Algebraic Geometry and Combinatorics Seminar: Equivariant gamma-positivity of Chow rings of matroids

Abstract: Chow and augmented Chow rings of matroids played important roles in the settlement of the Heron-Rota-Welsh conjecture and the Dowling-Wilson top-heavy conjecture. Their Hilbert series have been extensively studied ever since. It was shown by Ferroni, Mathern, Steven, and Vecchi, and independently by Wang, that the Hilbert series of Chow rings of matroids are gamma-positive. However, no interpretation for the gamma-coefficients was known. In a recent paper, Angarone, Nathanson, and Reiner further conjectured that the Chow rings of matroids are equivariant gamma-positive under the action of groups of automorphisms of matroids. In this talk, I will present a somewhat surprising proof of this conjecture, showing that both Chow rings and augmented Chow rings of matroids are equivariant gamma-positive. Moreover, our method yields explicit interpretations for the equivariant gamma-coefficients and specializes to an interpretation of the ordinary gamma-coefficients. In the special case of uniform matroids, our results extend Shareshian and Wachs' Schur-gamma-positivity for the cohomologies of the permutahedral and the stellahedral varieties.