Georgia Institute of Technology College of Sciences

Chow rings 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 since then. 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 using inductive arguement followed from the semismall decompositions of the Chow ring of matroids. However, they do not have an interpretation for the coefficients in the $\gamma$-expansion. Recently, Angarone, Nathanson, and Reiner further conjectured that Chow rings of matroids are equivariant $\gamma$-positive under the action of groups of matroid automorphisms. In this talk, I will give a proof of this conjecture without using semismall decomposition, showing that both Chow rings and augmented Chow rings of matroids are equivariant $\gamma$-positive. Moreover, we obtain explicit descriptions for the coefficients of the equivariant $\gamma$-expansions. Then we consider the special case of uniform matroids which extends Shareshian and Wachs Schur-$\gamma$-positivity of Frobenius characteristics of the cohomologies of the permutahedral and the stellahedral varieties.