Equivariant gamma-positivity of matroid Chow rings

Abstract

In this paper, we prove that the Chow ring and augmented Chow ring of a matroid are equivariantly $\gamma$-positive under the action of any group of automorphisms. Our approach provides an explicit combinatorial interpretation of the coefficients in the equivariant $\gamma$-expansion, which is new even in the non-equivariant setting. This result confirms a conjecture of Angarone, Nathanson, and Reiner, and extends the author's previous work on the positivity of equivariant Charney--Davis quantities for matroids. Specializing our formulas to uniform matroids, we obtain representation-theoretic interpretations that extend the Schur-$\gamma$-positivity results of Shareshian and Wachs for Eulerian and binomial Eulerian quasisymmetric functions. Finally, we address a problem posed by Athanasiadis by giving a combinatorial interpretation of a $(p,q)$-analog of the $\gamma$-expansion of the binomial Eulerian polynomial.