},t)=\grFrob_q(\widetilde{A}(U_{n-1,n}(q))_{\mathbb{C}},t) =Q_n(\x,t).\)

The fact that $\grFrob(\widetilde{A}(U_{n,n})_{\mathbb{C}},t)=\widetilde{Q}_n(\x,t)$ is proved in <cit.> by constructing an $\S_n$-equivariant bijection between the basis $\widetilde{FY}(U_{n,n})$ and the extended codes in [19].

Combining Theorem <ref> (i) and Corollary <ref> (ii) in the case of $r=n-1$ gives the following recurrence relation for $Q_n(\x,t)$

\[
\]

which also appears as Corollary 4.1 $(r=1)$ in [25].

§.§ Hilbert series: $q$-uniform matroids

In this subsection, we present our results for the Hilbert series of the Chow rings and the augmented Chow rings of $U_{r,n}(q)$. The Hilbert series can be obtained by computing the dimension of the corresponding $GL_n(q)$-representations, which turns out to be the stable principal specialization of the corresponding $q$-Frobenius series. In the non-augmented case, we recover Hameister, Rao, and Simpson's formulas in (<ref>), (<ref>), and (<ref>). We note that all our results hold as stated in the case $q=1$ if we view $U_{r,n}(1)\coloneqq U_{r,n}$.

Recall from Section <ref> that for $D_n\subset\S_n\subset\widetilde{\S}_n$ we have the $q$-Eulerian polynomials

\begin{align*}
& d_n(q,t)\coloneqq\sum_{\sigma\in D_n}q^{\maj(\sigma)-\exc(\sigma)}t^{\exc(\sigma)},\\
& A_n(q,t)\coloneqq\sum_{\sigma\in\S_n}q^{\maj(\sigma)-\exc(\sigma)}t^{\exc(\sigma)},\\
& \widetilde{A}_n(q,t)=\sum_{\sigma\in\widetilde{\S}_n}q^{\maj(\sigma)-\exc(\sigma)}t^{\exc(\sigma)+1},
\end{align*}

and table <ref> that summarizes the relations between the combinatorial models and their enumerators lifted to different extensions.

Applying the operator $\prod_{i=1}^n(1-q^i)\ps(-)$ to the two identities in Theorem <ref>, we obtain the following two expressions for the Hilbert series of $A(U_{r,n}(q))$. The second expression recovers Theorem 1.1 in [15] (See (<ref>)).

The Hilbert series $\Hilb(A(U_{r,n}(q)),t)$ has the following two expressions

(i) \(\displaystyle \sum_{j=0}^{r-1}\qbin{n}{j}d_j(q,t)(1+t+\ldots+t^{r-j-1})\),

(ii) ([15]) \(\displaystyle\sum_{\sigma\in\S_n}q^{\maj(\sigma)-\exc(\sigma)}t^{\exc(\sigma)}-\sum_{j=r}^{n-1}\sum_{\substack{\sigma\in\S_n\\ \fix(\sigma)\ge n-j}}q^{\maj(\sigma)-\exc(\sigma)}t^{j-\exc(\sigma)}.\)

The special case of $q=1$ of Theorem <ref> (i) was also found by Ferroni, Matherne, Stevens, and Vecchi  [11].

Applying the operator $\prod_{i=1}^n(1-q^i)\ps(-)$ to Corollary <ref>, we recover Hameister, Rao, and Simpson's formulas (<ref>), (<ref>).

For the special case that $r=n$ and $r=n-1$ we have

(i) \(\displaystyle \Hilb(A(U_{n,n}(q),t) =\sum_{\sigma\in\S_n}q^{\maj(\sigma)-\exc(\sigma)}t^{\exc(\sigma)}=A_n(q,t).\)

(ii)\(\displaystyle \Hilb(A(U_{n-1,n}(q)),t) =\sum_{\sigma\in D_n}q^{\maj(\sigma)-\exc(\sigma)}t^{\exc(\sigma)-1}=t^{-1}d_n(q,t).\)

Applying the operator $\prod_{i=1}^n(1-q^i)\ps(-)$ to the two identities in Theorem <ref>, we obtain the following new expressions for the Hilbert series of $\widetilde{A}(U_{r,n}(q))$.

The Hilbert series $\Hilb(\widetilde{A}(U_{r,n}(q)),t)$ has the following two expressions

(i) \(\displaystyle 1+t\sum_{j=0}^{r-1}\qbin{n}{j}A_j(q,t)(1+t+\ldots+t^{r-j-1})\),

(ii) \(\displaystyle\sum_{\sigma\in\widetilde{\S}_n}q^{\maj(\sigma)-\exc(\sigma)}t^{\exc(\sigma)+1}-\sum_{j=r}^{n-1}\sum_{\substack{
\sigma\in\widetilde{\mathfrak{S}}_n\\
\fixTwo(\sigma)\ge n-j}}q^{\maj(\sigma)-\exc(\sigma)}t^{j-\exc(\sigma)}\).

The special case of $q=1$ of Theorem <ref>(i) was also found by Ferroni, Matherne, Stevens, and Vecchi  [11].

Applying the operator $\prod_{i=1}^n(1-q^i)\ps(-)$ to Corollary <ref> yields the following:

For the special case that $r=n$ and $r=n-1$ we have the following

(i)\(\displaystyle \Hilb(\widetilde{A}(U_{n,n}(q)),t)=\sum_{\sigma\in\widetilde{\S}_n}q^{\maj(\sigma)-\exc(\sigma)}t^{\exc(\sigma)+1}=\widetilde{A}_n(q,t)\),

§.§ $(-1)$-evaluation of the Hilbert series and Charney-Davis quantities

In this subsection, we state our results regarding the $(-1)$-evaluations of the Hilbert series for Chow rings and augmented Chow rings of $U_{r,n}(q)$ or equivalently, their Charney-Davis quantities. In fact, we obtain equivariant formulas for general matroids.
The proofs of the two general theorems, Theorem <ref> and Theorem <ref>, will be postponed to Section <ref>. The special cases of uniform matroids and $q$-uniform matroids follow from the general formulas right away.

Let $M$ be a matroid on $E=[n]$ of rank $r$ with the lattice of flats $\L(M)$. We say a subset $I\subseteq E$ is an independent subset if $\rk(I)=|I|$.
Then a permutation $g\in\S_n$ is an automorphism of $M$ if $g(I)$ is independent if and only if $I$ is independent.
We denote by $\Aut(M)$ the group of automorphisms of $M$.

For any $g\in\Aut(M)$, the action of $g$ preserves the rank of the flats, i.e., $\rk_M(g(F))=\rk_M(F)$ for any flat $F\in\L(M)$. It is easy to check that any subgroup $G$ of $\Aut(M)$ acts on both $A(M)$ and $\widetilde{A}(M)$.

Let $R_{\mathbb{C}}(G)$ be the representation ring of the subgroup $G$ of $\Aut(M)$. Consider ourselves working in the polynomial ring $R_{\mathbb{C}}(G)[t]$ over the $\mathbb{Z}$-algebra $R_{\mathbb{C}}(G)$. Then evaluating $t$ at $-1$ in the $G$-equivariant Hilbert series of $A(M)_{\mathbb{C}}$ gives us the following.

Let $M$ be a (loopless) matroid on $[n]$ of rank $r$,

\[
\sum_{i=0}^{r-1}(-1)^i A^i(M)_{\mathbb{C}}\cong_{G}
\begin{cases}
0   &   \text{ if $r$ is even}\\
& \text{ if $r$ is odd}
\end{cases}.
\]

If $M$ is the uniform matroid $U_{r,n}$, then $\Aut(U_{r,n})=\S_n$. Note that $\L(U_{r,n})$ is the rank selected poset ${(B_n)}_{[r-1]}$ of the Boolean lattice $B_n$. Then by Theorem <ref>,

\[
\tilde{H}_{\frac{r-3}{2}}\left(\L(U_{r,n})_{\{2,4,\ldots,r-1\}}\right)=\tilde{H}_{\frac{r-3}{2}}\left({B_n}_{\{2,4,\ldots,r-1\}}\right)\cong_{\S_n} S^{H_{\{2,4,\ldots,r-1\},n}}.
\]

Applying Theorem <ref> to uniform matroids $U_{r,n}$, we obtain the following new results.

