Chow Rings of Vector Space Matroids

TL;DR

This paper investigates the Chow rings of uniform and vector space matroids, linking their Hilbert series to permutation statistics and providing formulas for Charney-Davis quantities, advancing understanding of matroid invariants.

Contribution

It offers explicit formulas for the Hilbert series and Charney-Davis quantities of these matroids, connecting algebraic invariants to permutation statistics and special number sequences.

Findings

Hilbert series expressed via permutation statistics

Full rank case yields maj-exc q-Eulerian polynomials

Formulas for Charney-Davis quantities using determinants and q-secant numbers

Abstract

The Chow ring of a matroid (or more generally, atomic latice) is an invariant whose importance was demonstrated by Adiprasito, Huh and Katz, who used it to resolve the long-standing Heron-Rota-Welsh conjecture. Here, we make a detailed study of the Chow rings of uniform matroids and of matroids of finite vector spaces. In particular, we express the Hilbert series of such matroids in terms of permutation statistics; in the full rank case, our formula yields the maj-exc $q$-Eulerian polynomials of Shareshian and Wachs. We also provide a formula for the Charney-Davis quantities of such matroids, which can be expressed in terms of either determinants or $q$-secant numbers.