# Chow rings of matroids as permutation representations

Robert Angarone, Anastasia Nathanson, Victor Reiner  University of Minnesota -
Twin Cities, Minneapolis MN 55455<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Given a matroid and a group of its matroid automorphisms, we study the induced
group action on the Chow ring of the matroid. This turns out to always be a
permutation action. Work of Adiprasito, Huh and Katz showed that the Chow ring
satisfies Poincaré duality and the Hard Lefschetz theorem. We lift these to
statements about this permutation action, and suggest further conjectures in
this vein.

###### Key words and phrases:

matroid, Chow ring, Koszul, log-concave, unimodal, Kahler package, Burnside
ring, equivariant, Polya freqency, real-rooted

###### 2010 Mathematics Subject Classification:

05B35, 05E18, 05E14,

##  1\. Introduction

A matroid $\mathcal{M}$ is a combinatorial abstraction of lists of vectors
$v_{1},v_{2},\ldots,v_{n}$ in a vector space, recording only the information
about which subsets of the vectors are linearly independent or dependent,
forgetting their coordinates– see Section 2.1 for definitions and references.
In groundbreaking work, Adiprasito, Huh and Katz [AHK18] affirmed long-
standing conjectures of Rota–Heron–Welsh and Mason about vectors and matroids
via a new methodology. Their work employed a certain graded
$\mathbb{Z}$-algebra $A(\mathcal{M})$ called the Chow ring of $\mathcal{M}$,
introduced by Feichtner and Yuzvinsky [FY04] as a generalization of the Chow
ring of DeConcini and Procesi’s wonderful compactifications for hyperplane
arrangement complements. A remarkable integral Gröbner basis result proven by
Feichtner and Yuzvinsky [FY04, Thm. 2] shows that for a matroid $\mathcal{M}$
of rank $r+1$ with Chow ring $A(\mathcal{M})=\bigoplus_{k=0}^{r}A^{k}$, each
homogeneous component is free abelian: $A^{k}\cong\mathbb{Z}^{a_{k}}$ for some
Hilbert function $(a_{0},a_{1},\ldots,a_{r})$. A key step in the work of
Adiprasito, Huh and Katz shows not only symmetry and unimodality for the
Hilbert function

(1) $\displaystyle a_{k}=a_{r-k}\text{ for }r\leq k/2$ (2) $\displaystyle
a_{0}\leq a_{1}\leq\cdots\leq
a_{\lfloor\frac{r}{2}\rfloor}=a_{\lceil\frac{r}{2}\rceil}\geq\cdots\geq
a_{r-1}\geq a_{r},$

but in fact proves that $A(\mathcal{M})$ enjoys a trio of properties referred
to as the Kähler package, reviewed in Section 2.4 below. The first of these
properties is Poincaré duality, proving (1) via a natural $\mathbb{Z}$-module
isomorphism $A^{r-k}\cong\mathrm{Hom}_{\mathbb{Z}}(A^{k},\mathbb{Z})$. The
second property, called the Hard Lefschetz Theorem, shows that after tensoring
over $\mathbb{Z}$ with $\mathbb{R}$ to obtain
$A(\mathcal{M})_{\mathbb{R}}=\bigoplus_{k=0}A^{k}_{\mathbb{R}}$, one can find
Lefschetz elements $\omega$ in $A^{1}_{\mathbb{R}}$ such that multiplication
by $\omega^{r-2k}$ give $\mathbb{R}$-linear isomorphisms
$A^{k}_{\mathbb{R}}\rightarrow A^{r-k}_{\mathbb{R}}$ for $k\leq\frac{r}{2}$.
In particular, multiplication by $\omega$ mapping
$A^{k}_{\mathbb{R}}\rightarrow A_{\mathbb{R}}^{k+1}$ is injective for
$k<\frac{r}{2}$, strengthening the unimodality (2).

We are interested here in how these Poincaré duality and Hard Lefschetz
properties interact with the group $G\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{\mathrm{Aut}}(\mathcal{M})$ of symmetries of the
matroid $\mathcal{M}$. It is not hard to check (see Section 2.1 below) that
$G$ acts via graded $\mathbb{Z}$-algebra automorphisms on $A(\mathcal{M})$,
giving $\mathbb{Z}G$-module structures on each $A^{k}$, and
$\mathbb{R}G$-module structures on each $A^{k}_{\mathbb{R}}$. One can also
check (see the proof of Corollary 2.14 below) that $A^{r}\cong\mathbb{Z}$ with
trivial $G$-action. From this, the Poincaré duality pairing immediately gives
rise to a $\mathbb{Z}G$-module isomorphism

(3) $A^{r-k}\cong\mathrm{Hom}_{\mathbb{Z}}(A^{k},\mathbb{Z})$

where $g$ in $G$ acts on $\varphi$ in
$\mathrm{Hom}_{\mathbb{Z}}(A^{k},\mathbb{Z})$ via $\varphi\mapsto\varphi\circ
g^{-1}$; similarly $A^{r-k}\cong\mathrm{Hom}_{\mathbb{R}}(A^{k},\mathbb{R})$
as $\mathbb{R}G$-modules. Furthermore, it is not hard to check (see Corollary
2.15 below) that one can pick an explicit Lefschetz element $\omega$ as in
[AHK18] which is $G$-fixed, giving $\mathbb{R}G$-module isomorphisms and
injections

$\displaystyle A_{\mathbb{R}}^{k}$
$\displaystyle\overset{\sim}{\longrightarrow}A_{\mathbb{R}}^{r-k}\quad\text{
for }r\leq\frac{k}{2}$ (4) $\displaystyle a$ $\displaystyle\longmapsto
a\cdot\omega^{r-2k}$ $\displaystyle A_{\mathbb{R}}^{k}$
$\displaystyle\hookrightarrow A_{\mathbb{R}}^{k+1}\quad\text{ for
}r<\frac{k}{2}$ (5) $\displaystyle a$ $\displaystyle\longmapsto a\cdot\omega.$

Our goal in this paper is to use Feichtner and Yuzvinsky’s Gröbner basis
result to prove a combinatorial strengthening of the isomorphisms and
injections (3), (1), (1). To this end, recall (or see Section 2.1 below) that
the matroid $\mathcal{M}$ can be specified by its family $\mathfrak{F}$ of
flats; in the case where $\mathcal{M}$ is realized by a list of vectors
$v_{1},v_{2},\ldots,v_{n}$ in a vector space, a subset
$F\subseteq\\{1,2,\ldots,n\\}=\mathrel{\mathop{\mathchar 58\relax}}E$ is a
flat when $\\{v_{j}\\}_{j\in F}$ is linearly closed, meaning that every vector
$v_{i}$ for $i$ in $E$ that lies in the linear span of $\\{v_{j}\\}_{j\in F}$
already has $i$ in $F$. Then the Chow ring $A(\mathcal{M})$ is presented as a
quotient of the polynomial ring $S\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{Z}[x_{F}]$ having one variable $x_{F}$ for each nonempty
flat $F$ in $\mathfrak{F}\setminus\\{\varnothing\\}$. The presentation takes
the form $A(\mathcal{M})\mathrel{\mathop{\mathchar 58\relax}}=S/(I+J)$ where
$I,J$ are certain ideals of $S$ defined more precisely in Definition 2.2
below.

Feichtner and Yuzvinsky exhibited (see Theorem 2.8, Corollary 2.9 below) a
Gröbner basis for $I+J$ that leads to the following standard monomial
$\mathbb{Z}$-basis for $A(\mathcal{M})$, which we call the FY-monomials of
$\mathcal{M}$:

$\mathrm{FY}\mathrel{\mathop{\mathchar
58\relax}}=\\{x_{F_{1}}^{m_{1}}x_{F_{2}}^{m_{2}}\cdots
x_{F_{\ell}}^{m_{\ell}}\mathrel{\mathop{\mathchar
58\relax}}(\varnothing=\mathrel{\mathop{\mathchar 58\relax}}F_{0})\subsetneq
F_{1}\subsetneq F_{2}\subsetneq\cdots\subsetneq F_{\ell},\text{ and
}m_{i}\leq\operatorname{\mathrm{rk}}(F_{i})-\operatorname{\mathrm{rk}}(F_{i-1})-1\\}$

Here $\operatorname{\mathrm{rk}}(F)$ denotes the matroid rank of the flat $F$
(= the dimension of the linear span of $\\{v_{j}\\}_{j\in F}$ when
$\mathcal{M}$ is realized by a list of vectors). The subset $\mathrm{FY}^{k}$
of FY-monomials $x_{F_{1}}^{m_{1}}\cdots x_{F_{\ell}}^{m_{\ell}}$ of total
degree $m_{1}+\cdots+m_{\ell}=k$ then gives a $\mathbb{Z}$-basis for $A^{k}$.
One can readily check (see Corollary 2.10 below) that the group
$G=\operatorname{\mathrm{Aut}}(\mathcal{M})$ permutes the $\mathbb{Z}$-basis
$\mathrm{FY}^{k}$ for $A^{k}$, endowing $A^{k}$ with the structure of a
permutation representation, or $G$-set. Our main result, proven in Section 3,
is this strengthening of the isomorphisms and injections (3), (1), (1).

######  Theorem 1.1.

For every matroid $\mathcal{M}$ of rank $r+1$, there exist

  * (i)

$G$-equivariant bijections $\pi\mathrel{\mathop{\mathchar
58\relax}}\mathrm{FY}^{k}\overset{\sim}{\longrightarrow}\mathrm{FY}^{r-k}$ for
$k\leq\frac{r}{2}$, and

  * (ii)

$G$-equivariant injections $\lambda\mathrel{\mathop{\mathchar
58\relax}}\mathrm{FY}^{k}\hookrightarrow\mathrm{FY}^{k+1}$ for
$k<\frac{r}{2}$.

######  Example 1.2.

Let $\mathcal{M}=U_{4,5}$ be the uniform matroid of rank $4$ on
$E=\\{1,2,3,4,5\\}$, associated to a list of $5$ generic vectors
$v_{1},v_{2},v_{3},v_{4},v_{5}$ in a $4$-dimensional vector space, so that any
quadruple $v_{i},v_{j},v_{k},v_{\ell}$ is linearly independent. One has these
flats of various ranks:

rank | flats $F\in\mathfrak{F}$
---|---
$0$ | $\varnothing$
$1$ | $1,2,3,4,5$
$2$ | $12,13,14,15,23,24,25,34,35,45$
$3$ | $123,124,125,134,135,145,234,235,245,345$
$4$ | $E=12345$

The Chow ring $A(\mathcal{M})=S/(I+J)$, where
$S=\mathbb{Z}[x_{i},x_{jk},x_{\ell mn},x_{E}]$ with
$\\{i\\},\\{j,k\\},\\{\ell,m,n\\}$ running through all one, two and three-
element subsets of $E=\\{1,2,3,4,5\\}$, and

$I=\Big{(}x_{F}x_{F^{\prime}}\Big{)}_{F\not\subset
F^{\prime},F^{\prime}\not\subset F},\qquad
J=\bigg{(}x_{i}+\sum_{\begin{subarray}{c}1\leq j<k\leq 5\\\
i\in\\{j,k\\}\end{subarray}}x_{jk}+\sum_{\begin{subarray}{c}1\leq\ell<m<n\leq
5\\\ i\in\\{\ell,m,n\\}\end{subarray}}x_{\ell
mn}\,\,\,+x_{E}\bigg{)}_{i=1,2,3,4,5}.$

The FY-monomial bases for $A^{0},A^{1},A^{2},A^{3}$ are shown here, together
with the $G$-equivariant maps $\lambda$:

$\begin{array}[]{cccccccc}\bf{FY^{0}}&&\bf{FY^{1}}&&\bf{FY^{2}}&&&\bf{FY^{3}}\\\
&&&&&&&\\\
1&\overset{\lambda}{\longmapsto}&x_{E}&\overset{\lambda}{\longmapsto}&x_{E}^{2}&&&x_{E}^{3}\\\
&&&&&&&\\\ &&x_{ijk}&\overset{\lambda}{\longmapsto}&x_{ijk}^{2}&&&\\\ &&1\leq
i<j<k\leq 5&&&&&\\\ &&&&&&&\\\
&&x_{ij}&\overset{\lambda}{\longmapsto}&x_{ij}\cdot x_{E}&&&\\\ &&1\leq
i<j\leq 5&&&&&\\\ \end{array}$

Thus $A(\mathcal{M})$ has Hilbert function
$(a_{0},a_{1},a_{2},a_{3})=(1,21,21,1)$. Here the bijection
$\pi\mathrel{\mathop{\mathchar
58\relax}}\mathrm{FY}^{0}\rightarrow\mathrm{FY}^{3}$ necessarily maps
$1\longmapsto x_{E}^{3}$, and the bijection $\pi\mathrel{\mathop{\mathchar
58\relax}}\mathrm{FY}^{1}\rightarrow\mathrm{FY}^{2}$ coincides with the map
$\lambda\mathrel{\mathop{\mathchar
58\relax}}\mathrm{FY}^{1}\rightarrow\mathrm{FY}^{2}$ above.

Before proving Theorem 1.1 in Section 3, the background Section 2 reviews
matroids and Chow rings, and collects a few simple observations. Section 4
recasts the maps $\lambda$ from Theorem 1.1(ii) in the language of unimodality
within the Burnside ring of $G$-permutation representations. This motivates
some analogous conjectures about Burnside rings that would extend recent log-
concavity and total positivity conjectures for Hilbert functions of Chow rings
$A(\mathcal{M})$. Section 5 poses some additional questions and conjectures.

## Acknowledgements

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

##  2\. Background

###  2.1. Matroids

There are many equivalent definitions of a matroid; the one most useful here
uses their _flats_. For matroid basics and background, good references are
Oxley [Oxl11] and Ardila [Ard21].

######  Definition 2.1.

A matroid $\mathcal{M}=(E,\mathfrak{F})$ consists of a (finite) ground set $E$
and a collection of subsets $\mathfrak{F}=\\{F\\}\subseteq 2^{E}$ called
flats, satisfying these axioms:

  1. (F1)

$E\in\mathfrak{F}$.

  2. (F2)

If $F,F^{\prime}\in\mathfrak{F}$ , then $F\cap F^{\prime}\in\mathfrak{F}$.