For any positive integer $n$ and $1\le r\le n$,

\[
\sum_{i=0}^{r-1}(-1)^i A^i(U_{r,n})_{\mathbb{C}}\cong_{\S_n}
\begin{cases}
0   &   \text{ if $r$ is even}\\
& \text{ if $r$ is odd}
\end{cases}.
\]

or equivalently as graded Frobenius series

\[
\grFrob(A(U_{r,n})_{\mathbb{C}},-1)=
\begin{cases}
0   &   \text{ if $r$ is even}\\
& \text{ if $r$ is odd}
\end{cases}.
\]

where $s_{D}$ is the Schur function of skew shape $D$.

Apply $\prod_{i=1}^n(1-q^i)\ps(-)$ to the above symmetric function equality. We have

\[
\prod_{i=1}^n(1-q^i)\ps(s_{H_{\{2,4,\ldots,r-1}\},n})=\sum_{\substack{\sigma\in\S_n \\ \DES(\sigma)=\{2,4,\ldots,r-1\}}}q^{\maj(\sigma^{-1})}=\sum_{\substack{\sigma\in\S_n \\ \DES(\sigma)=\{2,4,\ldots,r-1\}}}q^{\inv(\sigma)};
\]

see <cit.> and its proof for detailed discussion of the above identity. In this way, we show that Hameister, Rao, and Simpson's $(-1)$-evaluation of the Hilbert series (<ref>) (and (<ref>)) can be significantly simplified to an elegant combinatorial interpretation, which is new even in the case of $q=1$.

For any positive integer $n$ and $1\le r\le n$,

\[
\Hilb(A(U_{r,n}(q)),-1)=\begin{cases}
0  & \mbox{ if $r$ is even}\\
\displaystyle (-1)^{\frac{r-1}{2}}\!\!\!\!\!\sum_{\substack{\sigma\in\S_n\\ \DES(\sigma)=\{2,4,6,\ldots,r-1\}}}q^{\inv(\sigma)} & \mbox{ if $r$ is odd}
\end{cases}.
\]

When $r$ is odd, by Example 2.2.4 in [31], the evaluation can be expressed as one determinant

\[
\begin{array}{cccccc}
\frac{1}{[2]_q!} &         1        &   0   & \ldots & 0  & 0\\
\frac{1}{[4]_q!} & \frac{1}{[2]_q!} &   1   & \ldots & 0  & 0\\
\frac{1}{[6]_q!} & \frac{1}{[4]_q!} &\frac{1}{[2]_q!} &\ldots & 0 & 0\\
\vdots           & \vdots           & \vdots& \ddots & \vdots & \vdots\\
\frac{1}{[r-1]_q!} & \frac{1}{[r-3]_q!} & \frac{1}{[r-5]_q!} & \ldots & \frac{1}{[2]_q!} & 1\\
\frac{1}{[n]_q!} & \frac{1}{[n-2]_q!}& \frac{1}{[n-4]_q!} & \ldots & \frac{1}{[n-r+3]_q!} & \frac{1}{[n-r+1]_q!}
\end{array}
\right|.
\]

The combinatorial interpretation in Corollary <ref> when $r$ is odd can also be obtained from (<ref>) by realizing the alternating sum as an inclusion-exclusion process.

Combining with Corollary <ref>, Corollary <ref> generalizes Foata and Han's result in <cit.>.

In the augmented case, the $(-1)$-evalution of the $G$-equivariant Hilbert series of $\widetilde{A}(M)_{\mathbb{C}}$ gives the following.

Let $M$ be a matroid on $[n]$ of rank $r$. Then

\[
\sum_{i=0}^{r}(-1)^i\widetilde{A}^i(M)_{\mathbb{C}}\cong_{G}
\begin{cases}
(-1)^{\frac{r}{2}}\tilde{H}_{\frac{r-2}{2}}\left(\L(M)_{\{1,3,\ldots,r-1\}}\right)   &   \text{ if $r$ is even}\\
0   & \text{ if $r$ is odd}
\end{cases}
\]

When $M$ is the uniform matroid, by an argument similar to the non-augmented case, we obtain the following new results.

For any positive integer $n$ and $1\le r\le n$,

\[
\sum_{i=0}^{r}(-1)^i \widetilde{A}^i(U_{r,n})_{\mathbb{C}}\cong_{\S_n}
\begin{cases}
(-1)^{\frac{r}{2}}S^{H_{\{1,3,\ldots,r-1\},n}}   &   \text{ if $r$ is even}\\
0    & \text{ if $r$ is odd}
\end{cases}.
\]

or equivalently as graded Frobenius series

\[
\grFrob(\widetilde{A}(U_{r,n})_{\mathbb{C}},-1)=
\begin{cases}
(-1)^{\frac{r}{2}}s_{H_{\{1,3,\ldots,r-1\},n}}   &   \text{ if $r$ is even}\\
0,   & \text{ if $r$ is odd}
\end{cases}
\]

Apply $\prod_{i=1}^n(1-q^i)\ps(-)$ to the above symmetric function equality. Similar to the non-augmented case, we have

\[
\prod_{i=1}^n(1-q^i)\ps(s_{H_{\{1,3,\ldots,r-1}\}})=\sum_{\substack{\sigma\in\S_n \\ \DES(\sigma)=\{1,3,\ldots,r-1\}}}q^{\maj(\sigma^{-1})}=\sum_{\substack{\sigma\in\S_n \\ \DES(\sigma)=\{1,3,\ldots,r-1\}}}q^{\inv(\sigma)};
\]

Therefore, we have the following identity analogous to Corollary <ref>

For any positive integer $n$ and $1\le r\le n$,

\[
\Hilb(\widetilde{A}(U_{r,n}(q)),-1)=\begin{cases}
\displaystyle (-1)^{\frac{r}{2}}\sum_{\substack{\sigma\in\S_n\\ \DES(\sigma)=\{1,3,5,\ldots,r-1\}}}q^{\inv(\sigma)} & \mbox{ if $r$ is even}\\
0  & \mbox{if $r$ is odd}
\end{cases}.
\]

When $r$ is even, by Example 2.2.4 in [31], the evaluation can be expressed as the determinant

\[
\begin{array}{cccccc}
\frac{1}{[1]_q!} &         1        &   0   & \ldots & 0  & 0\\
\frac{1}{[3]_q!} & \frac{1}{[2]_q!} &   1   & \ldots & 0  & 0\\
\frac{1}{[5]_q!} & \frac{1}{[4]_q!} &\frac{1}{[2]_q!} &\ldots & 0 & 0\\
\vdots           & \vdots           & \vdots& \ddots & \vdots & \vdots\\
\frac{1}{[r-1]_q!} & \frac{1}{[r-2]_q!} & \frac{1}{[r-4]_q!} & \ldots & \frac{1}{[2]_q!} & 1\\
\frac{1}{[n]_q!} & \frac{1}{[n-1]_q!}& \frac{1}{[n-3]_q!} & \ldots & \frac{1}{[n-r+3]_q!} & \frac{1}{[n-r+1]_q!}
\end{array}
\right|.
\]

In the case of even $r$, it is easy to see by the principle of inclusion-exclusion that the combinatorial interpretation can also be expressed as the following form analogous to the one in (<ref>)

\begin{equation} \label{eq:FormTwoSign}
\end{equation}

where $E^*_{2k+1}(q)=\sum_{\sigma\in\Alt_{2k+1}}q^{\inv(\sigma)}$ and $
\Alt_{2k+1}=\{\sigma\in\S_{2k+1}:\sigma_1>\sigma_2<\sigma_3\ldots<\sigma_{2k+1}\}$ is the set of alternating permutations in $\S_{2k+1}$. Note that $E_n(q)$ and $E^*_n(q)$ are related by

\[
\]

We may interpret the $(-1)$-evaluation results in terms of the Charney-Davis quantity introduced in Section <ref>.

