Matroid analogues of Gal's conjecture

Abstract

Well-known conjectures of Charney--Davis, Gal, and Nevo--Petersen predict increasingly strong positivity phenomena for the h-vectors of flag simplicial spheres. In this paper, we formulate and prove matroid analogues of these conjectures in the setting of Chow polynomials of matroids with building sets. Our proofs rely on toric geometry and make crucial use of tropical intersection theory. We begin by introducing complete building sets, a class encompassing all maximal building sets and other important families such as minimal building sets of braid matroids. For matroids with complete building sets, we analyze the Chow rings of the associated toric varieties, and prove that their Hilbert--Poincar\'e polynomials are gamma-positive. From this analysis, we derive a combinatorial formula for the coefficients of the gamma-expansion, and use it to explicitly construct a simplicial complex $\Gamma$, whose f-vector coincides with the gamma-vector. This establishes a matroid analogue of the Nevo--Petersen conjecture. When the building set is maximal, we further prove that $\Gamma$ is balanced, confirming the strongest such analogue in this case. As an application, we obtain a new combinatorial formula for the gamma-expansion of the Poincar\'e polynomial of the Deligne--Mumford--Knudsen compactification $\overline{\mathcal{M}}_{0,n}$, and derive several novel numerical inequalities for its coefficients. We also study the toric varieties of matroids with flag building sets, another class containing maximal building sets as well as several other prominent families. We prove that the Hilbert--Poincar\'e polynomials of these toric varieties are gamma-positive. This result establishes matroid analogues of the Charney--Davis and Gal conjectures, and simultaneously extends several recent gamma-positivity results for Chow polynomials.