  3. (F3)

For any $F\in\mathfrak{F}$, and any $i\in E\setminus F$, there is a unique
$F^{\prime}\in\mathfrak{F}$ containing $i$ which covers $F$ in this sense:
$F\subsetneq F^{\prime}$, and no other $F^{\prime\prime}$ has $F\subsetneq
F^{\prime\prime}\subsetneq F$.

These axioms combinatorially abstract properties from the case of a realizable
matroid $\mathcal{M}$ associated to a list of vectors
$v_{1},v_{2},\ldots,v_{n}$ in a vector space over some field $\mathbb{F}$.
This realizable matroid $\mathcal{M}$ has ground set $E=\\{1,2,\ldots,n\\}$,
and a subset $F\subseteq E$ is a flat in $\mathfrak{F}$ if and only if this
inclusion is an equality:

$F\subseteq\\{i\in E\mathrel{\mathop{\mathchar
58\relax}}v_{i}\in\mathrm{span}_{\mathbb{F}}\\{v_{j}\\}_{j\in F}\\}.$

Axioms (F1),(F2) imply that the inclusion order $(\mathfrak{F},\subseteq)$
will be a lattice, with meets $F\wedge F^{\prime}=F\cap F^{\prime}$ and joins
$F\vee F^{\prime}=\bigcap_{F^{\prime\prime}\supseteq
F,F^{\prime}}F^{\prime\prime}$. Axiom (F3) further implies that the lattice
$\mathfrak{F}$ will be geometric, that is, both atomic (every $F$ is the join
of the atoms below it) and upper-semimodular: it has a rank function
$\operatorname{\mathrm{rk}}\mathrel{\mathop{\mathchar
58\relax}}\mathfrak{F}\rightarrow\\{0,1,2,\ldots\\}$ satisfying

$\operatorname{\mathrm{rk}}(F\vee
F^{\prime})+\operatorname{\mathrm{rk}}(F\wedge
F^{\prime})\leq\operatorname{\mathrm{rk}}(F)+\operatorname{\mathrm{rk}}(F^{\prime}).$

The rank of the matroid $\mathcal{M}$ itself is defined to be
$\operatorname{\mathrm{rk}}(E)$, and we assume throughout that
$\operatorname{\mathrm{rk}}(E)=r+1$.

An automorphism of the matroid $\mathcal{M}$ is any permutation
$g\mathrel{\mathop{\mathchar 58\relax}}E\to E$ of the ground set $E$ that
carries flats to flats: for all $F$ in $\mathfrak{F}$ one has $g(F)$ in
$\mathfrak{F}$. Let $G=\operatorname{\mathrm{Aut}}(\mathcal{M})$ denote the
group of all automorphisms of $\mathcal{M}$. Since such automorphisms respect
the partial order via inclusion on $\mathfrak{F}$, they also preserve the rank
function: $\operatorname{\mathrm{rk}}(g(F))=\operatorname{\mathrm{rk}}(F)$ for
all $g$ in $G$ and $F$ in $\mathfrak{F}$.

###  2.2. Chow Rings

As defined in the Introduction, Feichtner and Yuzvinsky [FY04] introduced the
Chow ring $A(\mathcal{M})$ of a matroid $\mathcal{M}$. Their goal was to model
the Chow ring of DeConcini and Procesi’s wonderful compactification of the
complement $V\setminus\bigcup_{i=1}^{n}H_{i}$ of an arrangement of hyperplanes
$H_{1},\ldots,H_{n}$, inside a vector space $V$. Here $\mathcal{M}$ is the
matroid in the dual space $V^{*}$ realized by linear forms
$f_{1},\ldots,f_{n}$ in $V^{*}$ with $H_{i}=\ker(f_{i})$ for $i=1,2,\ldots,n$.

######  Definition 2.2.

The _Chow ring_ $A(\mathcal{M})$ of a matroid $\mathcal{M}$ is the quotient
$Z$-algebra

$A(\mathcal{M})\mathrel{\mathop{\mathchar 58\relax}}=S/(I+J)$

where $S=\mathbb{Z}[x_{F}]$ is a polynomial ring having one variable $x_{F}$
for each nonempty flat $\varnothing\neq F\in\mathfrak{F}$, and where $I,J$ are
the following ideals of $S$:

  * •

$I$ is generated by products $x_{F}x_{F^{\prime}}$ for non-nested flats
$F,F^{\prime}$ (neither $F\subseteq F^{\prime}$ nor $F^{\prime}\subseteq F$),

  * •

$J$ is the ideal of $S$ generated by the linear elements
$\displaystyle\sum_{a\in F\in\mathfrak{F}}x_{F}$ for each atom $a$ (= flat of
rank $1$) in $\mathfrak{F}$.

The fact that the presentation of the Chow ring $A(\mathcal{M})$ only uses the
information about the partial order on the lattice of flats $\mathfrak{F}$ has
some consequences.

  * •

$A(\mathcal{M})$ depends only upon the associated simple matroid of
$\mathcal{M}$. That is, one may assume $\mathcal{M}$ has

    * –

no loops, meaning that one has empty intersection
$\varnothing=\bigcap_{F\in\mathfrak{F}}F$, and

    * –

no elements $i\neq j$ in $E$ which are parallel in the sense
$\\{F\in\mathfrak{F}\mathrel{\mathop{\mathchar 58\relax}}i\in
F\\}=\\{F\in\mathfrak{F}\mathrel{\mathop{\mathchar 58\relax}}j\in F\\}$.

Thus without loss of generality, one may assume that $\mathcal{M}$ is a simple
matroid throughout.

  * •

Any element $g$ in $G=\operatorname{\mathrm{Aut}}(\mathcal{M})$ will send the
generators of the ideals $I,J$ to other such generators. Thus $I+J$ is a
$G$-stable ideal, and $G$ acts on $A(\mathcal{M})$.

######  Remark 2.3.

A few remarks are in order, comparing the above definition of the Chow ring
$A(\mathcal{M})$ to that given by Feichtner–Yuzvinsky [FY04], as well as the
one used in Adiprasito-Huh-Katz [AHK18].

Feichtner and Yuzvinsky in [FY04] consider Chow rings which are more general
in two ways: they are associated to the more general class of atomic lattices
(not necessarily geometric lattices), and they incorporate the choice of a
subset of the lattice called a building set. Here we are both assuming that
the lattice is geometric, so that it corresponds to a (simple) matroid
$\mathcal{M}$, and we are also considering only the case of the maximal
building set, which is the set of all nonempty flats in $\mathfrak{F}$. A
reason for this choice is so that the entire group
$G=\operatorname{\mathrm{Aut}}(\mathcal{M})$ acts on the Chow ring, which
requires the building set to be stable under $G$, which does not always occur.

In [AHK18], they do consider certain other choices of nonmaximal building sets
which we are ignoring. However, they also alter the presentation of
$A(\mathcal{M})$ to eliminate the variable $x_{E}$ from the polynomial ring
$S=\mathbb{Z}[x_{F}]_{\varnothing\subsetneq F\subseteq E}$. Specifically they
write $A(\mathcal{M})=\hat{S}/(\hat{I}+\hat{J})$ where
$\hat{S}=\mathbb{Z}[x_{F}]$ has a variable $x_{F}$ for each nonempty, proper
flat $F$ with $\varnothing\subsetneq F\subsetneq E$. Then $\hat{I}$ is again
the ideal generated by the monomials $x_{F}x_{F^{\prime}}$ where
$F,F^{\prime}$ are not-nested, but now $\hat{J}$ is generated by the linear
elements

$\displaystyle\sum_{a\in F\neq
E\in\mathfrak{F}}x_{F}-\displaystyle\sum_{a^{\prime}\in F\neq
E\in\mathfrak{F}}x_{F}$

for each pair of distinct atoms $a\neq a^{\prime}$ in $\mathfrak{F}$. It is
not hard to check that this presentation of $A(\mathcal{M})$ is equivalent to
Definition 2.2: mutually inverse isomorphisms are induced by the map
$\hat{S}\rightarrow S$ sending $x_{F}\longmapsto x_{F}$, and the backward map
$S\rightarrow\hat{S}$ sending $x_{F}\longmapsto x_{F}$ for $F\neq E$ and
$x_{E}\longmapsto-\displaystyle\sum_{a\in F\neq E\in\mathfrak{F}}x_{F}$ for
any atom $a$ in $\mathfrak{F}$.

Note if one considers $S=\mathbb{Z}[x_{F}]$ as a graded $\mathbb{Z}$-algebra
in which $\deg(x_{F})=1$ for all nonempty flats $F$, then the ideals $I,J$ are
generated by homogeneous elements: all quadratic generators for $I$, all
linear generators for $J$. Hence the quotient $A(\mathcal{M})=S/(I+J)$
inherits the structure of a graded $\mathbb{Z}$-algebra
$A(\mathcal{M})=\bigoplus_{k=0}^{\infty}A^{k}$. Since the action of
$G=\operatorname{\mathrm{Aut}}(\mathcal{M})$ on $A(\mathcal{M})$ preserves
degrees, both $A(\mathcal{M})$ and each homogeneous component $A^{k}$ become
$\mathbb{Z}G$-modules.

It is not yet clear that $A(\mathcal{M})$ has only finitely many non-vanishing
components $A^{k}$, nor that it vanishes beyond $A^{r}$ where
$r=\operatorname{\mathrm{rk}}(\mathcal{M})-1$. For this and other purposes, we
next consider Feichtner and Yuzvinsky’s remarkable Gröbner basis for $I+J$
mentioned in the Introduction.

###  2.3. In praise of the Feichtner–Yuzvinsky Gröbner basis

We recall here one version of Gröbner basis theory with respect to a monomial
order on a polynomial ring $\mathbb{Z}[x_{1},\ldots,x_{n}]$, as used in
[FY04].

######  Definition 2.4.

A linear order $\prec$ on the set of all monomials
$\\{\mathbf{x}^{\alpha}\mathrel{\mathop{\mathchar
58\relax}}=x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}\\}$ in
$\mathbb{Z}[x_{1},\ldots,x_{n}]$ is called a monomial ordering if it is a
well-ordering (every subset of monomials has a $\prec$-minimum element) and
$\mathbf{x}^{\alpha}\prec\mathbf{x}^{\beta}$ implies
$\mathbf{x}^{\alpha}\cdot\mathbf{x}^{\gamma}\prec\mathbf{x}^{\beta}\cdot\mathbf{x}^{\gamma}$
for all $\alpha,\beta,\gamma$ in $\\{0,1,2,\ldots\\}^{n}$.

######  Example 2.5.

After fixing a linear order on the variables $x_{1}<\cdots<x_{n}$, the
associated lexicographic order $\prec$ has
$\mathbf{x}^{\alpha}\prec\mathbf{x}^{\beta}$ if there exists some
$k=1,2,\ldots,n$ with
$\alpha_{1}=\beta_{1},\alpha_{2}=\beta_{2},\ldots,\alpha_{k-1}=\beta_{k-1}$,
but $\alpha_{k}<\beta_{k}$.

######  Definition 2.6.

For $f=\sum_{\alpha}c_{\alpha}\mathbf{x}^{\alpha}$ in
$\mathbb{Z}[x_{1},\ldots,x_{n}]$, let
$\mathrm{in}_{\prec}(f)=\mathbf{x}^{\beta}$ denote the $\prec$-largest
monomial in $f$ having $c_{\beta}\neq 0$. Say $f$ is $\prec$-monic if the
initial monomial $\mathrm{in}_{\prec}(f)=\mathbf{x}^{\beta}$ has its
coefficient $c_{\beta}=\pm 1$.

Having fixed a monomial order $\prec$, given an ideal
$I\subset\mathbb{Z}[x_{1},\ldots,x_{n}]$, say that a subset
$\\{g_{1},\ldots,g_{t}\\}\subset I$ is a monic Gröbner basis for $I$ (with
respect to $\prec$) if each $g_{i}$ is $\prec$-monic, and every $f$ in $I$ has
$\mathrm{in}_{\prec}(f)$ divisible by at least one of the intial monomials
$\\{\mathrm{in}_{\prec}(g_{1}),\ldots,\mathrm{in}_{\prec}(g_{t})\\}$. Call
$\mathbf{x}^{\alpha}$ a standard monomial for $\\{g_{1},\ldots,g_{t}\\}$ with
respect to $\prec$ if $\mathbf{x}^{\alpha}$ is divisible by none of
$\\{\mathrm{in}_{\prec}(g_{1}),\ldots,\mathrm{in}_{\prec}(g_{t})\\}$.

It should be noted that some ideals $I\subset\mathbb{Z}[x_{1},\ldots,x_{n}]$
have no monic Gröbner basis, e.g. $I=(2)\subset\mathbb{Z}[x]$. However,
whenever $I$ does have a monic Gröbner basis, it has the following strong
consequence, proven exactly as for Gröbner bases over field cofficients; see,
e.g., Cox, Little, O’Shea [CLO15, §2.5, 5.3].

######  Proposition 2.7.

If $\\{g_{1},\ldots,g_{t}\\}$ is a monic Gröbner basis for $I\subset
S=\mathbb{Z}[x_{1},\ldots,x_{n}]$ with respect to $\prec$, then

  * •

$I=(g_{1},\ldots,g_{t})$, that is, $\\{g_{1},\ldots,g_{t}\\}$ generate $I$ as
an ideal.

  * •

The quotient ring $S/I$ is a free $\mathbb{Z}$-module, with a
$\mathbb{Z}$-basis given by the standard monomials $\\{x^{\alpha}\\}$.

The following crucial result appears as [FY04, Thm. 2]. To state it, define an
FY-monomial order on $S=\mathbb{Z}[x_{F}]_{\varnothing\neq F\in\mathfrak{F}}$
to be any monomial order based on a linear order of the variables with
$x_{F}>x_{F^{\prime}}$ if $F\subsetneq F^{\prime}$.