Given a graded $\mathbb{C}G$-module  $A=\bigoplus_{i=0}^n A^i$, we let $\Hilb^G(A,t)=\sum_{i=0}^n A^it^i$ be a polynomial in $R_{\mathbb{C}}(G)[t]$. Similarly to the non-equivariant case, we may define the $G$-equivariant Charney-Davis quantity of $A$ to be

\[
(-1)^{\frac{n}{2}}\Hilb^G(A,-1) & \text{ if $n$ is even}\\
\Hilb^G(A,-1)  & \text{ if $n$ is odd}
\end{cases}.
\]

Then Theorem <ref> and Theorem <ref> imply that the $G$-equivariant Charney-Davis quantity of $A(M)$ and $\widetilde{A}(M)$ are nonnegative in $R_{\mathbb{C}}(G)$, i.e., they are genuine representations of $G$.

Let $M$ be a (loopless) matroid on $[n]$ of rank $r$. Then

\[
0   &   \text{ if $r$ is even}\\
\tilde{H}_{\frac{r-3}{2}}\left(\L(M)_{\{2,4,\ldots,r-1\}}\right)
& \text{ if $r$ is odd}
\end{cases}
\]

\[
\begin{cases}
\tilde{H}_{\frac{r-2}{2}}\left(\L(M)_{\{1,3,\ldots,r-1\}}\right)   &   \text{ if $r$ is even}\\
0   & \text{ if $r$ is odd}
\end{cases}.
\]

By Corollary <ref>, the Charney-Davis quantity of $A(U_{r,n}(q))$ is a polynomial in $q$ with nonnegative coefficients. Letting $q=1$, we see that the Charney-Davis quantity of $A(U_{r,n})$ is nonnegative. Both of them admit combinatorial interpretations.

For any positive integer $n$ and $1\le r\le n$, we have

\[
0  & \mbox{ if $r$ is even}\\
\displaystyle\sum_{\substack{\sigma\in\S_n\\ \DES(\sigma)=\{2,4,6,\ldots,r-1\}}}q^{\inv(\sigma)} & \mbox{ if $r$ is odd}
\end{cases}
\]

\[
0  & \mbox{ if $r$ is even}\\
|\{\sigma\in\S_n: \DES(\sigma)=\{2,4,\ldots,r-1\}\}| & \mbox{ if $r$ is odd}
\end{cases}.
\]

When $r=n$, the Hilbert series of $A(U_{n,n})=A(B_n)$ is exactly the $h$-polynomial of the permutahedron. So Corollary <ref> recovers the well-known fact that the Charney-Davis quantity of the permutahedron is the Euler number $E_{n}$ when $n$ is odd, and is 0 otherwise.

By Corollary <ref>, the Charney-Davis quantity of $\widetilde{A}(U_{r,n}(q))$ is a polynomial in $q$ with nonnegative coefficients. Letting $q=1$, we see that the Charney-Davis quantity of $\widetilde{A}(U_{r,n})$ is nonnegative. Both admit combinatorial interpretations that are somehow compatible with the non-augmented cases.

For any positive integer $n$ and $1\le r\le n$, we have

\[
\displaystyle\sum_{\substack{\sigma\in\S_n\\ \DES(\sigma)=\{1,3,5,\ldots,r-1\}}}q^{\inv(\sigma)}  & \mbox{ if $r$ is even}\\
0 & \mbox{ if $r$ is odd}
\end{cases}.
\]

\[
|\{\sigma\in\S_n: \DES(\sigma)=\{1,3,\ldots,r-1\}\}|  & \mbox{ if $r$ is even}\\
0 & \mbox{ if $r$ is odd}
\end{array}\right..
\]

When $r=n$, the Hilbert series of $\widetilde{A}(U_{n,n})=\widetilde{A}(B_n)$ is exactly the $h$-polynomial of the stellahedron. So Corollary <ref> recovers the fact that the Charney-Davis quantity of the stellahedron is the Euler number $E_{n}$ when $n$ is even, and is 0 otherwise.

§ PROOFS FOR THE FROBENIUS SERIES OF THE CHOW RINGS

In this section, we will prove Theorem <ref> which give formulas for the Frobenius series of $A(U_{r,n})_{\mathbb{C}}$ and the graded $q-$Frobenius series of $A(U_{r,n}(q))_{\mathbb{C}}$. We have seen that the $q$-Frobenius series of $A(U_{r,n}(q))_{\mathbb{C}}$ agrees with the Frobenius series of $A(U_{r,n})_{\mathbb{C}}$. Therefore, it suffices to just consider $\grFrob(A(U_{r,n})_{\mathbb{C}},t)$. Our proof of Theorem <ref> is parallel to Hameister, Rao, and Simpson's proof of (<ref>), which is given in [15].

Before we start proving our main results, let us establish a few facts first. By
Theorem 6.9 in [19], we have the graded Frobenius series

\begin{equation} \label{eq:ChowBnIsQ}
\grFrob(A(\mathsf{B}_n)_{\mathbb{C}},t)=\grFrob(\mathbb{C}\Code_n,t)=Q_n(\x,t)=\sum_{\sigma\in\S_n}F_{\DEX(\sigma),n}(\x)t^{\exc(\sigma)}.
\end{equation}

Liao obtained the first equality by constructing an $\S_n$-equivariant bijection between the set $\Code_n$ of Stembridge codes and the FY-basis

\[
FY(\mathsf{B}_n)=\left\{x_{F_1}^{a_1}x_{F_2}^{a_2}\ldots x_{F_k}^{a_k}:\substack{~\emptyset=F_0\subsetneq F_1\subsetneq F_2\subsetneq\ldots\subsetneq F_k\subseteq [n],\\
1\le a_i\le |F_i|-|F_{i-1}|-1}\right\},
\]

which induces an isomorphism $\mathbb{C}\Code_n\cong_{\S_n} A(\mathsf{B}_n)_{\mathbb{C}}$ as permutation representations.

On the other hand, Shareshian and Wachs [25] introduced another variant of Eulerian quasisymmetric functions

\[
Q_{n,j,k}(\x)\coloneqq\sum_{\substack{\sigma\in\S_n\\ \exc(\sigma)=j \\ \fix(\sigma)=k}}F_{\DEX(\sigma),n}(\x)
\]

and showed that its generating function has the closed form

\[
1+\sum_{n\ge 1,j,k\ge 0}Q_{n,j,k}(\x)t^j r^k z^n=\frac{(1-t)H(rz)}{H(tz)-tH(z)}.
\]

Recall the $\S_n$-equivariant bijection $\phi:FY(\mathsf{B}_n)\rightarrow\Code_n$ from Theorem 3.6 in [19], which transfers the degree of the monomial in $FY(\mathsf{B}_n)$ to the index of the corresponding Stembridge code. Let $\Code_{n,j,k}$ be the subset of $\Code_n$ consisting of Stembridge codes of index $j$ and $k$ zeros. It is easy to see that $\phi$ can be restricted to a bijection between $\Code_{n,j,k}$ and the subset

\begin{equation} \label{eq:FYjk}
FY^{j,k}(\mathsf{B}_n)\coloneqq\left\{ x_{F_1}^{a_1}\ldots x_{F_\ell}^{a_\ell}\in FY^j(\mathsf{B}_n): |[n]-F_\ell|=k\right\}.
\end{equation}

Let $A^{j,k}(\mathsf{B}_n)$ be the subspace of $A^j(\mathsf{B}_n)$ spanned by $FY^{j,k}(\mathsf{B}_n)$.
Then slightly modifying (adding one variable $r$ that counts the number of zeros) Stembridge's proof for the Frobenius characteristic of $\mathbb{C}\Code_{n,j}$ in <cit.> gives the following theorem.

We have

\[
1+\sum_{n\ge 1 ,j,k\ge 0}\ch\left(\mathbb{C}\Code_{n,j,k}\right)t^j r^k z^n=\frac{(1-t)H(rz)}{H(tz)-tH(z)},
\]

or equivalently, $\ch(\mathbb{C}\Code_{n,j,k})=Q_{n,j,k}(\x)$.

