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 ${\hat{I}}^3$-positive under the action of any group of automorphisms. Our approach provides an explicit combinatorial interpretation of the coefficients in the equivariant ${\hat{I}}^3$-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-${\hat{I}}^3$-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 ${\hat{I}}^3$-expansion of the binomial Eulerian polynomial.

19 pages. Main results Thm 3.4, Thm 3.8 were announced in AMS central sectional meeting on April 2024; improve exposition, add a section of new developments v.3