Séminaire Lotharingien de Combinatoire, 84B.52 (2020), 11 pp.

Spencer Backman, Christopher Eur and Connor Simpson

Simplicial Generation of Chow Rings of Matroids

Abstract. We introduce a new presentation of the Chow ring of a matroid whose variables 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 Poincaré 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 [Adiprasito-Huh-Katz, 2018], 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 [Adiprasito-Huh-Katz, 2018]. Thus our proof does not leave the realm of matroids.

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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