Then immediately from the $\S_n$-equivariant bijection between $\Code_{n,j,k}$ and $FY^{j,k}(\mathsf{B}_n)$ obtained from restricting $\phi$, we have the following corollary.

We have $\ch(A^{j,k}(\mathsf{B}_n)_{\mathbb{C}})=Q_{n,j,k}(\x)$.

We need the following two properties of $Q_{n,j,k}$. The first one is about the palindromicity of $Q_{n,j,k}$ and is a special case of Theorem 5.9 in [25].

For positive integers $n$, $0\le k \le n$, and $0\le j\le n-k$,

\[
\]

For all $n,j,k$,

\[
\]

Now let

\begin{align*}
\Delta_{n,r}(\x)
\end{align*}

For $1\le r\le n-1$, the polynomial $\Delta_{n,r}(\x)\in\Lambda_n[t]$ has two expressions:

\[
\sum_{i=0}^r h_{n-r+i}Q_{r-i}^0(\x,t)t^i
\]

\[
\sum_{\substack{\sigma\in\S_n \\ \fix(\sigma)\ge n-r}}F_{\DEX(\sigma),n}(\x)t^{r-\exc(\sigma)}.
\]

Note that for $0\le k\le n-1$, as sets we have

\[
FY^k(U_{n,1})\subseteq FY^k(U_{n,2})\subseteq\ldots\subseteq FY^k(U_{n,n}).
\]

The symmetric function $a_{n,r}^{(k)}(\x)$ is the Frobenius characteristic of the permutation representation generated by $FY^k(U_{r+1,n})\setminus FY^k(U_{r,n})$, which consists of the monomial

\[
x_{[n]}^r \quad \text{ if }k=r
\]

and monomials

\[
x_{F_1}^{a_1}\ldots x_{F_\ell}^{a_{\ell}}x_{[n]}^i\in FY^k(U_{r+1,n})  \quad\text{such that}\quad i=(r+1)-|F_\ell|-1=r-|F_\ell|
\]

for $i=0,1,\ldots,k-1$ if $1\le k\le r-1$. Hence $a_{n,r}^{(r)}(\x)=h_n$; and for $1\le k\le r-1$, one can construct an easy $\S_n$-equivariant bijection $x_{F_1}^{a_1}\ldots x_{F_\ell}^{a_{\ell}}x_{[n]}^i\mapsto x_{F_1}^{a_1}\ldots x_{F_\ell}^{a_{\ell}}$ that maps $FY^k(U_{r+1,n})\setminus FY^k(U_{r,n})$ to the set of monomials $x_{F_1}^{a_1}\ldots x_{F_\ell}^{a_{\ell}}$ such that $F_1\subsetneq\ldots\subsetneq F_{\ell}$ and $a_1,\ldots,a_\ell$ satisfy

* $|F_\ell|=r-i$,

* $a_j\le |F_j|-|F_{j-1}|-1$ for all $j$,

* $a_1+\ldots+a_\ell=k-i$,

for $i=0,1,\ldots, k-1$. Recalling our notation in (<ref>), the above set of monomials can be viewed as $\bigcup_{i=0}^{k-1} FY^{k-i,n-r+i}(\mathsf{B}_n)$; since $FY^{k-i,n-r+i}(\mathsf{B}_n)$ is a permutation basis for $A^{k-i,n-r+i}(\mathsf{B}_n)_{\mathbb{C}}$, by Corollary <ref>, we have the coefficient of $t^k$ in $\Delta_{n,r}(\x)$ is

\begin{equation}
a_{n,r}^{(k)}(\x)=\sum_{i=0}^{k-1} Q_{n,k-i,n-r+i}  \label{eq:hQzero}
\end{equation}

\begin{align*}
\Delta_{n,r}(\x)
&=\sum_{k=1}^r a_{n,r}^{(k)}(\x)t^k=h_nt^r+\sum_{k=1}^{r-1}\sum_{i=0}^{k-1} h_{n-r+i}Q_{r-i,k-i,0}t^k \quad(\text{by Lemma \ref{lem:ExtractHQ}})\\
&=h_nt^r+\sum_{i=0}^{r-2} h_{n-r+i}\left(\sum_{k=i+1}^{r-1} Q_{r-i,k-i,0}t^{k-i}\right)t^i
=h_nt^r+\sum_{i=0}^{r-2} h_{n-r+i}Q_{r-i}^0(\x,t)t^i\\
&=\sum_{i=0}^{r} h_{n-r+i}Q_{r-i}^0(\x,t)t^i  \quad (\text{since }Q_1^0(\x,t)=0)
\end{align*}

Note that $\Delta_{n,1}(\x)=h_n t$.

On the other hand, we can obtain a different expression from (<ref>) as follows

\begin{align*}
\Delta_{n,r}(\x)
&=h_nt^r+\sum_{k=1}^{r-1}\sum_{i=0}^{k-1} Q_{n,k-i,n-r+i}t^k\\
&=h_nt^r+\sum_{k=1}^{r-1}\sum_{i=0}^{k-1} Q_{n,r-k,n-r+i}t^k \quad(\text{by Lemma \ref{lem:PalinQ}})\\
&=h_nt^r+\sum_{k=1}^{r-1}\sum_{\substack{\sigma\in\S_n \\ \exc(\sigma)=r-k \\ \fix(\sigma)\ge n-r}} F_{\DEX(\sigma),n}(\x)t^k\\
&=F_{\DEX(e),n}(\x)t^{r} +\sum_{\substack{\sigma\in\S_n\setminus\{e\} \\ \fix(\sigma)\ge n-r}}F_{\DEX(\sigma),n}(\x)t^{r-\exc(\sigma)}\\
&=\sum_{\substack{\sigma\in\S_n \\ \fix(\sigma)\ge n-r}}F_{\DEX(\sigma),n}(\x)t^{r-\exc(\sigma)}
\end{align*}

where $e$ is the identity in $\S_n$.

For $1\le r\le n-1$, the difference $\Hilb(A(U_{r+1,n}(q)),t)-\Hilb(A(U_{r,n}(q)),t)$ has two expressions:

\[
\sum_{i=0}^r\qbin{n}{i}d_{i}(q,t)t^{r-i}
\]

\[
\sum_{\substack{\sigma\in\S_n \\ \fix(\sigma)\ge n-r}}q^{\maj(\sigma)-\exc(\sigma)}t^{r-\exc(\sigma)}.
\]

Since we know the Hilbert series for $A(U_{r,n}(q))$ is given by the stable principal specialization of $\grFrob(A(U_{r,n}),t)(=\grFrob_q(A(U_{r,n}(q)),t))$, the stable principal specialization of $\Delta_{n,r}(\x)$ gives us $\Hilb(A(U_{r+1,n}(q)),t)-\Hilb(A(U_{r,n}(q)),t)$. Recalling the relations stated in table <ref> and Corollary <ref>, the formulas follow from Theorem <ref> after taking stable principal specialization of it.

We are now ready to prove Theorem <ref>, which we restate here.

The series $\grFrob(A(U_{r,n})_{\mathbb{C}},t)$ and $\grFrob_q(A(U_{r,n}(q))_{\mathbb{C}},t)$ coincide and have two expressions:

(i) \(\displaystyle \sum_{j=0}^{r-1}h_{n-j}Q_j^0(\x,t)(1+t+\ldots+t^{r-j-1})\),

(ii) \(\displaystyle Q_n(\x,t)-\sum_{j=r}^{n-1}\sum_{\substack{\sigma\in\S_n\\ \fix(\sigma)\ge n-j}}F_{\DEX(\sigma),n}t^{j-\exc(\sigma)}\).

