Equivariant $γ$-positivity of Chow rings and augmented Chow rings of matroids

Abstract: In this paper, we prove the Chow ring and augmented Chow ring of a matroid is equivariant $\gamma$-positivity under the action of any group of automorphisms of the matroid. This verifies a conjecture of Angarone, Nathanson, and Reiner. Our method gives an explicit interpretation to the coefficients of the equivariant $\gamma$-expansion. Applying our theorem to uniform matroids, we extend and recover known results regarding the positivity of the equivariant Charney-Davis quantity of uniform matroids in the author's previous work and the Schur-$\gamma$-positivity of Eulerian and binomial Eulerian quasisymmetric functions first proved by Shareshian and Wachs. In the end, we answer a problem proposed by Athanasiadis about extending the $\gamma$-expansion of the binomial Eulerian polynomial.