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

We study the Hilbert series and the representations of ${\mathfrak{S}}_n$ and $G{L}_n({\mathbb{F}}_q)$ 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.