To prove expression (i), we apply the first expression from Theorem <ref> and obtain

\begin{align*}
\grFrob(A(U_{r,n})_{\mathbb{C}},t)
&=h_n+\sum_{j=1}^{r-1}\sum_{i=0}^j h_{n-j+i}Q_{j-i}^0(\x,t)t^i\\
&=h_n+\sum_{j=1}^{r-1}\sum_{i=0}^j h_{n-i}Q_{i}^0(\x,t)t^{j-i}\\
&=h_n+h_n(t+t^2+\ldots+t^{r-1})+\sum_{j=1}^{r-1}\sum_{i=1}^j h_{n-i}Q_{i}^0(\x,t)t^{j-i}\\
\end{align*}

To prove expression (ii), we apply the second expression from Theorem <ref> and obtain

\begin{align*}
\grFrob(A(U_{r,n})_{\mathbb{C}},t)
&=Q_n(\x,t)-\sum_{j=r}^{n-1}\sum_{\substack{\sigma\in\S_n\\ \fix(\sigma)\ge n-j}}F_{\DEX(\sigma),n}t^{j-\exc(\sigma)}.
\end{align*}

From Theorem <ref>, we can prove Corollary <ref> which we restate here.

For the special case that $r=n$ and $r=n-1$ we have

(i) \(\displaystyle \grFrob(A(U_{n,n})_{\mathbb{C}},t)=\grFrob_q(A(U_{n,n}(q))_{\mathbb{C}},t) =Q_n(\x,t).\)

(ii)\(\displaystyle \grFrob(A(U_{n-1,n})_{\mathbb{C}},t)=\grFrob_q(A(U_{n-1,n}(q))_{\mathbb{C}},t) =t^{-1}Q_n^0(\x,t).\)

When $r=n$, the matroid is Boolean. This is the case in (<ref>).

When $r=n-1$, by Theorem <ref> (ii), we have

\begin{align*}
\grFrob(A(U_{r,n})_{\mathbb{C}},t)
&=Q_n(\x,t)-\sum_{\substack{\sigma\in\S_n \\ \fix(\sigma)\ge 1}}F_{\DEX(\sigma),n}t^{n-1-\exc(\sigma)}.
\end{align*}

Since $Q_{n}(\x,t)=\sum_{j=0}^{n-1}Q_{n,j}(\x)t^{j}$ and $Q_{n,j}=Q_{n,n-1-j}$ by Lemma <ref>, we have

\begin{align*}
\grFrob(A(U_{n-1,n})_{\mathbb{C}},t)
&=\sum_{\sigma\in\S_n}F_{\DEX(\sigma),n}t^{n-1-\exc(\sigma)}-\sum_{\substack{\sigma\in\S_n \\ \fix(\sigma)\ge 1}}F_{\DEX(\sigma),n}t^{n-1-\exc(\sigma)}\\
&=\sum_{\substack{\sigma\in\S_n \\ \fix(\sigma)=0}}F_{\DEX(\sigma),n}t^{n-1-\exc(\sigma)}\\
&=\sum_{j=1}^{n-1}Q_{n,n-j,0}(\x)t^{n-1-j} \quad \text{(by Lemma \ref{lem:PalinQ})}\\
\end{align*}

§ PROOFS FOR THE FROBENIUS SERIES OF THE AUGMENTED CHOW RINGS

In this section, we will prove Theorem <ref> which is the augmented analog of Theorem <ref>. Similarly, since the $q$-Frobenius series of $\widetilde{A}(U_{r,n}(q))_{\mathbb{C}}$ agrees with the Frobenius series of $\widetilde{A}(U_{r,n})_{\mathbb{C}}$. It suffices to just consider $\grFrob(\widetilde{A}(U_{r,n})_{\mathbb{C}},t)$.

Before we prove our main results, let us also establish some facts we will use in the proof.  By Theorem 6.22 in [19], we have

\begin{equation} \label{eq:AugBnIsTildeQ}
\grFrob(\widetilde{A}(\mathsf{B}_n)_{\mathbb{C}},t)=\grFrob(\mathbb{C}\widetilde{\Code}_n,t)=\widetilde{Q}_n(\x,t)=\sum_{\sigma\in\widetilde{\S}_n}F_{\DEX(\sigma),n}(\x)t^{\exc(\sigma)+1},
\end{equation}

where $\widetilde{\Code}_n$ is the set of extended codes graded by their index.
The first equality in (<ref>) was obtained from Theorem 4.7 in [19] by constructing an $\S_n$-equivariant bijection between the basis $\widetilde{FY}(\mathsf{B}_n)$ and $\widetilde{\Code}_n$, which induces an isomorphism $\widetilde{A}(\mathsf{B}_n)_{\mathbb{C}}\cong_{\S_n} \mathbb{C}\widetilde{\Code}_n$ as permutation modules.

Define the following variant of binomial Eulerian quasisymmetric functions

\[
\widetilde{Q}_n(\x,t,r)\coloneqq h_n r^n+t\sum_{k=1}^n h_{n-k}Q_k(\x,t)r^{n-k}.
\]

When $r=1$, $\widetilde{Q}_n(\x,t,1)$ is the original binomial Eulerian quasisymmetric function $\widetilde{Q}_{n}(\x,t)$. Write $\widetilde{Q}_n(\x,t,r)=\sum_{j\ge 0, k\ge 0}\widetilde{Q}_{n,j,k}(\x)t^j r^k$.

For any integer $n\ge 1$,

\[
\widetilde{Q}_n(\x,t,r)=\sum_{\sigma\in\widetilde{\mathfrak{S}}_n}F_{\DEX(\sigma),n}(\x)t^{\exc(\sigma)+1}r^{\fixTwo(\sigma)}.
\]

Or equivalently, for all $0\le j,~k\le n$,

\[
\widetilde{Q}_{n,j,k}(\x)=\sum_{\substack{\sigma\in\widetilde{\S}_n\\ \exc(\sigma)=j-1\\ \fixTwo(\sigma)=k}} F_{\DEX(\sigma),n}(\x).
\]

In particular,

\[
\widetilde{Q}_{n,j,0}(\x)=\sum_{\substack{\sigma\in\S_n \\ \exc(\sigma)=j-1}}F_{\DEX(\sigma),n}(\x)=Q_{n,j-1}(\x).
\]

The proof is similar to the proof of Theorem 6.21 in [19]. By definition of $\widetilde{Q}_n(\x,t,r)$ and $h_n=F_{\emptyset,n}(\x)$, we have

\begin{align*}
\widetilde{Q}_n(\x,t,r)
&=F_{\emptyset,n}(\x)r^n+\sum_{k=1}^n\sum_{\substack{\pi\in\widetilde{\S}_n\\ \fixTwo(\pi)=n-k}}F_{\DEX(\sigma),n}(\x)t^{\exc(\pi)+1}r^{n-k}.\\
\end{align*}

The following proposition is the analog of Lemma <ref>.

For all $n,j,k$,

\[
\widetilde{Q}_{n,j,k}(\x)=h_k(\x)\widetilde{Q}_{n-k,j,0}(\x).
\]

By Theorem <ref>, we have

\begin{align*}
=&\sum_{\substack{\sigma\in\widetilde{\S}_{n-k} \\ \exc(\sigma)=j-1 \\ \fixTwo(\sigma)=0}}F_{\DES(0^{k}),k}(\x) F_{\DEX(\sigma),n-k}(\x)\\
=&\sum_{\substack{\sigma\in\widetilde{\S}_{n-k} \\ \exc(\sigma)=j-1 \\ \fixTwo(\sigma)=0}}F_{\DES(0^{k}),k}(\x) F_{\DES(\overline{\sigma}),n-k}(\x),
\end{align*}

where, recall from Definition <ref>, the barred permutation $\overline{\sigma}$ is obtained from $\sigma$ by adding a bar on $\sigma(i)$ for each $i\in\EXC(\sigma)$. Then the right-hand side of the above equality can be rewritten as

\begin{align*}
&\sum_{\substack{\sigma\in\widetilde{\S}_{n-k} \\ \exc(\sigma)=j-1 \\ \fixTwo(\sigma)=0}}F_{\DES(0^{k}),k}(\x) F_{\DES(\overline{\sigma}),n-k}\\
=&\sum_{\substack{\sigma\in\widetilde{\S}_{n-k} \\ \exc(\sigma)=j-1 \\ \fixTwo(\sigma)=0}}\sum_{\pi'\in 0^k\shuffle \overline{\sigma}}F_{\DES(\pi'),n}(\x)\\
=& \sum_{\substack{\pi\in\widetilde{\S}_n\\ \exc(\pi)=j-1 \\ \fixTwo(\sigma)=k}} F_{\DEX(\pi),n}(\x)\quad (\text{see the proof of Theorem }\ref{thm:FexpWideQ})\\
\end{align*}

