Simplicial generation of Chow rings of matroids

By Spencer Backman, Christopher Eur, Connor Simpson. Journal of the European Mathematical Society (Print), 2019

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We introduce a new presentation of the Chow ring of a matroid whose variables now admit a combinatorial interpretation via the theory of matroid quotients and display a geometric behavior analogous to that of nef classes on smooth projective varieties. We apply these properties to produce a bijection between a standard monomial basis of our presentation and a relative generalization of Schubert matroids. As a corollary we obtain the Poincare duality property for Chow rings of matroids. We then give a formula for the volume polynomial with respect to our presentation and show that it is log-concave in the positive orthant. We recover the portion of the Hodge theory of matroids in [AHK18] which implies the Heron-Rota-Welsh conjecture on the log-concavity of the coefficients of the characteristic polynomial. We emphasize that our work eschews the use of flipping, which is the key technical tool employed in [AHK18]. Thus our proof does not leave the realm of matroids.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6d7b3fc4874329be15cc9f83c0fa7711a72beaa8

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Citations in the corpus

A semi-small decomposition of the Chow ring of a matroid (2020, methodology): “ctive CHpMq-module homomorphism (Propositions2.21and2.22) F M : CHpM FqbCHpMFqÝÑCHpMq; „ F 1 x 1z b „ F2 x F2ÞÝÑx F „ F x F1 „ F2 x F2; 5Independent proofs of Poincare duality for´ CHpMqwere given in [BES] and [BDF]. The authors of [BES] also prove the degree 1 Hodge–Riemann relations for CHpMq. 8 T”