Chow rings and augmented Chow rings of uniform matroids and their $q$-analogs

TL;DR

This paper investigates the algebraic and representation-theoretic properties of Chow rings of uniform and q-uniform matroids, providing formulas, nonnegativity results, and combinatorial interpretations for their Hilbert series and group actions.

Contribution

It introduces explicit formulas for the Hilbert series and group representations of Chow rings of uniform and q-uniform matroids, including new combinatorial interpretations and nonnegativity results.

Findings

Computed Frobenius series for uniform and q-uniform matroids.

Established nonnegativity of the equivariant Charney--Davis quantity.

Provided combinatorial interpretations of Hilbert series and representations.

Abstract

We study the Hilbert series and the representations of $\mathfrak{S}_n$ and $GL_n(\mathbb{F}_q)$ on the (augmented) Chow rings of uniform matroids $U_{r,n}$ and $q$-uniform matroids $U_{r,n}(q)$. The Frobenius series for uniform matroids and their $q$-analogs are computed. As a byproduct, we recover Hameister, Rao, and Simpson's formula for the 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 the equivariant Charney--Davis quantity of the (augmented) Chow ring of a matroid is nonnegative (i.e., a genuine representation of a group of automorphisms of the matroid). When the matroid is a uniform matroid and the group is $\mathfrak{S}_n$, the representation either vanishes or is a Foulkes representation (i.e., a Specht module of a ribbon shape). Specializing 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.