######  Theorem 2.8.

Given a matroid $\mathcal{M}$ and any FY-monomial order on
$S=\mathbb{Z}[x_{F}]_{\varnothing\neq F\in\mathfrak{F}}$, the ideal $I+J$
presenting $A(\mathcal{M})=S/(I+J)$ has a monic Gröbner basis
$\\{g_{F,F^{\prime}}\\}$ indexed by $F\neq F^{\prime}$ in $\mathfrak{F}$, with
$g_{F,F^{\prime}}$ and their initial terms
$\mathrm{in}_{\prec}(g_{F,F^{\prime}})$ as shown here:

condition on $F\neq F^{\prime}$ in $\mathfrak{F}$ | $g_{F,F^{\prime}}$ | $\mathrm{in}_{\prec}(g_{F,F^{\prime}})$
---|---|---
$F,F^{\prime}$ non-nested | $x_{F}x_{F^{\prime}}$ | $x_{F}x_{F^{\prime}}$
$\varnothing\neq F\subsetneq F^{\prime}$ | $\displaystyle x_{F}\left(\sum_{\begin{subarray}{c}F^{\prime\prime}\in\mathfrak{F}\mathrel{\mathop{\mathchar 58\relax}}\\\ F^{\prime\prime}\supseteq F^{\prime}\end{subarray}}x_{F^{\prime\prime}}\right)^{\operatorname{\mathrm{rk}}(F^{\prime})-\operatorname{\mathrm{rk}}(F)}$ | $\displaystyle x_{F}\cdot x_{F^{\prime}}^{\operatorname{\mathrm{rk}}(F^{\prime})-\operatorname{\mathrm{rk}}(F)}$
$\varnothing=F\subsetneq F^{\prime}$ | $\displaystyle\left(\sum_{\begin{subarray}{c}F^{\prime\prime}\in\mathfrak{F}\mathrel{\mathop{\mathchar 58\relax}}\\\ F^{\prime\prime}\supseteq F^{\prime}\end{subarray}}x_{F^{\prime\prime}}\right)^{\operatorname{\mathrm{rk}}(F^{\prime})}$ | $\displaystyle x_{F^{\prime}}^{\operatorname{\mathrm{rk}}(F^{\prime})}$

######  Corollary 2.9.

([FY04, Cor. 1]) For a matroid $\mathcal{M}$ of rank $r+1$, the Chow ring
$A(\mathcal{M})$ has these properties:

  * (i)

$A(\mathcal{M})$ is free as a $\mathbb{Z}$-module, with $\mathbb{Z}$-basis
given by the set of FY-monomials

(6) $\mathrm{FY}\mathrel{\mathop{\mathchar
58\relax}}=\\{x_{F_{1}}^{m_{1}}x_{F_{2}}^{m_{2}}\cdots
x_{F_{\ell}}^{m_{\ell}}\mathrel{\mathop{\mathchar
58\relax}}(\varnothing=\mathrel{\mathop{\mathchar 58\relax}}F_{0})\subsetneq
F_{1}\subsetneq F_{2}\subsetneq\cdots\subsetneq F_{\ell},\text{ in
}\mathfrak{F},\text{ and
}m_{i}\leq\operatorname{\mathrm{rk}}(F_{i})-\operatorname{\mathrm{rk}}(F_{i-1})-1\\}.$

  * (ii)

$A(\mathcal{M})$ vanishes in degrees strictly above $r$, that is,
$A(\mathcal{M})=\bigoplus_{k=0}^{r}A^{k}$.

  * (iii)

$A^{r}$ has $\mathbb{Z}$-basis $\\{x_{E}^{r}\\}$, and hence one has a
$\mathbb{Z}$-module isomorphism $\deg\mathrel{\mathop{\mathchar
58\relax}}A^{r}\longrightarrow\mathbb{Z}$ sending $x_{E}^{r}\longmapsto 1$.

###### Proof.

Assertion (i) follows from Theorem 2.8 after checking that the FY-monomials in
(6) are exactly the standard monomials for the $\prec$-monic Gröbner basis
$\\{g_{F,F^{\prime}}\\}$, that is, the monomials divisible by no
$\mathrm{in}_{\prec}(g_{F,F^{\prime}})$.

For assertion (ii), note that the typical FY-monomial
$x_{F_{1}}^{m_{1}}x_{F_{2}}^{m_{2}}\cdots x_{F_{\ell}}^{m_{\ell}}$, has total
degree

$\sum_{i=1}^{\ell}m_{i}\leq\sum_{i=1}^{\ell}(\operatorname{\mathrm{rk}}(F_{i})-\operatorname{\mathrm{rk}}(F_{i-1})-1)=\operatorname{\mathrm{rk}}(F_{\ell})-\ell\leq(r+1)-1=r.$

For assertion (iii), note equality occurs only if $\ell=1$ and $F_{\ell}=E$,
in which case the FY-monomial is $x_{E}^{r}$. ∎

Note that for any matroid automorphism $g$, the fact that
$\operatorname{\mathrm{rk}}(g(F))=\operatorname{\mathrm{rk}}(F)$ for every
flat $F$ in $\mathfrak{F}$ implies that $g$ sends any FY-monomial to another
FY-monomial:

$x_{F_{1}}^{m_{1}}x_{F_{2}}^{m_{2}}\cdots
x_{F_{\ell}}^{m_{\ell}}\,\,\overset{g}{\longmapsto}\,\,x_{g(F_{1})}^{m_{1}}x_{g(F_{2})}^{m_{2}}\cdots
x_{g(F_{\ell})}^{m_{\ell}}.$

This has an immediate corollary, inspired by work of H.-C. Liao on Boolean
matroids [Lia22, Thm. 2.5].

######  Corollary 2.10.

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

######  Remark 2.11.

It is rare to find families of ideals $I$ inside polynomial rings $S$ that are
stable under a finite group $G$ acting on $S$, and which also have a
$G$-stable Gröbner basis $\\{g_{i}\\}$ with respect to some monomial order
$\prec$. This occurs, for example, with the Hibi rings studied in [Hib87], and
the more general $P$-partition rings studied by Feray and the third author in
[FR12, Thm. 6.3]. More often, when starting with a $G$-stable ideal $I$,
passing to an intial ideal destroys the $G$-symmetry. One then usually needs
alternative techniques to work with the quotient $S/I$, as discussed, e.g., by
Faugére [FS13] and Sturmfels [Stu93, §2.6].

######  Remark 2.12.

Although the $\mathbb{Z}G$-module structure on $A(\mathcal{M})$ and $A^{k}$
are canonical, their lifts to permutation representations on the sets
$\mathrm{FY}$ and $\mathrm{FY}^{k}$ are not. In general, one can have two
different $G$-permutation representations on sets $X,X^{\prime}$ (that is,
with no $G$-equivariant set bijection $X\overset{\sim}{\longrightarrow}X$) but
with a $\mathbb{Z}G$-module isomorphism
$\mathbb{Z}[X]\cong\mathbb{Z}[X^{\prime}]$; see, e.g., Conlon [Con69] and
Scott [Sco93].

###  2.4. The Kähler package

The following theorem on the Kähler package for $A(\mathcal{M})$ compiles some
of the main results of the work of Adiprasito, Huh and Katz [AHK18].

######  Theorem 2.13.

For a matroid $\mathcal{M}$ of rank $r+1$, the Chow ring $A(\mathcal{M})$
satisfies the Kähler package:

  * •

(Poincaré duality)
For every $k\leq\frac{r}{2}$, one has a perfect $\mathbb{Z}$-bilinear pairing

$\displaystyle A^{k}\times A^{r-k}$ $\displaystyle\longrightarrow\mathbb{Z}$
$\displaystyle(a,b)$ $\displaystyle\longmapsto\deg(a\cdot b)$

that is, $b\longmapsto\varphi_{b}(-)$ defined by $\varphi_{b}(a)=\deg(a\cdot
b)$ is a $\mathbb{Z}$-linear isomorphism
$A^{r-k}\cong\mathrm{Hom}_{\mathbb{Z}}(A^{k},\mathbb{Z})$.

  *   * •

(Hard Lefschetz)
Tensoring over $\mathbb{Z}$ with $\mathbb{R}$, the (real) Chow ring
$A_{\mathbb{R}}(\mathcal{M})=\sum_{k=0}^{r}A^{k}_{\mathbb{R}}$ contains
Lefschetz elements $\omega$ in $A^{1}_{\mathbb{R}}$, meaning that $a\mapsto
a\cdot\omega^{r-2k}$ is an $\mathbb{R}$-linear isomorphism
$A^{k}_{\mathbb{R}}\rightarrow A^{r-k}_{\mathbb{R}}$ for $k\leq\frac{r}{2}$.

In particular, multiplication by $\omega$ is an injection
$A^{k}_{\mathbb{R}}\rightarrow A_{\mathbb{R}}^{k+1}$ for $k<\frac{r}{2}$.

  *   * •

(Hodge-Riemann-Minkowski inequalities)
The Lefschetz elements $\omega$ define quadratic forms
$a\longmapsto(-1)^{k}\deg(a\cdot\omega^{r-2k}\cdot a)$ on $A^{k}_{\mathbb{R}}$
that become positive definite upon restriction to the kernel of the map
$A^{k}_{\mathbb{R}}\longrightarrow A^{r-k+1}_{\mathbb{R}}$ that sends
$a\longmapsto a\cdot\omega^{r-2k+1}$.

In fact, they show that one obtains a Lefschetz element $\omega$ whenever
$\omega=\sum_{\varnothing\neq F\in\mathfrak{F}}c_{F}x_{F}$ has coefficients
$c_{F}$ coming from restricting to $\mathfrak{F}$ any function $A\mapsto
c_{A}$ that maps $2^{E}\rightarrow\mathbb{R}$ and satisfies these two
properties:

  * (1)

the strict submodular inequality $c_{A}+c_{B}>c_{A\cap B}+c_{A\cup B}$ for all
$A\neq B$, and

  * (2)

$c_{\varnothing}=c_{E}=0$.

This has consequences for $G$ acting on $A(\mathcal{M})$ and each $A^{k}$. The
first will be refined by Theorem 1.1(i).

######  Corollary 2.14.

For any matroid $\mathcal{M}$, one has an isomorphism of $\mathbb{Z}G$-modules
$A^{r-k}\rightarrow A^{k}$ for each $k\leq\frac{r}{2}$.

###### Proof.

Corollary 2.9(iii) shows that $A^{r}$ has only one $\mathbb{Z}$-basis element
$x_{E}^{r}$, fixed by every $g$ in $G$, so the degree map
$\deg\mathrel{\mathop{\mathchar 58\relax}}A^{r}\longrightarrow\mathbb{Z}$ is
$G$-equivariant for the trivial $G$-action on the target $\mathbb{Z}$. Thus
the Poincaré duality isomorphism
$A_{r-k}\longrightarrow\mathrm{Hom}_{\mathbb{Z}}(A_{k},\mathbb{Z})$, sending
$b\longmapsto\varphi_{b}(-)$ with $\varphi_{b}(a)=\deg(a\cdot b)$, is also
$G$-equivariant.

It only remains to exhibit a $G$-equivariant isomorphism
$\mathrm{Hom}_{\mathbb{Z}}(A_{k},\mathbb{Z})\rightarrow A_{k}$. To this end,
use Corollary 2.10 to pick a $\mathbb{Z}$-basis $\\{e_{i}\\}$ permuted by $G$,
so that each element $g$ in $G$ acts by a permutation matrix $P(g)$ in this
basis. Letting $\\{f_{i}\\}$ be the dual $\mathbb{Z}$-basis for
$\mathrm{Hom}_{\mathbb{Z}}(A_{k},\mathbb{Z})$, one finds that $g$ acts via the
matrix $P(g^{-1})^{T}=(P(g)^{-1})^{T}=P(g)$, since $P(g)$ is a permutation
matrix. Hence the map $e_{i}\longmapsto f_{i}$ is such a $G$-equivariant
isomorphism $\mathrm{Hom}_{\mathbb{Z}}(A_{k},\mathbb{Z})\rightarrow A_{k}$. ∎

The next consequence will be refined by Theorem 1.1(ii).

######  Corollary 2.15.

One has $\mathbb{R}G$-module maps $A^{k}_{\mathbb{R}}\rightarrow
A_{\mathbb{R}}^{k+1}$ which are injective for $k<\frac{r}{2}$.

###### Proof.

There exist Lefschetz elements $\omega\in A^{1}_{\mathbb{R}}$ which are
$G$-fixed, such as those exhibited on [AHK18, p. 384] having
$\omega=\sum_{F}c_{F}x_{F}$ with $c_{F}=|F|\cdot|E\setminus F|$.
Multiplication by such $\omega$ give the asserted $\mathbb{R}G$-module
injections. ∎

##  3\. Proof of Theorem 1.1

We recall the statement of the theorem, involving the FY-monomial
$\mathbb{Z}$-basis for $A(\mathcal{M})$ in Corollary 2.9:

$\mathrm{FY}\mathrel{\mathop{\mathchar
58\relax}}=\\{x_{F_{1}}^{m_{1}}x_{F_{2}}^{m_{2}}\cdots
x_{F_{\ell}}^{m_{\ell}}\mathrel{\mathop{\mathchar
58\relax}}(\varnothing=\mathrel{\mathop{\mathchar 58\relax}}F_{0})\subsetneq
F_{1}\subsetneq F_{2}\subsetneq\cdots\subsetneq F_{\ell}\text{ in
}\mathfrak{F},\text{ and
}m_{i}\leq\operatorname{\mathrm{rk}}(F_{i})-\operatorname{\mathrm{rk}}(F_{i-1})-1\\}$

This also means that the FY-monomials $\mathrm{FY}^{k}$ of degree $k$ form a
$\mathbb{Z}$-basis for $A^{k}$ for each $k=0,1,2,\ldots,r$.

Theorem 1.1 For every matroid $\mathcal{M}$ of rank $r+1$, there exist

  * (i)

$G$-equivariant bijections $\pi\mathrel{\mathop{\mathchar
58\relax}}\mathrm{FY}^{k}\overset{\sim}{\longrightarrow}\mathrm{FY}^{r-k}$ for
$k\leq\frac{r}{2}$, and

  * (ii)

$G$-equivariant injections $\lambda\mathrel{\mathop{\mathchar
58\relax}}\mathrm{FY}^{k}\hookrightarrow\mathrm{FY}^{k+1}$ for
$k<\frac{r}{2}$.

We offer two (related) proofs, the first slightly more conceptual, the second
with more explicit maps.

