Simplicial generation of Chow rings of matroids

Spencer Backman, Christopher Eur, Connor Simpson

Published 2019-05-17Version 1

We introduce a new presentation for the Chow ring of a matroid with far-reaching geometric and combinatorial implications that include recovering a central portion of the Hodge theory of matroids developed by Adiprasito, Huh, and Katz in \cite{AHK18}. The generators of this presentation, which we call the simplicial presentation, are nef divisors that correspond to principal truncations of the matroid. Furthermore, in the simplicial presentation the Chow ring admits a monomial basis that corresponds to a set of matroids that we call relative nested quotients, which can be viewed as a relative analogue of Schubert matroids. This combinatorial property of the simplicial presentation allows us to prove directly, independently of \cite{AHK18}, that the Chow ring of a matroid satisfies Poincar\'e duality. Moreover, we give a combinatorial formula for the volume polynomial of the simplicial presentation, generalizing Postnikov's formula for the volumes of generalized permutohedra, and show that the volume polynomial is strongly log-concave in its positive orthant. As a consequence, we obtain a more direct proof of the portion of the Hodge theory of matroids in \cite{AHK18} that implies the Rota-Heron-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 previously employed in \cite{AHK18}. Thus our proof does not leave the realm of matroids.