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

Abstract

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.

38 pages, 3 figures. Improved readability and fixed typos. Final version accepted in IMRN

References in corpus (7)

• Total positivity, Grassmannians, and networks
• Hilbert-Poincaré series of matroid Chow rings and intersection cohomology
• Chow rings of matroids are Koszul
• Chow functions for partially ordered sets
• Stembridge codes, permutahedral varieties, and their extensions
• Equivariant gamma-positivity of matroid Chow rings
• The Chow and augmented Chow polynomials of uniform matroids