###  3.1. First proof

This proof organizes the FY-monomials according to the fibers of a map
$\mathrm{supp}_{+}\mathrel{\mathop{\mathchar 58\relax}}\mathrm{FY}\rightarrow
2^{\mathfrak{F}}.$

######  Definition 3.1.

For an FY-monomial $a=x_{F_{1}}^{m_{1}}x_{F_{2}}^{m_{2}}\cdots
x_{F_{\ell}}^{m_{\ell}}$, define its extended support
$\mathrm{supp}_{+}(a)\subset\mathfrak{F}$ by

$\mathrm{supp}_{+}(a)\mathrel{\mathop{\mathchar
58\relax}}=\\{F_{1},\ldots,F_{\ell}\\}\cup\\{E\\}=\begin{cases}\\{F_{1},\ldots,F_{\ell}\\}\cup\\{E\\}&\text{
if }F_{\ell}\subsetneq E,\\\ \\{F_{1},\ldots,F_{\ell}\\}&\text{ if
}F_{\ell}=E.\end{cases}$

Define a partial order $<_{+}$ on the FY-monomials in which $a<_{+}b$ if $a$
divides $b$ and $\mathrm{supp}_{+}(a)=\mathrm{supp}_{+}(b)$.

For integers $p<q$, let $[p,q]$ denote the usual linear order on the integers
$\\{p,p+1,\ldots,q-1,q\\}$. Given a sequence of such pairs $p_{i}<q_{i}$ for
$i=1,2,\ldots,m$, let

(7)
$\prod_{i=1}^{n}[p_{i},q_{i}]=[p_{1},q_{1}]\times[p_{2},q_{2}]\times\cdots\times[p_{m},q_{m}]$

denote their Cartesian product, partially ordered componentwise.

######  Proposition 3.2.

For any nested flag $\\{F_{1}\subsetneq\cdots\subsetneq F_{\ell}\subsetneq
E\\}$ in $\mathfrak{F}$ containing $E$, with convention

$\displaystyle F_{0}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\varnothing,$ $\displaystyle F_{\ell+1}$
$\displaystyle\mathrel{\mathop{\mathchar 58\relax}}=E$

the fiber $\mathrm{supp}_{+}^{-1}\\{F_{1},\ldots,F_{\ell},E\\}$ is the set of
monomials $\\{x_{F_{1}}^{m_{1}}x_{F_{2}}^{m_{2}}\cdots
x_{F_{\ell}}^{m_{\ell}}x_{E}^{m_{\ell+1}}\\}$ satisfying these inequalities:

$\displaystyle 1\leq
m_{i}\leq\operatorname{\mathrm{rk}}(F_{i})-\operatorname{\mathrm{rk}}(F_{i-1})-1$
$\displaystyle\text{ for }i=1,2,\ldots,\ell$ $\displaystyle 0\leq
m_{\ell+1}\leq\operatorname{\mathrm{rk}}(E)-\operatorname{\mathrm{rk}}(F_{\ell})-1=r-\operatorname{\mathrm{rk}}(F_{\ell}).$

Consequently, the minimum and maximum degree of monomials in
$\mathrm{supp}_{+}^{-1}\\{F_{1},\ldots,F_{\ell},E\\}$ are $\ell$ and $r-\ell$,
and one has a poset isomorphism

$\begin{array}[]{rcl}(\mathrm{supp}_{+}^{-1}\\{F_{1},\ldots,F_{\ell},E\\},<_{+})&\longrightarrow&\displaystyle\prod_{i=1}^{\ell}[1,\operatorname{\mathrm{rk}}(F_{i})-\operatorname{\mathrm{rk}}(F_{i-1})-1]\,\,\times\,\,[0,r-\operatorname{\mathrm{rk}}(F_{\ell})]\\\
&&\\\ x_{F_{1}}^{m_{1}}x_{F_{2}}^{m_{2}}\cdots
x_{F_{\ell}}^{m_{\ell}}x_{E}^{m_{\ell+1}}&\longmapsto&(m_{1},m_{2},\ldots,m_{\ell},m_{\ell+1}).\end{array}$

###### Proof.

Most assertions of the proposition are immediate from the definition of
$<_{+}$ and the map $\mathrm{supp}_{+}$. The minimum and maximum degrees of
monomials in $\mathrm{supp}_{+}^{-1}\\{F_{1},\ldots,F_{\ell},E\\})$ are
achieved by

$\displaystyle\deg(x_{F_{1}}^{1}x_{F_{2}}^{1}\cdots
x_{F_{\ell}}^{1}x_{E}^{0})$ $\displaystyle=\ell$
$\displaystyle\deg\left(\prod_{i=1}^{\ell}x_{F_{i}}^{\operatorname{\mathrm{rk}}(F_{i})-\operatorname{\mathrm{rk}}(F_{i})-1}\cdot
x_{E}^{\operatorname{\mathrm{rk}}(E)-\operatorname{\mathrm{rk}}(F_{\ell})-1)}\right)$
$\displaystyle=\sum_{i=1}^{\ell+1}(\operatorname{\mathrm{rk}}(F_{i})-\operatorname{\mathrm{rk}}(F_{i-1}-1)=\operatorname{\mathrm{rk}}(E)-(\ell+1)\
=r-\ell.\qed$

The first proof of Theorem 1.1 stems from the observation that all products of
chains, as in (7), have symmetric chain decompositions, which can then be
pulled back to each fiber
$\mathrm{supp}_{+}^{-1}\\{F_{1},\ldots,F_{\ell},E\\}.$

######  Definition 3.3.

A symmetric chain decomposition (SCD) of a finite ranked poset $P$ of rank $r$
is a disjoint decomposition $P=\bigsqcup_{i=1}^{t}C_{i}$ in which each $C_{i}$
is a totally ordered subset containing one element of each rank
$\\{\rho_{i},\rho_{i}+1,\ldots,r-\rho_{i}-1,r-\rho_{i}\\}$ for some
$\rho_{i}\in\\{0,1,2,\ldots,\lfloor\frac{r}{2}\rfloor\\}$.

It is not hard to check that when posets $P_{1},P_{2}$ each have an SCD, then
so does their Cartesian product. In particular, all products of chains have an
SCD; see, e.g., Anderson [And02, §3.1]. Fix one such SCD for each product
poset in (7), once and for all, and use the isomorphisms from Proposition 3.2
to induce an SCD on each fiber
$\mathrm{supp}_{+}^{-1}\\{F_{1},\ldots,F_{\ell},E\\}.$ An important point for
the proof is that the structure of these symmetric chains will depend only
upon the rank sets $\\{\operatorname{\mathrm{rk}}(F_{i})\\}_{i=1}^{\ell}$.

######  Example 3.4.

Assume $\mathcal{M}$ has $\operatorname{\mathrm{rk}}(E)=10=r+1$ with $r=9$,
and one has a pair of nested flats $F\subset F^{\prime}$ with
$\operatorname{\mathrm{rk}}(F)=3,\operatorname{\mathrm{rk}}(F^{\prime})=7$.
Then the poset $\mathrm{supp}_{+}^{-1}\\{F,F^{\prime},E\\}$ and one choice of
SCD for it look as follows:

$x_{F}x_{F^{\prime}}$$x_{F}x_{F^{\prime}}^{2}$$x_{F}^{2}x_{F^{\prime}}$$x_{F}x_{F^{\prime}}x_{E}$$x_{F}x_{F^{\prime}}^{3}$$x_{F}^{2}x_{F^{\prime}}^{2}$$x_{F}x_{F^{\prime}}^{2}x_{E}$$x_{F}^{2}x_{F^{\prime}}x_{E}$$x_{F}x_{F^{\prime}}x_{E}^{2}$$x_{F}^{2}x_{F^{\prime}}^{3}$$x_{F}x_{F^{\prime}}^{3}x_{E}$$x_{F}^{2}x_{F^{\prime}}^{2}x_{E}$$x_{F}x_{F^{\prime}}^{2}x_{E}^{2}$$x_{F}^{2}x_{F^{\prime}}x_{E}^{2}$$x_{F}^{2}x_{F^{\prime}}^{3}x_{E}$$x_{F}x_{F^{\prime}}^{3}x_{E}^{2}$$x_{F}^{2}x_{F^{\prime}}^{2}x_{E}^{2}$$x_{F}^{2}x_{F^{\prime}}^{3}x_{E}^{2}$$x_{F}x_{F^{\prime}}$$x_{F}x_{F^{\prime}}^{2}$$x_{F}^{2}x_{F^{\prime}}$$x_{F}x_{F^{\prime}}x_{E}$$x_{F}x_{F^{\prime}}^{3}$$x_{F}^{2}x_{F^{\prime}}^{2}$$x_{F}x_{F^{\prime}}^{2}x_{E}$$x_{F}^{2}x_{F^{\prime}}x_{E}$$x_{F}x_{F^{\prime}}x_{E}^{2}$$x_{F}^{2}x_{F^{\prime}}^{3}$$x_{F}x_{F^{\prime}}^{3}x_{E}$$x_{F}^{2}x_{F^{\prime}}^{2}x_{E}$$x_{F}x_{F^{\prime}}^{2}x_{E}^{2}$$x_{F}^{2}x_{F^{\prime}}x_{E}^{2}$$x_{F}^{2}x_{F^{\prime}}^{3}x_{E}$$x_{F}x_{F^{\prime}}^{3}x_{E}^{2}$$x_{F}^{2}x_{F^{\prime}}^{2}x_{E}^{2}$$x_{F}^{2}x_{F^{\prime}}^{3}x_{E}^{2}$

###### First proof of Theorem 1.1..

Let $a$ be any FY-monomial $a$ with $k\mathrel{\mathop{\mathchar
58\relax}}=\deg(a)$. Then $a$ lies in a unique fiber
$\mathrm{supp}_{+}^{-1}\\{F_{1},\ldots,F_{\ell},E\\}$ of the map
$\mathrm{supp}_{+}$, and on a unique chain $C_{i}$ in our fixed SCD of this
fiber. Writing

(8) $C_{i}=\\{a_{\rho}\lessdot a_{\rho+1}\lessdot\cdots\lessdot
a_{r-\rho-1}\lessdot a_{r-\rho}\\},$

where $\deg(a_{j})=j$ for $j=\rho,\rho+1,\ldots,r-\rho$, so that $a=a_{k}$,
then define the bijection $\pi$ and maps $\lambda$ via

$\displaystyle\pi(a)$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=a_{r-k},$ $\displaystyle\lambda(a)$
$\displaystyle\mathrel{\mathop{\mathchar 58\relax}}=a_{k+1}\text{ if
}k<\frac{r}{2}.$

To see that $\pi,\lambda$ are equivariant, note that having fixed
$\\{F_{1},\ldots,F_{\ell},F_{\ell+1}=E\\}$, both maps $f=\pi,\lambda$ will
send a monomial of the form $a=\prod_{i=1}^{\ell+1}x_{F_{i}}^{m_{i}}$ with
$m_{i}\geq 1$ to one of the form
$f(a)=\prod_{i=1}^{\ell+1}x_{F_{i}}^{m^{\prime}_{i}}$ in such a way that the
exponents $(m^{\prime}_{1},\ldots,m^{\prime}_{\ell},m^{\prime}_{\ell+1})$ are
determined uniquely as a function of the pair of data

$\left(\,\,(m_{1},\ldots,m_{\ell},m_{\ell+1})\,\,,\,\,\\{\operatorname{\mathrm{rk}}(F_{i})\\}_{i=1}^{\ell}\,\,\right).$

Thus for any $g$ in $G$, since
$\\{\operatorname{\mathrm{rk}}(g(F_{i}))\\}_{i=1}^{\ell}=\\{\operatorname{\mathrm{rk}}(F_{i})\\}_{i=1}^{\ell}$,
either of the maps $f=\pi,\lambda$ will send

$g(a)=\prod_{i=1}^{\ell+1}x_{g(F_{i})}^{m_{i}}\quad\longmapsto\quad
f(g(a))=\prod_{i=1}^{\ell+1}x_{g(F_{i})}^{m^{\prime}_{i}}=g(f(a)).\qed$

###  3.2. Proof via parenthesis pairing

This proof borrows an idea from the famous parenthesis-pairing SCD of Boolean
algebras due to Greene and Kleitman [GK76]. We begin with a two-step encoding
of FY-monomials within a fiber of the map $\mathrm{supp}_{+}$.

######  Definition 3.5.

Consider all FY-monomials $a=x_{F_{1}}^{m_{1}}\cdots
x_{F_{\ell}}^{m_{\ell}}x_{E}^{m_{\ell+1}}$ having a fixed extended support set
$\mathrm{supp}_{+}(a)=\\{F_{1}\subsetneq\cdots\subsetneq F_{\ell}\subsetneq
F_{\ell+1}=E\\}$, so $m_{1},\ldots,m_{\ell}\geq 1$ and $m_{\ell+1}\geq 0$. In
the first step, encode such monomials $a$ via a sequence $\mathfrak{D}(a)$ of
length $r$ in three symbols $\times,\bullet$, and a blank space, defined as
follows:

  * •

$\mathfrak{D}(a)$ has $\bullet$ in the positions
$\\{\operatorname{\mathrm{rk}}(F_{1}),\ldots,\operatorname{\mathrm{rk}}(F_{\ell})\\}$.

  * •

$\mathfrak{D}(a)$ has $\times$ in the first consecutive $m_{i}$ positions to
the left of $\operatorname{\mathrm{rk}}(F_{i})$ for each
$i=1,2,\ldots,\ell,\ell+1$.

  * •

$\mathfrak{D}(a)$ has a blank space in the remaining positions.