Recall the $\S_n$-equivariant bijection $\widetilde{\phi}:\widetilde{FY}(\mathsf{B}_n)\rightarrow\widetilde{\Code}_n$ from Theorem 4.6 in [19]. Let $\widetilde{\Code}_{n,j,k}$ be the subset of $\widetilde{\Code}_{n}$ consisting of extended codes with index $j$ and $k$ $\infty$s. It is easy to see that $\widetilde{\phi}$ can be restricted to an $\S_n$-equivariant bijection between $\widetilde{\Code}_{n,j-1,k}$ and the subset

\[ %\label{eq:WideFYjk}
\widetilde{FY}^{j,k}(\mathsf{B}_n)\coloneqq\left\{
x_{F_1}^{a_1}\ldots x_{F_\ell}^{a_\ell}\in\widetilde{FY}^j(\mathsf{B}_n): |[n]-F_\ell|=k
\right\}.
\]

Let $\widetilde{A}^{j,k}(\mathsf{B}_n)$ be the subspace of $\widetilde{A}^j(\mathsf{B}_n)$ spanned by $\widetilde{FY}^{j,k}(\mathsf{B}_n)$. Then slightly modifying the proof of Theorem 4.3 in [19] (adding one variable $r$ that counts the number of $\infty$s) for the Frobenius characteristics of $\mathbb{C}\widetilde{\Code}_{n,j}$ gives the following theorem

For $n\ge 1$, we have

\[
\sum_{j=0}^n\sum_{k=0}^n\ch(\mathbb{C}\widetilde{\Code}_{n,j-1,k})t^j r^k=\widetilde{Q}_{n}(\x,t,r),
\]

and equivalently, $\ch(\mathbb{C}\widetilde{\Code}_{n,j-1,k})=\widetilde{Q}_{n,j,k}(\x)$.

Then the $\S_n$-equivariant bijection between $\widetilde{\Code}_{n,j-1,k}$ and $\widetilde{FY}^{j,k}(\mathsf{B}_n)$ gives us the following corollary.

For any $n,j,k$ we have $\ch(\widetilde{A}^{j,k}(\mathsf{B}_n)_{\mathbb{C}})=\widetilde{Q}_{n,j,k}(\x)$.

Now similar to the non-augmented case, let

\begin{align*}
\widetilde{\Delta}_{n,r}(\x)
&\coloneqq \grFrob(\widetilde{A}(U_{r+1,n})_{\mathbb{C}},t)-\grFrob(\widetilde{A}(U_{r,n})_{\mathbb{C}},t)\\
\end{align*}

For $1\le r\le n-1$, the polynomial $\widetilde{\Delta}_{n,r}(\x)\in\Lambda_n[t]$ has two expressions:

\[
\sum_{j=0}^r h_{n-j}Q_j(\x,t)t^{r-j+1}
\]

\[
\sum_{\substack{\sigma\in\widetilde{\S}_n \\ \fixTwo(\sigma)\ge n-r }}F_{\DEX(\sigma),n}(\x)t^{r-\exc(\sigma)}.
\]

Similar to the case of Chow rings, for $1\le k\le n$ we have

\[
\widetilde{FY}^k(U_{n,1})\subsetneq \widetilde{FY}^k(U_{n,2})\subsetneq \ldots\subsetneq \widetilde{FY}^k(U_{n,n}).
\]

The symmetric function $\widetilde{a}_{n,r}^{(k)}(\x)$ is the Frobenius characteristic of the permutation module generated by $\widetilde{FY}^k(U_{r+1,n})\setminus\widetilde{FY}^k(U_{r,n})$, which consists of monomial

\[
x_{[n]}^{r+1} \quad \text{ if }k=r+1
\]

and monomials

\[
x_{F_1}^{a_1}\ldots x_{F_\ell}^{a_\ell}x_{[n]}^i\in \widetilde{FY}^k(U_{r+1,n}) \quad\text{ such that }\quad |F_\ell|=r-i
\]

for $i=0,1,\ldots,k-1$ if $1\le k\le r$. Hence, $\widetilde{a}_{n,r}^{(r+1)}(\x)=h_n$. For $1\le k\le r$, One can construct an $\S_n$-equivariant bijection from $\widetilde{FY}^k(U_{r+1,n})\setminus\widetilde{FY}^k(U_{r,n})$ to $\bigcup_{i=0}^{k-1}\widetilde{FY}^{k-i,n-r-i}(\mathsf{B}_n)$ defined by $x_{F_1}^{a_1}\ldots x_{F_\ell}^{a_\ell}x_{[n]}^i\mapsto x_{F_1}^{a_1}\ldots x_{F_\ell}^{a_\ell}$. By Corollary <ref>, we have

\[
\widetilde{a}_{n,r}^{(k)}=\sum_{i=0}^{k-1}\widetilde{Q}_{n,k-i,n-r+i}=\sum_{i=0}^{k-1}h_{n-r+i}\widetilde{Q}_{r-i,k-i,0} \quad\text{(by Proposition \ref{prop:ReductionExtCode})}.
\]

Then since for any $n$ and $0\le j\le n-1$,

\[
\widetilde{Q}_{n,j,0}=\sum_{\substack{\sigma\in\widetilde{\S}_{n}\\ \exc(\sigma)=j-1\\ \fixTwo(\sigma)=0}}F_{\DEX(\sigma),n}=\sum_{\substack{\sigma\in\S_{n}\\ \exc(\sigma)=j-1}}F_{\DEX(\sigma),n}=Q_{n,j-1},
\]

we have

\begin{align}
\widetilde{\Delta}_{n,r}(\x)
&=\sum_{k=1}^{r+1}\widetilde{a}_{n,r}^{(k)}(\x)t^k=h_nt^{r+1}+\sum_{k=1}^{r}\sum_{i=0}^{k-1}h_{n-r+i}\widetilde{Q}_{r-i,k-i,0}t^k \label{eq:differenceAug}\\
&=h_nt^{r+1}+\sum_{k=1}^r\sum_{i=0}^{k-1}h_{n-r+i}Q_{r-i,k-i-1}t^k \nonumber\\
&=h_nt^{r+1}+\sum_{i=0}^{r-1}h_{n-r+i}t^{i+1}\left(\sum_{k=i+1}^rQ_{r-i,k-i-1}t^{k-i-1}\right) \nonumber \\
&=h_nt^{r+1}+\sum_{i=0}^{r-1}h_{n-r+i}t^{i+1}Q_{r-i}(\x,t) \nonumber\\
&=\sum_{i=0}^r h_{n-r+i}Q_{r-i}(\x,t)t^{i+1} \nonumber\\
&=\sum_{j=0}^r h_{n-j}Q_j(\x,t)t^{r-j+1}.  \nonumber
\end{align}

On the other hand, by Theorem <ref> and Lemma <ref>,  $\widetilde{Q}_{r-i,k-i,0}=Q_{r-i,k-i-1}(\x)=Q_{r-i,r-k}(\x)=\widetilde{Q}_{r-i,r-k+1,0}$. Therefore, equation (<ref>) can be expressed in a different way as follows

\begin{align*}
\widetilde{\Delta}_{n,r}(\x)
&=h_nt^{r+1}+\sum_{k=1}^r\sum_{i=0}^{k-1}\widetilde{Q}_{n,r-k+1,n-r+i}t^k \quad\text{(by Proposition \ref{prop:ReductionExtCode}})\\
&=h_nt^{r+1}+\sum_{k=1}^r\sum_{\substack{\sigma\in\widetilde{\S}_n\\ \exc(\sigma)=r-k \\ \fixTwo(\sigma)\ge n-r}}F_{\DEX(\sigma),n}(\x)t^k\\
&=F_{\DEX(\theta),n}(\x)t^{r-(-1)}+\sum_{\substack{\sigma\in\widetilde{\S}_n\setminus\{\theta\} \\ \fixTwo(\sigma)\ge n-r }}F_{\DEX(\sigma),n}(\x)t^{r-\exc(\sigma)}\\
&=\sum_{\substack{\sigma\in\widetilde{\S}_n \\ \fixTwo(\sigma)\ge n-r }}F_{\DEX(\sigma),n}(\x)t^{r-\exc(\sigma)}.
\end{align*}

Similarly to Corollary <ref>, we may obtain the difference of Hilbert series for the augmented Chow rings of $q$-uniform matroids by taking stable principal specialization of Theorem <ref>.

