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

We study the natural representations of Sₙ and GLₙ (Fq) 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.