######  Example 3.6.

Continuing with the matroid $\mathcal{M}$ and its flats $F\subset F^{\prime}$
as discussed in Example 3.4. Here the monomials lie in the fiber
$\mathrm{supp}_{+}^{-1}\\{F,F^{\prime},E\\}$ where
$\\{\operatorname{\mathrm{rk}}(F),\operatorname{\mathrm{rk}}(F^{\prime}),\operatorname{\mathrm{rk}}(E)\\}=\\{3,7,10\\}$,
so $r=9$, and the positions
$\\{\operatorname{\mathrm{rk}}(F),\operatorname{\mathrm{rk}}(F^{\prime})\\}=\\{3,7\\}$
are shown in  green. The monomial $x_{F}x_{F^{\prime}}^{2}$ gets encoded as

$\begin{array}[]{ccccccccc}1&2&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}3}&4&5&6&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}7}&8&9\\\
\\\
&\times&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}&&\times&\times&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}&&\end{array}$

Note that one can recover $a$ from
$\mathrm{supp}_{+}(a)=\\{F_{1},\ldots,F_{\ell},E\\}$ and $\mathfrak{D}(a)$,
since $m_{i}$ can be read off as the number of $\times$ in $\mathfrak{D}(a)$
between positions $\operatorname{\mathrm{rk}}(F_{i-1})$ and
$\operatorname{\mathrm{rk}}(F_{i})$, with usual conventions
$F_{0}=\varnothing,F_{\ell+1}\mathrel{\mathop{\mathchar 58\relax}}=E$.

######  Definition 3.7.

The second step encodes $\mathfrak{D}(a)$ as a length $r$ parenthesis sequence
in $\\{(,)\\}^{r}$, having

  * •

a right parenthesis “$)$” in the positions of each $\bullet$ and each blank
space, and

  * •

a left parenthesis “(” in the positions of the $\times$.

######  Example 3.8.

Continuing Example 3.6, the diagram $\mathfrak{D}(x_{F}x_{F^{\prime}}^{2})$
gets encoded as the following sequence of parentheses:

$\begin{array}[]{ccccccccc}1&2&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}3}&4&5&6&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}7}&8&9\\\
\\\
&\times&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}&&\times&\times&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}&&\\\
)&(&)&)&(&(&)&)&)\end{array}$

Note that one can recover $\mathfrak{D}(a)$ from this $\\{(,)\\}^{r}$ sequence
as follows:

  * -

$\mathfrak{D}(a)$ has $\times$ occurring in the positions of the left
parentheses, and

  * -

$\mathfrak{D}(a)$ has the $\bullet$ occurring exactly in the positions of the
right parenthesis in a consecutive pair $()$, while blank spaces occur in the
position of all other right parentheses.

###### Second proof of Theorem 1.1..

Given an FY-monomial $a$ in $\mathrm{FY}^{k}$ with $k\leq\frac{r}{2}$, we will
use its parenthesis sequence in $\\{(,)\\}^{r}$ to place $a$ within a chain of
monomials as in (8), of the form

$C_{i}=\\{a_{\rho}\lessdot a_{\rho+1}\lessdot\cdots\lessdot
a_{r-\rho-1}\lessdot a_{r-\rho}\\},$

where $\deg(a_{j})=j$ for $j=\rho,\rho+1,\ldots,r-\rho$, so that $a=a_{k}$. To
this end, define the set of paired parentheses in $a$ (shown underlined in
Example 3.9) by including all consecutive pairs $()$, and after removing these
consecutive pairs, including the new $()$ pairs which have become consecutive,
repeating this recursively. After removing some number $\rho$ of pairs $()$
via this pairing process, the process ends when one reaches a sequence of
$r-2\rho$ remaining unpaired parentheses of this form:

(9)
$\underbrace{))\cdots))}_{k-\rho}\underbrace{{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}((\cdots((}}_{r-\rho-k}.$

The monomials in the chain $C_{i}$ are defined to be those whose set of paired
parentheses agree exactly with those of $a$, both in their postitions, and
left-right pairing structure– see the underlined parentheses in Example 3.9.

As in the first proof of Theorem 1.1, one then defines the bijections $\pi$
and maps $\lambda$ via

$\displaystyle\pi(a)$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=a_{r-k},$ $\displaystyle\lambda(a)$
$\displaystyle\mathrel{\mathop{\mathchar 58\relax}}=a_{k+1}\text{ if
}k<\frac{r}{2}.$

In other words, both maps $\lambda$ and $\pi$ when applied to $a$ will keep
all of the paired parentheses fixed, but

  * •

$\lambda$ changes the leftmost unpaired left parenthesis “)” into an unpaired
right parenthesis “(”, and

  * •

$\pi$ swaps the numbers $k-\rho$ and $r-\rho-k$ of unpaired right and left
parentheses in (9).

The equivariance of these two maps $\pi,\lambda$ is argued exactly as in the
first proof of the theorem. ∎

######  Example 3.9.

Here is an example of a symmetric chain from Example 3.4, explained via this
two-step encoding with $\mathfrak{D}(a)$ and $\\{(,)\\}^{r}$-sequences, as we
have seen preceding Examples 3.6 and 3.8. Paired parentheses are underlined,
and are fixed throughout the chain. Moving up the chain, unpaired right
parentheses change one-by-one to left parentheses (depicted  orange here), in
order from right to left:

$\begin{array}[]{ccccccccccl}&1&2&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}3}&4&5&6&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}7}&8&9\\\
&&&&&&&&&\\\
x_{F}^{2}x_{F^{\prime}}^{3}x_{E}&\times&\times&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}&\times&\times&\times&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}&&\times\\\
&{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}(}&\underline{(}&\underline{)}&{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}(}&\underline{(}&\underline{(}&\underline{)}&\underline{)}&{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}(}\\\
&&&&&&&&&\\\ \uparrow&&&&&&&&&\\\ &&&&&&&&&\\\
x_{F}x_{F^{\prime}}^{3}x_{E}&&\times&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}&\times&\times&\times&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}&&\times\\\
&)&\underline{(}&\underline{)}&{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}(}&\underline{(}&\underline{(}&\underline{)}&\underline{)}&{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}(}\\\
&&&&&&&&&\\\ \uparrow&&&&&&&&&\\\ &&&&&&&&&\\\
x_{F}x_{F^{\prime}}^{2}x_{E}&&\times&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}&&\times&\times&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}&&\times\\\
&)&\underline{(}&\underline{)}&)&\underline{(}&\underline{(}&\underline{)}&\underline{)}&{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}(}\\\
&&&&&&&&&\\\ \uparrow&&&&&&&&&\\\ &&&&&&&&&\\\
x_{F}x_{F^{\prime}}^{2}&&\times&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}&&\times&\times&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}&&\\\
&)&\underline{(}&\underline{)}&)&\underline{(}&\underline{(}&\underline{)}&\underline{)}&)\\\
\end{array}$

Here $r=9$ and the number of parenthesis pairs is $\rho=3$, so that this chain
$C_{i}$ is symmetrically placed within the degrees of $A(\mathcal{M})$,
containing monomials of degrees $[\rho,r-\rho]=[3,6]=\\{3,4,5,6\\}$ out of the
list of possible degrees $[0,r]=[0,9]=\\{0,1,2,\mathbf{3,4,5,6},7,8,9\\}$.

##  4\. Conjectures on equivariant and Burnside ring inequalities

We have mentioned that the unimodality statement (2), asserting for
$k<\frac{r}{2}$ that one has

$a_{k}\leq a_{k+1},$

is weaker than the statement in Corollary 2.15 asserting that there are
injective $\mathbb{R}G$-module maps

$A^{k}_{\mathbb{R}}\rightarrow A_{\mathbb{R}}^{k+1},$

which is weaker than Theorem 1.1(ii) asserting that there are injective
$G$-equivariant maps of the $G$-sets

$\mathrm{FY}^{k}\hookrightarrow\mathrm{FY}^{k+1}.$

In this section, we wish to consider not only unimodality for
$(a_{0},a_{1},\ldots,a_{r})$, but other properties like log-concavity, the
Pólya frequency property, and how to similarly lift them to statements
regarding $\mathbb{R}G$-modules and $G$-permutation representations. In
phrasing this, it helps to consider certain algebraic objects.

###  4.1. Virtual character rings and Burnside rings

######  Definition 4.1.

(Virtual character ring) For a finite group $G$, its virtual (complex)
character ring $R_{\mathbb{C}}(G)$ is the free $\mathbb{Z}$-submodule of the
ring of (conjugacy) class functions $\\{f\mathrel{\mathop{\mathchar
58\relax}}G\rightarrow\mathbb{C}\\}$ with pointwise addition and
multiplication, having as a $\mathbb{Z}$-basis the irreducible complex
characters $\\{\chi_{1},\ldots,\chi_{M}\\}$, where $M$ is the number of
conjugacy classes of $G$. Thus every virtual character $\chi$ in
$R_{\mathbb{C}}(G)$ has a unique expansion $\chi=\sum_{i=1}^{M}a_{i}\chi_{i}$
for some $a_{i}\in\mathbb{Z}$. If $a_{i}\geq 0$ for $i=1,2,\ldots,M$, call
$\chi$ a genuine character, and write $\chi\geq_{R_{\mathbb{C}}(G)}0$.
Similarly, we write $\chi\geq_{R_{\mathbb{C}}(G)}\chi^{\prime}$ when
$\chi-\chi^{\prime}\geq_{R_{\mathbb{C}}(G)}0$.

######  Definition 4.2.

(Burnside ring) For a finite group $G$, to define its Burnside ring $B(G)$ one
starts with a free $\mathbb{Z}$-module having as basis the isomorphism classes
$[X]$ of finite $G$-sets $X$. Then $B(G)$ is the quotient $\mathbb{Z}$-module
that mods out by the span of all elements $[X\sqcup Y]-([X]+[Y])$.
Multiplication in $B(G)$ is induced from the rule $[X]\cdot[Y]=[X\times Y]$.
It turns out that $B(G)$ has a $\mathbb{Z}$-basis $\\{[G/G_{i}]\\}_{i=1}^{N}$
as $G_{1},\ldots,G_{N}$ run through representatives of the $G$-conjugacy
classes of subgroups of $G$. Thus every element $b$ of $B(G)$ has a unique
expansion $b=\sum_{i=1}^{N}a_{i}[G/G_{i}]$ for some $a_{i}\in\mathbb{Z}$. If
$a_{i}\geq 0$ for $i=1,2,\ldots,N$, call $b$ a genuine permutation
representation, and write $b\geq_{B(G)}0$. Similarly, write
$b\geq_{B(G)}b^{\prime}$ when $b-b^{\prime}\geq_{B(G)}0$.

Note that there are natural ring maps

$\displaystyle B(G)$ $\displaystyle\longrightarrow R_{\mathbb{C}}(G),$
$\displaystyle R_{\mathbb{C}}(G)$ $\displaystyle\longrightarrow\mathbb{Z}.$

The first map sends the class $[X]$ of a $G$-set $X$ to the character
$\chi_{\mathbb{C}[X]}$ of its $G$-permutation representation $\mathbb{C}[X]$,
having character values $\chi_{\mathbb{C}[X]}(g)=\\#\\{x\in
X\mathrel{\mathop{\mathchar 58\relax}}g(x)=x\\}$. The second map sends a
virtual character $\chi$ to its value $\chi(e)$ on the identity $e$ of $G$.
These maps carry genuine elements $b\geq_{B(G)}0$ in $B(G)$ to genuine
characters $\chi\geq_{R_{\mathbb{C}}(G)}0$ which are then carried to
nonnegative integers $\mathbb{Z}_{\geq 0}$. In this way, inequalities in
$B(G)$ lift inequalities in $R_{\mathbb{C}}(G)$, which lift inequalities in
$\mathbb{Z}$.

######  Example 4.3.

We saw that the unimodality inequality (2) $a_{k}\leq_{\mathbb{Z}}a_{k+1}$
lifts to the inequality
$A_{\mathbb{R}}^{k}\leq_{R_{\mathbb{C}}(G)}A_{\mathbb{R}}^{k+1}$ in Corollary
2.15, which lifts to the inequality
$[\mathrm{FY}^{k}]\leq_{B(G)}[\mathrm{FY}^{k+1}]$ in Theorem 1.1(ii).

It should also be noted that, just as one can multiply inequalities in
$\mathbb{Z}$ like $a<b$ and $c<d$ to get new inequalities $ac<bd$, the same
works in $R_{\mathbb{C}}(G)$ and in $B(G)$. This is because
$\chi,\chi^{\prime}\geq_{R_{\mathbb{C}}(G)}0$ implies
$\chi\cdot\chi^{\prime}\geq_{R_{\mathbb{C}}(G)}0$, and similarly
$b,b^{\prime}\geq_{B(G)}0$ implies $b\cdot b^{\prime}\geq_{B(G)}0$.

###  4.2. PF sequences and log-concavity

For a sequence of positive real numbers $(a_{0},a_{1},\ldots,a_{r})$, the
property of unimodality lies at the bottom of a hierarchy of concepts

(10)
$\begin{array}[]{rccccccccccccccc}\text{unimodal}&\Leftarrow&PF_{2}&\Leftarrow&&PF_{3}&&\Leftarrow&&PF_{4}&&\Leftarrow&\cdots&\Leftarrow&&PF_{\infty}\\\
&&\|&&&&&&&&&&&&&\|\\\ &&\text{(strongly) log-concave}&&&&&&&&&&&&&PF\\\
\end{array}$

which we next review, along with their equivariant and Burnside ring
extensions. For background on the non-equivariant versions, see Brenti [Bre89]
and Stanley [Sta89]. For the equivariant versions, see Gedeon, Proudfoot and
Young [GPY17], Matherne, Miyata, Proudfoot and Ramos [Mat+23], Gui [Gui22],
Gui and Xiong [GX22], and Li [Li22].