For $1\le r\le n-1$, the difference $\Hilb(\widetilde{A}(U_{r+1,n}(q)),t)-\Hilb(\widetilde{A}(U_{r,n}(q)),t)$ has the closed form

\[
\sum_{i=0}^r\qbin{n}{i}t^{r+1-i}A_i(q,t),
\]

and the combinatorial interpretation

\[
\sum_{\substack{\sigma\in\widetilde{\S}_n\\ \fixTwo(\sigma)\ge n-r}}q^{\maj(\sigma)-\exc(\sigma)}t^{r-\exc(\sigma)}.
\]

We are ready to prove Theorem <ref>, which we restate here.

The series $\grFrob(\widetilde{A}(U_{r,n})_{\mathbb{C}},t)$ and $\grFrob_q(\widetilde{A}(U_{r,n}(q))_{\mathbb{C}},t)$ coincide and have two expressions:

(i) \(\displaystyle h_n+t\sum_{j=0}^{r-1}h_{n-j}Q_j(\x,t)(1+t+\ldots+t^{r-j-1})\),

(ii) \(\displaystyle \widetilde{Q}_n(\x,t)-\sum_{j=r}^{n-1}\sum_{\substack{\sigma\in\widetilde{\S}_n\\ \fixTwo(\sigma)\ge n-j}}F_{\DEX(\sigma),n}t^{j-\exc(\sigma)}\).

For (i), applying the first expression of Theorem <ref>, we have

\begin{align*}
\grFrob(\widetilde{A}(U_{r,n})_{\mathbb{C}},t)
&=h_n(1+t)+\sum_{j=1}^{r-1}\sum_{i=0}^j h_{n-i}Q_i(\x,t)t^{j-i+1}\\
&=h_n(1+t)+\sum_{j=1}^{r-1}h_n t^{j+1}+\sum_{j=1}^{r-1}\sum_{i=1}^j h_{n-i}Q_i(\x,t)t^{j-i+1}\\
&=h_n+t\sum_{j=0}^{r-1}h_n t^{j}+t\sum_{i=1}^{r-1}h_{n-i}Q_i(\x,t)\sum_{j=i}^{r-1}t^{j-i}\\
\end{align*}

For (ii), applying the second expression from Theorem <ref>, we have

\begin{align*}
\grFrob(\widetilde{A}(U_{r,n})_{\mathbb{C}},t)
&=\widetilde{Q}_n(\x,t)-\sum_{j=r}^{n-1}\sum_{\substack{\sigma\in\widetilde{\S}_n \\ \fixTwo(\sigma)\ge n-j}}F_{\DEX(\sigma),n}(\x)t^{j-\exc(\sigma)}.
\end{align*}

From Theorem <ref>, we can prove Corollary <ref>, which we restate below.

For the special cases that $r=n$ and $r=n-1$, we have

(i) \(\displaystyle \grFrob(\widetilde{A}(U_{n,n})_{\mathbb{C}},t)=\grFrob_q(\widetilde{A}(U_{n,n}(q))_{\mathbb{C}},t) =\widetilde{Q}_n(\x,t),\)

(ii)\(\displaystyle \grFrob(\widetilde{A}(U_{n-1,n})_{\mathbb{C}},t)=\grFrob_q(\widetilde{A}(U_{n-1,n}(q))_{\mathbb{C}},t) =Q_n(\x,t).\)

Note that (i) is the same as (<ref>).

When $r=n$, nothing is being subtracted on the right-hand side of Theorem <ref> (ii); hence (i) follows.

When $r=n-1$, by Theorem <ref> (ii), we have

\begin{equation}
\grFrob(\widetilde{A}(U_{n-1,n})_{\mathbb{C}},t)=\widetilde{Q}_n(\x,t)-\sum_{\substack{\sigma\in\widetilde{\S}_n\\ \fixTwo(\sigma)\ge 1}}F_{\DEX(\sigma),n}t^{n-1-\exc(\sigma)}. \label{eq:AugSpecial}
\end{equation}

Recall from Corollary 3.6 in [26] that $\widetilde{Q}_n(\x,t)$ as a polynomial in $\Lambda_n[t]$ is palindromic; hence

\[
\widetilde{Q}_n(\x,t)=\sum_{\sigma\in\widetilde{\S}_n}F_{\DEX(\sigma),n}t^{\exc(\sigma)+1}=\sum_{\sigma\in\widetilde{\S}_n}F_{\DEX(\sigma),n}t^{n-1-\exc(\sigma)}.
\]

Then (<ref>) can be rewritten as

\begin{align*}
\grFrob(\widetilde{A}(U_{n-1,n})_{\mathbb{C}},t)
&=\sum_{\substack{\sigma\in\widetilde{\S}_n\\ \fixTwo(\sigma)=0}}F_{\DEX(\sigma),n}t^{n-1-\exc(\sigma)}\\
&=\sum_{\sigma\in\S_n}F_{\DEX(\sigma),n}t^{\exc(\sigma)} \quad \text{(palindromic by Lemma \ref{lem:PalinQ})}\\
\end{align*}

§ PROOFS OF EQUIVARIANT GENERALIZATION OF $(-1)$-EVALUATION FOR GENERAL MATROIDS

Let $G$ by any subgroup of $\Aut(M)$. In this section, we will prove Theorem <ref> and Theorem <ref>, which are our formulas for the $(-1)$-evalutions of the $G$-equivariant Hilbert series of (augmented) Chow rings of general matroids.

Recalling from Section <ref>, consider $G$ acting on the rank selected subposet $\L(M)_S$ of $\L(M)$ for $S\subseteq [r-1]$.
Then $G$ permutes the maximal chains in $\L(M)_S$. Let $\alpha_{\L(M)}(S)$ denote this permutation representation of $G$. For any $S\subseteq [r-1]$, define a virtual representation of $G$ by

\[
\beta_{\L(M)}(S)\coloneqq\sum_{T\subseteq S}(-1)^{|S|-|T|}\alpha_{\L(M)}(T).
\]

Then by Theorem <ref> (cf. Theorem 3.4.9 in [34]), we have that $\beta_{\L(M)}(S)$ is a genuine representation of $G$ and is isomorphic to the top reduced homology of $\L(M)_S$, i.e.,

\begin{equation}  \label{eq:IsHomology}
\beta_{\L(M)}(S)\cong_{G}\tilde{H}_{|S|-1}(\L(M)_S).
\end{equation}

On the other hand, the action of $G$ on $\L(M)$ also induces actions on the Chow ring $A(M)$ and the augmented Chow ring $\widetilde{A}(M)$. Recall from (<ref>) and Corollary <ref> that the Feichtner-Yuzvinsky basis for the Chow ring and the augmented Chow ring are

\[
FY(M)=\left\{x_{F_1}^{a_1}x_{F_2}^{a_2}\ldots x_{F_\ell}^{a_\ell}:\substack{~\emptyset=F_0\subsetneq F_1\subsetneq F_2\subsetneq\ldots\subsetneq F_\ell\subseteq \L(M) \text{ for } 0\le \ell\le n \\
1\le a_i\le \rk_M(F_i)-\rk_M(F_{i-1})-1}\right\}
\]

\[
\widetilde{FY}(M)\coloneqq\left\{x_{F_1}^{a_1}x_{F_2}^{a_2}\ldots x_{F_\ell}^{a_\ell}: \substack{\emptyset\subsetneq F_1\subsetneq F_2\subsetneq\ldots\subsetneq F_\ell \text{ for }0\le \ell\le n \\
1\le a_1\le\rk_M(F_1),~ a_i\le\rk_M(F_i)-\rk_M(F_{i-1})-1 \text{ for }i\ge 2}
\right\}.
\]

Notice that the elements are monomials indexed by chains in $\L(M)$. It is easy to see that the subsets of degree $k$ monomials $FY^k(M)$ and $\widetilde{FY}^k(M)$ form permutation bases for the $G$-module $A^k(M)$ and $\widetilde{A}^k(M)$ respectively for all $k$.

