Induced log-concavity of equivariant matroid invariants

TL;DR

This paper introduces the concept of induced log-concavity for sequences of group representations and proves this property for certain equivariant matroid invariants, providing new evidence for conjectures on matroid Kazhdan-Lusztig polynomials.

Contribution

It develops the theory of induced log-concavity for equivariant matroid invariants and proves it for Kazhdan-Lusztig polynomials of specific classes of matroids, linking representation theory and combinatorics.

Findings

Proves induced log-concavity of equivariant Kazhdan-Lusztig polynomials for $q$-niform and uniform matroids.

Establishes log-concavity of Kazhdan-Lusztig polynomials for $q$-niform and uniform matroids.

Demonstrates induced log-concavity of equivariant characteristic and inverse Kazhdan-Lusztig polynomials.

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. For an equivariant matroid equipped with a symmetric group action or a finite general linear group action, we transform the problem of proving the induced log-concavity of matroid invariants to that of proving the Schur positivity of symmetric functions. 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, thus providing further positive evidence for Elias, Proudfoot and Wakefield's log-concavity conjecture on the matroid Kazhdan-Lusztig polynomials. From the latter we obtain the log-concavity of Kazhdan-Lusztig polynomials of uniform matroids, which was recently proved by Xie and Zhang by using a computer algebra approach. 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.