######  Definition 4.4.

Say a sequence of positive reals $(a_{0},a_{1},\ldots,a_{r})$ is unimodal if
there is some index $m$ with

$a_{0}\leq a_{1}\leq\cdots\leq a_{m-1}\leq a_{m}\geq a_{m+1}\geq\cdots\geq
a_{r-1}\geq a_{r}.$

Say the sequence is strongly111The word “strongly” here is superfluous, since
we assumed each $a_{k}>0$, so they are strongly log-concave if and only they
are weakly log-concave: $a_{k}^{2}\geq a_{k-1}a_{k+1}$. The distinction
becomes important for the equivariant analogue; see [Mat+23, §2]. log-concave
(or $PF_{2}$) if $0\leq i\leq j\leq k\leq\ell\leq r$ and $i+\ell=j+k$ implies

$a_{i}a_{\ell}\leq a_{j}a_{k},\text{ or equivalently,
}\det\left[\begin{matrix}a_{j}&a_{\ell}\\\ a_{i}&a_{k}\end{matrix}\right]\geq
0.$

For $\ell=2,3,4,\ldots$, say that the sequence is $PF_{\ell}$ if the
associated (infinite) Toeplitz matrix

$T(a_{0},\ldots,a_{r})\mathrel{\mathop{\mathchar
58\relax}}=\left[\begin{matrix}a_{0}&a_{1}&a_{2}&\cdots&a_{r-1}&a_{r}&0&0&\cdots\\\
0&a_{0}&a_{1}&\cdots&a_{r-2}&a_{r-1}&a_{r}&0&\cdots\\\
0&0&a_{0}&\cdots&a_{r-3}&a_{r-2}&a_{r-1}&a_{r}&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{matrix}\right]$

has all nonnegative square minor subdeterminants of size $m\times m$ for
$1\leq m\leq\ell$. Say that the sequence is a Pólya frequency sequence (or
$PF_{\infty}$, or just $PF$) if it is $PF_{\ell}$ for all $\ell=2,3,\ldots$.

One can check the implication ($PF_{2}$ implies unimodality) from (10) using
the assumption $a_{k}>0$ for all $k$. It also turns out that
$(a_{0},a_{1},\ldots,a_{r})$ is $PF$ if and only if the polynomial
$a_{0}+a_{1}t+a_{2}t^{2}+\cdots+a_{r}t^{r}$ has only (negative) real roots;
see [Bre89, §2.2, 4.5].

######  Definition 4.5.

For a finite group $G$ and (genuine, nonzero) $\mathbb{C}G$-modules
$(A^{0},A^{1},\ldots,A^{r})$, define the analogous notions of equivariant
unimodality, equivariant strong log-concavity, equivariant $PF_{r}$ or
$PF_{\infty}$ by replacing the numerical inequalities in Definition 4.4 by
inequalities in the representation ring $R_{\mathbb{C}}(G)$.

Similarly, for (nonempty) $G$-sets $(X_{0},X_{1},\ldots,X_{r})$, define the
notions of Burnside unimodality, Burnside strong log-concavity, Burnside
$PF_{r}$ or $PF_{\infty}$ by replacing them with inequalities in the Burnside
ring $B(G)$.

######  Example 4.6.

We’ve seen for Chow rings $A(\mathcal{M})=\bigoplus_{k=0}^{r}A^{k}$ of rank
$r+1$ matroids $\mathcal{M}$, and
$G=\operatorname{\mathrm{Aut}}(\mathcal{M})$,

  * •

the sequence $(a_{0},a_{1},\ldots,a_{r})$ with
$a_{k}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{\mathrm{rk}}_{\mathbb{Z}}A_{k}$ is unimodal,

  * •

after tensoring with $\mathbb{C}$, the sequence of $\mathbb{C}G$-modules
$(A^{0}_{\mathbb{C}},A^{1}_{\mathbb{C}},\ldots,A^{r}_{\mathbb{C}})$ is
equivariantly unimodal, and

  * •

the sequence of $G$-sets
$(\mathrm{FY}^{0},\mathrm{FY}^{1},\ldots,\mathrm{FY}^{r})$ is Burnside
unimodal.

######  Conjecture 4.7.

In the same Chow ring context as Example 4.6, one has that

  * (i)

(Ferroni-Schröter [FS22, Conj. 10.19]) $(a_{0},\ldots,a_{r})$ is
$PF_{\infty}$.

  * (ii)

$(A^{0}_{\mathbb{C}},\ldots,A^{r}_{\mathbb{C}})$ is equivariantly
$PF_{\infty}$.

  * (iii)

$(\mathrm{FY}^{0},\ldots,\mathrm{FY}^{r})$ is Burnside $PF_{2}$ (Burnside log-
concave), that is,

$[\mathrm{FY}^{i}][\mathrm{FY}^{\ell}]\leq_{B(G)}[\mathrm{FY}^{j}][\mathrm{FY}^{k}]\qquad\text{for
}\text{for }\;i\leq j\leq k\leq\ell\;\text{ with }\;i+\ell=j+k.$

Of course, in Conjecture 4.7, assertion (ii) implies assertion (i). However
assertion (iii) would only imply the weaker $PF_{2}$ part of the conjectural
assertion (ii), and only imply the $PF_{2}$ part of Ferroni and Schröter’s
assertion (i), but not their $PF_{\infty}$ assertions. Even the $PF_{2}$
property for $(a_{0},\ldots,a_{r})$ is still conjectural; see [FS22, §10.3]
and [Fer+23, §3.7] for a discussion of the current evidence for Conjecture
4.7(i).

######  Example 4.8.

We explain here why Conjecture 4.7 does not assert that
$(\mathrm{FY}^{0},\ldots,\mathrm{FY}^{r})$ is Burnside $PF_{\infty}$. In fact,
$(\mathrm{FY}^{0},\ldots,\mathrm{FY}^{r})$ fails even to be Burnside $PF_{3}$,
already when $\mathcal{M}$ is a rank $4$ Boolean matroid. Its Chow ring
$A(\mathcal{M})=A^{0}\oplus A^{1}\oplus A^{2}\oplus A^{3}$ has $A_{0},A_{3}$
carrying the trivial $\mathbb{C}\mathfrak{S}_{4}$-module, and $A^{1},A^{2}$
carrying isomorphic permutation representations, each having three orbits,
whose three $\mathfrak{S}_{4}$-stabilizer groups are the Young subgroups
$\mathfrak{S}_{4},\mathfrak{S}_{3}\times\mathfrak{S}_{1},\mathfrak{S}_{2}\times\mathfrak{S}_{2}$.
The red $3\times 3$ minor of the Toeplitz matrix shown here

$\left[\begin{matrix}a_{0}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a_{1}}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a_{2}}&a_{3}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&0&\cdots\\\
0&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a_{0}}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a_{1}}&a_{2}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a_{3}}&0&\cdots\\\
0&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a_{0}}&a_{1}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a_{2}}&a_{3}&\cdots\\\
0&0&0&a_{0}&a_{1}&a_{2}&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\\ \end{matrix}\right]$

has determinant
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a_{1}^{2}a_{2}-a_{1}a_{3}-a_{2}^{2}}.$
Hence the Burnside $PF_{3}$ condition would require that the following genuine
$\mathfrak{S}_{4}$-character should come from a genuine permutation
representation

${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\left(\chi_{A^{1}}\right)^{2}\cdot\chi_{A^{2}}-\chi_{A^{1}}\chi_{A^{3}}-\left(\chi_{A^{2}}\right)^{2}}=29\chi^{(1,1,1,1)}+124\chi^{(2,1,1)}+103\chi^{(2,2)}+172\chi^{(3,1)}+76\chi^{(4)},$

where here $\chi^{\lambda}$ denotes the irreducible
$\mathfrak{S}_{n}$-representation [Sag01], [Sta24, §7.18] indexed by the
partition $\lambda$ of $n$; this expansion was computed using Sage/Cocalc. But
one can check that this is not a permutation representation, as its character
value on the conjugacy class of $4$-cycles in $\mathfrak{S}_{4}$ is
$+76-172+124-29=-1<0$.

######  Remark 4.9.

Although Example 4.8 shows that even the Boolean matroids contradict
strengthening Conjecture 4.7(iii) to Burnside $PF_{3}$, they seem to satisfy a
different strengthening of strong log-concavity:

######  Conjecture 4.10.

For a Boolean matroid $\mathcal{M}$ of rank $n$ and $i\leq j\leq k\leq\ell$
with $i+\ell=j+k$, not only is the element
$[\mathrm{FY}^{j}][\mathrm{FY}^{k}]-[\mathrm{FY}^{i}][\mathrm{FY}^{\ell}]\geq_{B(\mathfrak{S}_{n})}0,$
so that it is a genuine permutation representation, but furthermore one whose
orbit-stabilizers are all Young subgroups
$\mathfrak{S}_{\lambda}\mathrel{\mathop{\mathchar
58\relax}}=\mathfrak{S}_{\lambda_{1}}\times\mathfrak{S}_{\lambda_{2}}\times\cdots\times\mathfrak{S}_{\lambda_{\ell}}$.

We note here a small amount of evidence for Conjecture 4.7(ii), (iii), namely
their strong log-concavity assertions hold for the case $i=0$. For assertion
(ii), this is an easy consequence of the fact that the Chow ring
$A(\mathcal{M})$ is generated by the variables $\\{y_{F}\\}$ spanning its
degree one component $A^{1}$, which shows that this $G$-equivariant
multiplication map surjects:

(11) $A^{j}\otimes A^{k}\twoheadrightarrow A^{j+k}\left(\cong A^{0}\otimes
A^{j+k}\right).$

We next check that the stronger assertion of Conjecture 4.7(iii) also holds in
the special case $i=0$.

######  Proposition 4.11.

For matroids $\mathcal{M}$ and $j,k\geq 0$, one has a $G$-equivariant
injection
$\mathrm{FY}^{j+k}\hookrightarrow\mathrm{FY}^{j}\times\mathrm{FY}^{k}.$

###### Proof.

Given $a=x_{F_{1}}^{m_{1}}\cdots x_{F_{\ell}}^{m_{\ell}}$ in
$\mathrm{FY}^{j+k}$, so that $j+k=\sum_{i=1}^{\ell}m_{i}$, let $p$ be the
smallest index such that

(12) $\sum_{i=1}^{p-1}m_{i}<j\leq\sum_{i=1}^{p}m_{i}$

and factor the monomial $a=b\cdot c$ where

$a=\underbrace{x_{F_{1}}^{m_{1}}\cdots
x_{F_{p-1}}^{m_{p-1}}x_{F_{p}}^{\delta}}_{b\mathrel{\mathop{\mathchar
58\relax}}=}\quad\cdot\quad\underbrace{x_{F_{p}}^{m_{p}-\delta}x_{F_{p+1}}^{m_{p+1}}\cdots
x_{F_{\ell}}^{m_{\ell}}}_{c\mathrel{\mathop{\mathchar 58\relax}}=}$

with $\delta\mathrel{\mathop{\mathchar 58\relax}}=j-\sum_{i=1}^{p-1}m_{i}$ (so
$\delta>0$ by (12)), and $m_{p}-\delta\geq 0$. One can check that, since $a$
lies in $\mathrm{FY}^{j+k}$, one will also have $b,c$ lying in
$\mathrm{FY}^{j},\mathrm{FY}^{k}$, respectively. It is easily seen that the
map $a\longmapsto(b,c)$ is injective, since its inverse sends
$(b,c)\longmapsto bc$. It is also not hard to check that it is
$G$-equivariant.∎

######  Remark 4.12.

Note that one can iterate the map in the previous proof to construct
$G$-equivariant injections
$\prod_{i=1}^{q}\mathrm{FY}^{\alpha_{i}}\hookrightarrow\prod_{j=1}^{p}\mathrm{FY}^{\beta_{j}}$
whenever $\beta=(\beta_{1},\beta_{2},\ldots,\beta_{p})$ is a composition
refining $\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{q}).$

As another small piece of evidence for Conjecture 4.7 (ii), (iii), we show
that $(\mathrm{FY}^{0},\ldots,\mathrm{FY}^{r})$ is Burnside $PF_{2}$ for
$r\leq 5$, that is, for matroids of rank at most $6$.

######  Proposition 4.13.

For any matroid $\mathcal{M}$ with
$\operatorname{\mathrm{rk}}(\mathcal{M})\leq 6$, the sequence
$(\mathrm{FY}^{0},\ldots,\mathrm{FY}^{r})$ is Burnside $PF_{2}$.

###### Proof sketch..

We check it for $\operatorname{\mathrm{rk}}(\mathcal{M})=6$, and
$\operatorname{\mathrm{rk}}(\mathcal{M})\leq 5$ is similar. Theorem 1.1(i)
shows that in $B(G)$,

$\Big{(}[\mathrm{FY}^{0}],\,[\mathrm{FY}^{1}],\,[\mathrm{FY}^{2}],\,[\mathrm{FY}^{3}],\,[\mathrm{FY}^{4}],\,[\mathrm{FY}^{5}]\Big{)}=\Big{(}1,\,[\mathrm{FY}^{1}],\,[\mathrm{FY}^{2}],\,[\mathrm{FY}^{2}],\,[\mathrm{FY}^{1}],\,1\Big{)}.$

Hence one must check nonnegativity in $B(G)$ for all 2 x 2 minors in this
infinite Toeplitz matrix:

$\begin{bmatrix}1&[\mathrm{FY}^{1}]&[\mathrm{FY}^{2}]&[\mathrm{FY}^{2}]&[\mathrm{FY}^{1}]&1&0&0&\ldots\\\
0&1&[\mathrm{FY}^{1}]&[\mathrm{FY}^{2}]&[\mathrm{FY}^{2}]&[\mathrm{FY}^{1}]&1&0&\ldots\\\
0&0&1&[\mathrm{FY}^{1}]&[\mathrm{FY}^{2}]&[\mathrm{FY}^{2}]&[\mathrm{FY}^{1}]&1&\ldots\\\
0&0&0&1&[\mathrm{FY}^{1}]&[\mathrm{FY}^{2}]&[\mathrm{FY}^{2}]&[\mathrm{FY}^{1}]&\ldots\\\
0&0&0&0&1&[\mathrm{FY}^{1}]&[\mathrm{FY}^{2}]&[\mathrm{FY}^{2}]&\ldots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\\
\end{bmatrix}$

From periodicity of the matrix, one may assume without loss of generality that
the $2\times 2$ minor has its top-left entry in the first row. If the minor
has a $0$ as either its lower-left or upper-right entry, then its determinant
is a product
$[\mathrm{FY}^{i}][\mathrm{FY}^{j}]=[\mathrm{FY}^{i}\times\mathrm{FY}^{j}]\geq_{B(G)}0$.
This already leaves only finitely many $2\times 2$ minors to check.
Additionally, if it has $1$ as its lower left entry, then it was shown to be
Burnside-nonnegative in Theorem 4.11. All of the remaining $2\times 2$ minors
we claim are Burnside-nonnegative because they compare two (possibly non-
consecutive) terms in this chain of inequalities:

$1\overset{(a)}{\leq}_{B(G)}[\mathrm{FY}^{1}]\overset{(b)}{\leq}_{B(G)}[\mathrm{FY}^{2}]\overset{(c)}{\leq}_{B(G)}[\mathrm{FY}^{1}][\mathrm{FY}^{1}]\overset{(d)}{\leq}_{B(G)}[\mathrm{FY}^{1}][\mathrm{FY}^{2}]\overset{(e)}{\leq}_{B(G)}[\mathrm{FY}^{2}][\mathrm{FY}^{2}]$

where inequalities (a),(b) follow from Theorem 1.1(ii), inequality (c) follows
from Theorem 4.11, and inequality (d),(e) come from multiplying inequality (b)
by $[\mathrm{FY}^{1}]$ and multiplying inequality (a) by $[\mathrm{FY}^{2}]$.
∎

######  Remark 4.14.

When $\operatorname{\mathrm{rk}}(\mathcal{M})\geq 7$, one encounters the first
$2\times 2$ minor in $B(G)$ for $G=\operatorname{\mathrm{Aut}}(\mathcal{M})$

$\det\left[\begin{matrix}[\mathrm{FY}^{2}]&[\mathrm{FY}^{3}]\\\
[\mathrm{FY}^{1}]&[\mathrm{FY}^{2}]\end{matrix}\right]=[\mathrm{FY}^{2}][\mathrm{FY}^{2}]-[\mathrm{FY}^{1}][\mathrm{FY}^{3}]=[\mathrm{FY}^{2}\times\mathrm{FY}^{2}]-[\mathrm{FY}^{1}\times\mathrm{FY}^{3}]$

whose Burnside nonnegativity does not already follow from our previous
results.

