Induced log-concavity of equivariant matroid invariants

Abstract

Inspired by the notion of equivariant log-concavity, we introduce the concept of induced log-concavity for a sequence of representations of a finite group. In this paper we prove the induced log-concavity of the equivariant Kazhdan–Lusztig polynomials of $q$-niform matroids equipped with the action of a finite general linear group, as well as that of the equivariant Kazhdan–Lusztig polynomials of uniform matroids equipped with the action of a symmetric group. As a consequence of the former, we obtain the log-concavity of Kazhdan–Lusztig polynomials of $q$-niform matroids, in support of Elias, Proudfoot, and Wakefield’s log-concavity conjecture on the matroid Kazhdan–Lusztig polynomials. We also establish the induced log-concavity of the equivariant characteristic polynomials and the equivariant inverse Kazhdan–Lusztig polynomials for $q$-niform matroids and uniform matroids.