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

2024/06/28 by Hsin-Chieh Liao, Liao, Hsin-Chieh
Computer Science · Decision Sciences · Mathematics · #05B35 #05E05 #05E14 #05E18 #Advanced Algebra and Logic #Combinatorics (math.CO) #FOS: Mathematics #Fuzzy and Soft Set Theory #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2406.19660

openalex publication_date 2024/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

We study the Hilbert series and the representations of \mathfrakS_n and GL_n(\mathbbF_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 \mathfrakS_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.

Related

• The Chow and augmented Chow polynomials of uniform matroids
• Chow polynomials of rank-uniform labeled posets
• Equivariant gamma-positivity of matroid Chow rings
• Straightening laws for Chow rings of matroids
• Chow polynomials of uniform matroids are real-rooted