###  4.3. Koszulity

The surjection in (11) that proved a special case of Conjecture 4.7(ii) turns
out to be the $2\times 2$ special case of more general equivariant
$\ell\times\ell$ Toeplitz minor inequalities for Chow rings $A(\mathcal{M})$.
These inequalities follow from general theory of Koszul algebras, along with a
recent result of Maestroni and McCullough [MM22] showing $A(\mathcal{M})$ is
Koszul. After reviewing these results, we state a conjecture that would
generalize Proposition 4.11 and upgrade these Toeplitz minor inequalities from
the representation ring $R_{\mathbb{C}}(G)$ to the Burnside ring $B(G)$. A
good reference on Koszul algebras is Polishchuk and Positselski [PP05].

######  Definition 4.15.

Let $\mathbb{F}$ be a field, and $A$ a finitely generated standard graded
associative $\mathbb{F}$-algebra. This means $A$ is a quotient $A=T/I$ where
$T=\mathbb{F}\langle x_{1},\ldots,x_{n}\rangle$ is the free associative
algebra on $n$ noncommuting variables $x_{1},\ldots,x_{n}$, considered to all
have $\deg(x_{i})=1$, and $I$ is a homogeneous two-sided ideal in $T$.

Writing222Apologies to the reader that we are writing $A_{k}$ here rather than
$A^{k}$ as we did for the Chow rings. $A=\bigoplus_{k=0}^{\infty}A_{k}$, let
$A_{+}\mathrel{\mathop{\mathchar 58\relax}}=\bigoplus_{k=1}^{\infty}A_{k}$ be
the maximal graded two-sided ideal of $A$. Regard the field $\mathbb{F}$ as an
$A$-module via the quotient surjection $A\twoheadrightarrow
A/A_{+}\cong\mathbb{F}$. In other words, each $x_{i}$ acts as $0$ on
$\mathbb{F}$.

Say that $A$ is a Koszul algebra if the above surjection
$A\twoheadrightarrow\mathbb{F}$ extends to a linear graded free $A$-resolution
of $\mathbb{F}$ as an $A$-module, meaning that the $i^{th}$ resolvent
$F_{i}=A(-i)^{\beta_{i}}$ for some $\beta_{i}\geq 0$:

$0\leftarrow\mathbb{F}\leftarrow A\leftarrow A(-1)^{\beta_{1}}\leftarrow
A(-2)^{\beta_{2}}\leftarrow A(-3)^{\beta_{3}}\leftarrow\cdots$

There are several equivalent ways to say when $A$ is Koszul, such as requiring
that the polynomial grading of $\mathrm{Tor}^{A}_{i}(\mathbb{F},\mathbb{F})$
is concentrated in degree $i$. Equivalently, this means that if one starts
with the bar complex $\mathcal{B}_{A}$ as an $A$-resolution of $\mathbb{F}$,
and then tensors over $A$ with $\mathbb{F}$, one obtains a complex
$\mathbb{F}\otimes_{A}\mathcal{B}_{A}$ of graded $\mathbb{F}$-vector spaces
whose $i^{th}$ homology is concentrated in degree $i$. The latter
characterization leads to the following result of Polishchuk and Positselski.

######  Theorem 4.16.

[PP05, Chap. 2, Prop. 8.3] For any Koszul algebra $A$, and any composition
$(\alpha_{1},\ldots,\alpha_{r})$ of $m=\sum_{i}\alpha_{i}$, there exists a
subcomplex $(C_{*},d)$ of $\mathbb{F}\otimes_{A}\mathcal{B}_{A}$ of the form
$0\rightarrow C_{\ell}\rightarrow C_{\ell-1}\rightarrow\cdots\rightarrow
C_{1}\rightarrow 0$ starting with $C_{\ell}=A_{\alpha_{1}}\otimes\cdots\otimes
A_{\alpha_{\ell}}$ at left, ending with
$C_{1}=A_{\alpha_{1}+\cdots+\alpha_{\ell}}=A_{m}$ at right, and with $i^{th}$
term

$C_{i}=\bigoplus_{\beta}A_{\beta_{1}}\otimes\cdots\otimes A_{\beta_{i}}$

where $\beta$ in the direct sum runs over all compositions with $i$ parts that
coarsen $\alpha$. This complex $(C_{*},d)$ is exact except at the left end
$C_{\ell}$, meaning that this complex is exact:

(13) $0\rightarrow\ker(d_{\ell})\rightarrow C_{\ell}\rightarrow
C_{\ell-1}\rightarrow\cdots\rightarrow C_{1}\rightarrow 0$

The complex $(C_{*},d)$ is also $G$-equivariant for any group $G$ of graded
$\mathbb{F}$-algebra automorphisms of $A$.

Taking the alternating sum of the Euler characteristics term-by-term in (13)
yields the following, where here we conflate a $\mathbb{F}G$-module $A_{k}$
with its character $\chi_{A_{k}}$.

######  Corollary 4.17.

In the above setting, the character of the $\mathbb{F}G$-module
$\ker(d_{\ell}\mathrel{\mathop{\mathchar 58\relax}}C_{\ell}\rightarrow
C_{\ell-1})$ has this expression

$\displaystyle\chi_{\ker(d_{\ell})}=\sum_{i=1}^{\ell}(-1)^{\ell-i}\chi_{C_{i}}$
$\displaystyle=\sum_{i=1}^{\ell}(-1)^{\ell-i}\sum_{\begin{subarray}{c}\beta\colon\ell(\beta)=i\\\
\beta\>{\rm coarsens}\>\alpha\end{subarray}}A_{\beta_{1}}\otimes\cdots\otimes
A_{\beta_{i}}$
$\displaystyle=\det\left[\begin{matrix}A_{\alpha_{1}}&A_{\alpha_{1}+\alpha_{2}}&A_{\alpha_{1}+\alpha_{2}+\alpha_{3}}&\cdots&A_{m}\\\
A_{0}&A_{\alpha_{2}}&A_{\alpha_{2}+\alpha_{3}}&\cdots&A_{m-\alpha_{1}}\\\
0&A_{0}&A_{\alpha_{3}}&\cdots&A_{m-(\alpha_{1}+\alpha_{2})}\\\ 0&0&&&\vdots\\\
\vdots&\vdots&&&A_{\alpha_{\ell-1}+\alpha_{\ell}}\\\
0&0&\cdots&A_{0}&A_{\alpha_{\ell}}\end{matrix}\right]$

as an $\ell\times\ell$ Toeplitz matrix minor for the sequence of
$\mathbb{F}G$-modules $(A_{0},A_{1},A_{2},\ldots)$. In particular, when
$\mathbb{F}=\mathbb{C}$, then all Toeplitz minors of this form are genuine
characters in $R_{\mathbb{C}}(G)$.

######  Example 4.18.

When $\ell=2$ so that $\alpha=(j,k)$, the exact sequence (13) looks like

$0\rightarrow\ker(d_{2})\rightarrow A_{j}\otimes A_{k}\rightarrow
A_{j+k}\rightarrow 0$

giving this character identity

$\det\left[\begin{matrix}A_{j}&A_{j+k}\\\
A_{0}&A_{k}\end{matrix}\right]=\chi_{\ker(d_{2})}\quad(\geq_{R_{\mathbb{C}}(G)}0\text{
if }\mathbb{F}=\mathbb{C}).$

When $\ell=3$ so that $\alpha=(a,b,c)$ , the exact sequence (13) looks like

$0\rightarrow\ker(d_{3})\rightarrow A_{a}\otimes A_{b}\otimes
A_{c}\rightarrow\begin{matrix}A_{a+b}\otimes A_{c}\\\ \oplus\\\ A_{a}\otimes
A_{b+c}\end{matrix}\rightarrow A_{a+b+c}\rightarrow 0$

giving this character identity

$\det\left[\begin{matrix}A_{a}&A_{a+b}&A_{a+b+c}\\\ A_{0}&A_{b}&A_{b+c}\\\
0&A_{0}&A_{c}\end{matrix}\right]=\chi_{\ker(d_{3})}\quad(\geq_{R_{\mathbb{C}}(G)}0\text{
if }\mathbb{F}=\mathbb{C}).$

When $\ell=4$ so that $\alpha=(a,b,c,d)$, the exact sequence (13) looks like

$0\rightarrow\ker(d_{4})\rightarrow A_{a}\otimes A_{b}\otimes A_{c}\otimes
A_{d}\rightarrow\begin{matrix}A_{a+b}\otimes A_{c}\otimes A_{d}\\\ \oplus\\\
A_{a}\otimes A_{b+c}\otimes A_{d}\\\ \oplus\\\ A_{a}\otimes A_{b}\otimes
A_{c+d}\end{matrix}\rightarrow\begin{matrix}A_{a+b+c}\otimes A_{d}\\\
\oplus\\\ A_{a+b}\otimes A_{c+d}\\\ \oplus\\\ A_{a}\otimes
A_{b+c+d}\end{matrix}\rightarrow A_{a+b+c+d}\rightarrow 0$

giving this character identity

$\det\left[\begin{matrix}A_{a}&A_{a+b}&A_{a+b+c}&A_{a+b+c+d}\\\
A_{0}&A_{b}&A_{b+c}&A_{b+c+d}\\\ 0&A_{0}&A_{c}&A_{c+d}\\\
0&0&A_{0}&A_{d}\end{matrix}\right]=\chi_{\ker(d_{4})}\quad(\geq_{R_{\mathbb{C}}(G)}0\text{
if }\mathbb{F}=\mathbb{C}).$

One can now apply this to the case of Chow rings of matroids, via this result.

######  Theorem 4.19.

(Maestroni and McCullough [MM22]) For any matroid $\mathcal{M}$, the Chow ring
$A(\mathcal{M})$ is Koszul.

This gives the following promised generalization of (11).

######  Corollary 4.20.

For a matroid $\mathcal{M}$ of rank $r+1$ with Chow ring
$A(\mathcal{M})=\bigoplus_{k=0}^{r}A_{k}$, and any composition
$\alpha=(\alpha_{1},\ldots,\alpha_{\ell})$ with $m\mathrel{\mathop{\mathchar
58\relax}}=\sum_{i}\alpha\leq r$, the $\ell\times\ell$ Toeplitz minor
determinant as shown in in Corollary 4.17 is a genuine character in
$R_{\mathbb{C}}(G)$ for $G=\operatorname{\mathrm{Aut}}(\mathcal{M})$.

Here is the conjectural lift of the previous corollary to Burnside rings,
whose $2\times 2$-case is Proposition 4.11.

######  Conjecture 4.21.

In the same context as Corollary 4.20, the analogous Toeplitz minors of
$G$-sets have

$\det\left[\begin{matrix}[\mathrm{FY}^{\alpha_{1}}]&[\mathrm{FY}^{\alpha_{1}+\alpha_{2}}]&[\mathrm{FY}^{\alpha_{1}+\alpha_{2}+\alpha_{3}}]&\cdots&[\mathrm{FY}^{m}]\\\
[\mathrm{FY}^{0}]&[\mathrm{FY}^{\alpha_{2}}]&[\mathrm{FY}^{\alpha_{2}+\alpha_{3}}]&\cdots&[\mathrm{FY}^{m-\alpha_{1}}]\\\
0&[\mathrm{FY}^{0}]&[\mathrm{FY}^{\alpha_{3}}]&\cdots&[\mathrm{FY}^{m-(\alpha_{1}+\alpha_{2})}]\\\
0&0&&&\vdots\\\
\vdots&\vdots&&&[\mathrm{FY}^{\alpha_{\ell-1}+\alpha_{\ell}}]\\\
0&0&\cdots&[\mathrm{FY}^{0}]&[\mathrm{FY}^{\alpha_{\ell}}]\end{matrix}\right]\geq_{B(G)}0.$

As a bit of further evidence for Conjecture 4.21, we check it for the Toeplitz
minors with $\alpha=(1,1,1)$.

