Straightening laws for Chow rings of matroids

TL;DR

The paper addresses foundational questions for Chow rings of matroids, including augmented variants, by developing elementary straightening-law techniques to obtain a standard monomial basis, Poincaré duality, and dragon-Hall-Rado-type identities. The authors construct a straightening procedure that expresses any monomial as a combination of standard monomials and use it to derive a flat-indexed direct-sum decomposition, degree maps, and duality without heavy inductive arguments. A key feature is the use of maps between Chow rings and a lower-triangular degree-matrix argument to prove linear independence and duality, together with a grading by flats and a Kohnen-type algebra with straightening laws framework. The approach yields new recursions for Hilbert series, clarifies the ASL structure underlying Chow rings, and provides alternative proofs via degeneration to Stanley-Reisner rings, with potential implications for Gröbner bases and combinatorial topology of matroids.

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.

Theorems & Definitions (55)

• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Proposition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• proof : Proof of Proposition \ref{['prop:SMspanning']}