Chow rings of vector space matroids

The Chow ring of a matroid (or more generally, atomic lattice) 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.

Keywords

matroid, Eulerian, lattice, Chow ring

2010 Mathematics Subject Classification

05B35

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This research was carried out as part of the 2017 summer REU programat the School of Mathematics, University of Minnesota, Twin Cities, andwas supported by NSF RTG grant DMS-1148634 and by NSF grant DMS-1351590.

Received 28 November 2018

Accepted 6 February 2020

Published 4 January 2021