######  Theorem 4.22.

For any matroid $\mathcal{M}$, the Chow ring $A(\mathcal{M})$ has

$\displaystyle\det\left|\begin{matrix}[\mathrm{FY}^{1}]&[\mathrm{FY}^{2}]&[\mathrm{FY}^{3}]\\\
[\mathrm{FY}^{0}]&[\mathrm{FY}^{1}]&[\mathrm{FY}^{2}]\\\
0&[\mathrm{FY}^{0}]&[\mathrm{FY}^{1}]\end{matrix}\right|\geq_{B(G)}0.$

###### Proof.

Multiplying out the determinant, one needs to prove the following inequality
in $B(G)$:

$[\mathrm{FY}^{1}\times\mathrm{FY}^{1}\times\mathrm{FY}^{1}]-\left(\begin{matrix}[\mathrm{FY}^{2}\times\mathrm{FY}^{1}]\\\
+\\\
[\mathrm{FY}^{1}\times\mathrm{FY}^{2}]\end{matrix}\right)+[\mathrm{FY}^{3}]\,\,\geq_{B(G)}0,$

or equivalently, one must show the inequality

$[\quad(\mathrm{FY}^{2}\times\mathrm{FY}^{1})\,\,\sqcup\,\,(\mathrm{FY}^{1}\times\mathrm{FY}^{2})\quad]\,\,\leq_{B(G)}\,\,[\quad(\mathrm{FY}^{1}\times\mathrm{FY}^{1}\times\mathrm{FY}^{1})\,\,\sqcup\,\,\mathrm{FY}^{3}\quad].$

For this, it suffices to provide an injective $G$-equivariant map

$(\mathrm{FY}^{1}\times\mathrm{FY}^{2})\sqcup(\mathrm{FY}^{2}\times\mathrm{FY}^{1})\,\,\hookrightarrow\,\,(\mathrm{FY}^{1}\times\mathrm{FY}^{1}\times\mathrm{FY}^{1})\sqcup\mathrm{FY}^{3}.$

Such a map is summarized schematically in Figures 1 and 2, with certain
abbreviation conventions: the variables $x,y,z$ always abbreviate the
variables $x_{F_{1}},x_{F_{2}},x_{F_{3}}$ for a generic nested flag of flats
$F_{1}\subset F_{2}\subset F_{3}$, while the variable $w$ abbreviates $x_{F}$
for a flat $F$ incomparable to any of $F_{1},F_{2},F_{3}$.

The Figures 1 and 2 describe for each type of element in
$\mathrm{FY}^{1}\times\mathrm{FY}^{2}$ and
$\mathrm{FY}^{2}\times\mathrm{FY}^{1}$ an appropriate image in either
$\mathrm{FY}^{1}\times\mathrm{FY}^{1}\times\mathrm{FY}^{1}$ or
$\mathrm{FY}^{3}$. Loosely speaking, the maps try to send elements to
$\mathrm{FY}^{3}$ whenever possible, that is, whenever their product is a
valid element of $\mathrm{FY}^{3}$. When this fails, we find an image in
$\mathrm{FY}^{1}\times\mathrm{FY}^{1}\times\mathrm{FY}^{1}$, carefully trying
to keep track of which images have been used by noting various conditions on
the $x,y,z,$ and $w$, involving their ranks and sometimes their coranks,
denoted $\operatorname{\mathrm{cork}}(F)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{\mathrm{rk}}(E)-\operatorname{\mathrm{rk}}(F)$.
Conditions in gray are forced by the form of the given tuple, while conditions
in black are assumed to separate the map into disjoint cases.∎

${{\mathrm{FY}^{1}\times\mathrm{FY}^{1}\times\mathrm{FY}^{1}}}$${{\mathrm{FY}^{1}\times\mathrm{FY}^{2}}}$${{\mathrm{FY}^{3}}}$${{(x,y,z)}}$${{(x,yz)}}$${xyz}$${{(w,x,y)}}$${{(w,xy)}}$${{(x,y,x)}}$${{(x,xy)}}$${x^{2}y}$${{(y,x,y)}}$${{(y,xy)}}$${xy^{2}}$${{(x,y,E)}}$${{(x,y^{2})}}$${{(x,E,y)}}$${{(y,x,x)}}$${{(y,x^{2})}}$${{(w,x,x)}}$${{(w,x^{2})}}$${{(E,x,y)}}$${{(E,xy)}}$${xyE}$${{(x,E,y)}}$${{(x,yE)}}$${{(y,x,E)}}$${{(y,xE)}}$${{(E,w,x)}}$${{(w,xE)}}$${{(x,x,x)}}$${{(x,x^{2})}}$${x^{4}}$${{(x,x,E)}}$${{(x,xE)}}$${xE^{2}}$${{(x,E,E)}}$${{(x,E^{2})}}$${{(E,x,x)}}$${{(E,x^{2})}}$${x^{2}E}$${{(E,E^{2})}}$${E^{3}}$$\scriptstyle{d(x,y)=1}$$\scriptstyle{d(x,y)\geq
2}$$\scriptstyle{\operatorname{\mathrm{rk}}(x)=2}$$\scriptstyle{\operatorname{\mathrm{rk}}(x)\geq
3}$$\scriptstyle{d(x,y)=2}$$\scriptstyle{d(x,y)\geq
3}$$\scriptstyle{\operatorname{\mathrm{cork}}(y)\geq
2}$$\scriptstyle{\operatorname{\mathrm{cork}}(y)=1}$$\scriptstyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\operatorname{\mathrm{rk}}(x)\geq
3}$$\scriptstyle{\operatorname{\mathrm{cork}}(y)=1}$$\scriptstyle{\operatorname{\mathrm{cork}}(y)\geq
2}$$\scriptstyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\operatorname{\mathrm{cork}}(y)\geq
2}$$\scriptstyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\operatorname{\mathrm{cork}}(x)\geq
2}}$$\scriptstyle{\operatorname{\mathrm{rk}}(x)=3}$$\scriptstyle{\operatorname{\mathrm{rk}}(x)\geq
4}$$\scriptstyle{\operatorname{\mathrm{cork}}(x)=2}$$\scriptstyle{\operatorname{\mathrm{cork}}(x)\geq
3}$$\scriptstyle{\operatorname{\mathrm{cork}}(x)=1}$$\scriptstyle{\operatorname{\mathrm{cork}}(x)=2}$
Figure 1. Part of the injective set map
$\mathrm{FY}^{1}\times\mathrm{FY}^{2}\hookrightarrow\mathrm{FY}^{1}\times\mathrm{FY}^{1}\times\mathrm{FY}^{1}\cup\mathrm{FY}^{3}$
${{\mathrm{FY}^{1}\times\mathrm{FY}^{1}\times\mathrm{FY}^{1}}}$${{\mathrm{FY}^{2}\times\mathrm{FY}^{1}}}$${{\mathrm{FY}^{3}}}$${{(y,z,x)}}$${{(yz,x)}}$${{(x,y,w)}}$${{(xy,w)}}$${{(x,y,x)}}$${{(xy,x)}}$${{(x,x,y)}}$${{(y,x,y)}}$${{(xy,y)}}$${{(x,y,y)}}$${{(y,y,x)}}$${{(y^{2},x)}}$${{(x,x,y)}}$${{(x^{2},y)}}$${{(x,x,w)}}$${{(x^{2},w)}}$${{(E,y,x)}}$${{(xy,E)}}$${{(x,y,E)}}$${{(y,E,x)}}$${{(yE,x)}}$${{(E,x,y)}}$${{(xE,y)}}$${{(E,x,w)}}$${{(xE,w)}}$${{(x,x,x)}}$${{(x^{2},x)}}$${{(E,x,E)}}$${{(x,x,E)}}$${{(xE,x)}}$${{(x,E,x)}}$${{(E,E,x)}}$${{(E^{2},x)}}$${{(x,x,E)}}$${{(x^{2},E)}}$${{(E,x,x)}}$${{(E,E,E)}}$${{(E^{2},E)}}$$\scriptstyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{d(y,z)\geq
2}}$$\scriptstyle{\operatorname{\mathrm{rk}}(x)\geq
3}$$\scriptstyle{\operatorname{\mathrm{rk}}(x)=2}$$\scriptstyle{d(x,y)\geq
3}$$\scriptstyle{d(x,y)=2}$$\scriptstyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\operatorname{\mathrm{rk}}(y)\geq
3}$$\scriptstyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\operatorname{\mathrm{rk}}(x)\geq
3}$$\scriptstyle{\operatorname{\mathrm{cork}}(y)\geq
2}$$\scriptstyle{\operatorname{\mathrm{cork}}(y)=1}$$\scriptstyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\operatorname{\mathrm{cork}}(y)\geq
2}}$$\scriptstyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\operatorname{\mathrm{cork}}(x)\geq
2}}$$\scriptstyle{\operatorname{\mathrm{rk}}(x)\geq
4}$$\scriptstyle{\operatorname{\mathrm{rk}}(x)=3}$$\scriptstyle{\operatorname{\mathrm{cork}}(x)\geq
3}$$\scriptstyle{\operatorname{\mathrm{cork}}(x)=2}$$\scriptstyle{\operatorname{\mathrm{cork}}(x)=1}$$\scriptstyle{\operatorname{\mathrm{cork}}(x)\geq
2}$ Figure 2. The rest of the injective set map
$\mathrm{FY}^{2}\times\mathrm{FY}^{1}\hookrightarrow\mathrm{FY}^{1}\times\mathrm{FY}^{1}\times\mathrm{FY}^{1}\cup\mathrm{FY}^{3}$

##  5\. Further questions and conjectures

In addition to Conjectures 4.7, 4.10, 4.21 above, we collect here are a few
more questions and conjectures.

###  5.1. Other building sets with symmetry?

Here we have focused on the Chow ring of a matroid $\mathcal{M}$ using its
maximal building set. However, Feichtner and Yuzvinsky [FY04] give a
presentation for their Chow ring with respect to any building set. Their
result [FY04, Thm. 2] also provide a Gröbner basis for the ideal presenting
the Chow ring that again has the pleasant properties of Theorem 2.8 and
Corollary 2.9: whenever the building set is stable under some subgroup $G$ of
$\operatorname{\mathrm{Aut}}(\mathcal{M})$, the initial ideal for their
Gröbner basis is $G$-stable, as is the standard monomial basis for the Chow
ring. One relevant example of such a building set is the minimal building set,
which is stable under the full automorphism group
$\operatorname{\mathrm{Aut}}(\mathcal{M})$, and which arises, for example, in
the study of the moduli space $\overline{M}_{0,n}$ of genus $0$ curves with
$n$ marked points; see, e.g., Dotsenko [Dot22], Gibney and Maclagan [GM10],
and Keel [Kee92].

Furthermore, the Chow ring of $\mathcal{M}$ with respect to any building set
still satisfies the Kähler package. This follows333The authors thank Chris Eur
for pointing them to this result. combining the results of [AHK18] for the
maximal building set, and a theorem of Ardila, Denham and Huh [ADH23, Thm.
1.6] asserting that having the Kähler package depends only on the support of
the Bergman fan of $\mathcal{M}$, not on how it is subdivided according to the
building set. By the same arguments as in Corollaries 2.14, 2.15, the
equivariant versions of Poincaré duality and the Hard Lefschetz theorem will
also hold. This raises the following question.

######  Question 5.1.

Does the analogue of Theorem 1.1 hold for the Chow ring of a matroid
$\mathcal{M}$ with respect to any $G$-stable building set? In particular, what
about the minimal building set?

###  5.2. Explicit formulas for Chow rings as permutation representations?

In [Ste92, Lem. 3.1], Stembridge provides a generating function for the
symmetric group representations on each graded component of the Chow ring for
all Boolean matroids; see also Liao [Lia22]. Furthermore, Stembridge’s
expression exhibits them as permutation representations, whose orbit-
stabilizers are all Young subgroups in the symmetric group.

######  Question 5.2.

Can one provide such explicit expressions as permutation representations for
other families of matroids with symmetry?

###  5.3. Equivariant $\gamma$-positivity?

Hilbert functions $(a_{0},a_{1},\ldots,a_{r})$ for Chow rings of rank $r+1$
matroids are not only symmetric and unimodal, but satisfy the stronger
condition of $\gamma$-positivity: one has nonnegativity for all coefficients
$\gamma=(\gamma_{0},\gamma_{1},\ldots,\gamma_{\lfloor\frac{r}{2}\rfloor})$
appearing in the unique expansion

(14)
$\sum_{i=0}^{r}a_{i}t^{i}=\sum_{i=0}^{\lfloor\frac{r}{2}\rfloor}\gamma_{i}\,\,t^{i}(1+t)^{r-2i}.$

See Athanasiadis [Ath18] for a nice survey on $\gamma$-positivity. It has been
shown, independently by Ferroni, Matherne, Schröter and Vecchi [Fer+23, Thm.
3.25] and by Wang (see [Fer+23, p. 29]), that the $\gamma$-positivity for
Hilbert series of Chow rings of matroids follows from results of Braden, Huh,
Matherne, Proudfoot and Wang [Bra+22] on semismall decompositions. One also
has the notion of equivariant $\gamma$-positivity for a sequence of
$G$-representations $(A_{0},A_{1},\ldots,A_{r})$, due originally to Shareshian
and Wachs [SW20, §5] (see also [Ath18, §5.2], [Fer+23, Def. 4.13]): upon
replacing each $a_{i}$ in (14) with the element $[A_{i}]$ of
$R_{\mathbb{C}}(G)$, one asks that the uniquely defined coefficients
$\gamma_{i}$ in $R_{\mathbb{C}}(G)$ have
$\gamma_{i}\geq_{R_{\mathbb{C}}(G)}0$. Computations suggests the following.

######  Conjecture 5.3.

For any matroid $\mathcal{M}$ of rank $r+1$ and its Chow ring
$A(\mathcal{M})=\bigoplus_{i}A^{i}$, the sequence of $G$-representations
$(A^{0}_{\mathbb{C}},A^{1}_{\mathbb{C}},\ldots,A^{r}_{\mathbb{C}})$ is
equivariantly $\gamma$-positive.

For example, [SW20, Cor. 5.4] verifies Conjecture 5.3 for Boolean matroids.
However, one can check that the stronger conjecture of Burnside
$\gamma$-nonnegativity for
$(\mathrm{FY}^{0},\mathrm{FY}^{1},\ldots,\mathrm{FY}^{r})$ would fail already
for the Boolean matroid of rank $3$: here $\mathrm{FY}^{0},\mathrm{FY}^{2}$
carry the trivial $\mathfrak{S}_{3}$ permutation representation $\mathbf{1}$ ,
while $\mathrm{FY}^{1}$ carries the defining $\mathfrak{S}_{3}$-permutation
representation on the set $X=\\{1,2,3\\}$, so $\gamma_{0}=[\mathbf{1}]$, but
$\gamma_{1}=[X]-[\mathbf{1}]\not\geq_{B(\mathfrak{S}_{3})}0$.

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