Straightening laws for Chow rings of matroids

TL;DR

This paper presents elementary proofs of key theorems about Chow rings of matroids using a straightening law, offering new insights into their structure and decompositions.

Contribution

It introduces a straightening law-based approach to prove fundamental properties of Chow rings of matroids, including their basis, duality, and formulas, applicable to augmented rings.

Findings

Elementary proofs of standard monomial basis, Poincare duality, and Hall-Rado formula

Decomposition of Chow rings into flats-based pieces

Applicable to augmented Chow rings

Abstract

We give elementary and non-inductive proofs of three fundamental theorems about Chow rings of matroids: the standard monomial basis, Poincare duality, and the dragon-Hall-Rado formula. Our approach, which also works for augmented Chow rings of matroids, is based on a straightening law. This approach also gives a decomposition of the Chow ring of a matroid into pieces indexed by flats.