Therefore, as representations of $G$, both $A(M)$ and $\widetilde{A}(M)$ are the direct sums of permutation representations of the form $\alpha_{\L(M)}(S)$ for some $S\subseteq [r-1]$.

Now we are ready to prove Theorem <ref>, which we restate here.

Let $M$ be a (loopless) matroid on $[n]$ of rank $r$,

\[
\sum_{i=0}^{r-1}(-1)^i A^i(M)_{\mathbb{C}}\cong_{G}
\begin{cases}
0   &   \text{ if $r$ is even}\\
& \text{ if $r$ is odd}
\end{cases}.
\]

Following from the restriction of exponents of the FY-basis elements in $FY(M)$, we have

\begin{equation} \label{eq:RepA(M)}
\sum_{i=0}^{r-1}A^i(M)_{\mathbb{C}}t^i
\cong_G \sum_{S=\{r_1<\ldots<r_\ell\}\subseteq [r]}\prod_{j=1}^\ell \frac{t(1-t^{r_i-r_{i-1}-1})}{1-t}\alpha_{\L(M)}(S)
\end{equation}

with the convention that $r_0\coloneqq 0$.
Notice that here we allow $S$ to be any subset of $[r]$, because the definiton of $\alpha_{\L(M)}(S)$ can be easily extended to $S$ containing $r$ and $\alpha_{\L(M)}(S)\cong_G\alpha_{\L(M)}(S\setminus\{r\})$ for any $S$ containing $r$. Then (<ref>) can be rewritten as

\begin{align*}
\cong_G  &\sum_{\emptyset\neq S=\{r_1<\ldots<r_\ell\}\subseteq [r-1]}\prod_{j=1}^{\ell}\frac{t(1-t^{r_j-r_{j-1}-1})}{1-t}\left(1+\frac{t(1-t^{r-r_{\ell}-1})}{1-t}\right)\alpha_{\L(M)}(S)\\
& +\left(1+\frac{t(1-t^{r-1})}{1-t}\right)\alpha_{\L(M)}(\emptyset).
\end{align*}

Evaluating the above identity in $R_{\mathbb{C}}(G)[t]$ at $t=-1$, for a term in the summation to be nonzero, all $r_i$'s are forced to be even. The product $\prod_{j=1}^{\ell}\frac{t(1-t^{r_j-r_{j-1}-1})}{1-t}$ evaluated at $t=-1$ gives $(-1)^{\ell}$.  If $r$ is also even, then both $1+\frac{t(1-t^{r-1})}{1-t}$ and $1+\frac{t(1-t^{r-r_{\ell}-1})}{1-t}$ evaluated at $t=-1$ give

\[
\]

Therefore, $\sum_{i=0}^{r-1}(-1)^i A^i(M)_{\mathbb{C}}=0$ if $r$ is even.

If $r$ is odd, then both $\frac{t(1-t^{r-1})}{1-t}$ and $\frac{t(1-t^{r-r_{\ell}-1})}{1-t}$ vanish after evaluated at $t=-1$. In this case, as a virtual representation of $G$, the evaluation at $t=-1$ gives

\begin{align*}
\sum_{i=0}^{r-1}(-1)^i A^i(M)_{\mathbb{C}}
&\cong_G \sum_{S\subseteq\{2,4,\ldots,r-1\}}(-1)^{|S|}\alpha_{\L(M)}(S)\\
&\cong_G (-1)^{\frac{r-1}{2}}\tilde{H}_{\frac{r-3}{2}}\left(\L(M)_{\{2,4,\ldots,r-1\}}\right) \quad (\text{by }(\ref{eq:IsHomology})).
\end{align*}

Next, we prove the augmented analog of Theorem <ref>, Theorem <ref>, which we restate below.
Recall that we have a similar $G$-action on the FY-basis $\widetilde{FY}(M)$ for $\widetilde{A}(M)$.

Let $M$ be a matroid on $[n]$ of rank $r$. Then

\[
\sum_{i=0}^{r}(-1)^i\widetilde{A}^i(M)_{\mathbb{C}}\cong_{G}
\begin{cases}
(-1)^{\frac{r}{2}}\tilde{H}_{\frac{r-2}{2}}\left(\L(M)_{\{1,3,\ldots,r-1\}}\right)   &   \text{ if $r$ is even}\\
0   & \text{ if $r$ is odd}
\end{cases}.
\]

The proof is similar to that of Theorem <ref>. Following from the restriction of the exponents of the FY-basis elements in $\widetilde{FY}(M)$, we have

\begin{align*}
\cong_G \sum_{S=\{r_1<\ldots<r_\ell\}\subseteq [r]}\frac{t(1-t^{r_1})}{1-t}\prod_{j=2}^\ell \frac{t(1-t^{r_i-r_{i-1}-1})}{1-t}\alpha_{\L(M)}(S)\\
=& \sum_{\emptyset\neq S=\{r_1<\ldots<r_\ell\}\subseteq [r-1]}\frac{t(1-t^{r_1})}{1-t}\prod_{j=2}^{\ell}\frac{t(1-t^{r_j-r_{j-1}-1})}{1-t}\left(1+\frac{t(1-t^{r-r_{\ell}-1})}{1-t}\right)\alpha_{\L(M)}(S)\\
& +\left(1+\frac{t(1-t^{r})}{1-t}\right)\alpha_{\L(M)}(\emptyset).
\end{align*}

After evaluating at $t=-1$, the right-hand side of the last identity is nonvanishing only when all $r_i$'s are odd and $r$ is even. Thus, in this case, the $(-1)$-evaluation gives

\begin{align*}
\sum_{i=0}^{r}(-1)^i\widetilde{A}(M)_{\mathbb{C}}
&\cong_G \sum_{S\subseteq\{1,3,\ldots,r-1\}}(-1)^{|S|}\alpha_{\L(M)}(S)\\
&\cong_G (-1)^{\frac{r}{2}}\tilde{H}_{\frac{r-2}{2}}(\L(M)_{\{1,3,\ldots,r-1\}}) \quad (\text{by }(\ref{eq:IsHomology}))
\end{align*}

§ FINAL REMARK ON THE UPCOMING WORK

Within the framework of palindromic polynomials, there is a generalization of the nonnegativity of the Chaney-Davis quantity called $\gamma$-positivity or $\gamma$-nonnegativity of a palindromic polynomial.
It has been shown independently by Ferroni, Matherne, Stevens, Vecchi <cit.> and Wang (see <cit.>) that the Hilbert series of $A(M)$ and $\widetilde{A}(M)$ are $\gamma$-positive, which follows from Braden, Huh, Matherne, Proudfoot, and Wang [6] on semismall decomposition of Chow rings.

One also has a notion of equivariant $\gamma$-positivity for a graded representation $A=\bigoplus_{i=0}^r A^i$, which is first considered by Shareshian and Wachs <cit.>.
Consider the subgroup $G$ of $Aut(M)$ acting on $A(M)$ and $\widetilde{A}(M)$. It has been conjectured by Angarone, Nathanson, and Reiner [2] that $A(M)_{\mathbb{C}}=\bigoplus_{i=0}^{r-1}A^i(M)$ is $G$-equivariant $\gamma$-positive. In the upcoming work [17], we generalize our results in Section <ref> showing that both $A(M)$ and $\widetilde{A}(M)$ are $G$-equivariant $\gamma$-positive. This confirms a conjecture of Angarone, Nathanson, and Reiner in [2].

§ ACKNOWLEDGEMENTS

The author thanks Michelle Wachs for her guidance and encouragement at every stage of this project, and Patty Commins and Richard Stanley for helpful conversations on unipotent representations of $GL_n(q)$.

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