Equivariant inverse $Z$-polynomials of matroids

TL;DR

This paper explores the properties of equivariant inverse Z-polynomials of matroids, proving their representation-theoretic nature, palindromicity, and unimodality, with explicit formulas for specific classes and conjectures for general cases.

Contribution

It introduces the equivariant inverse Z-polynomial for matroids, proves key properties, derives explicit formulas for uniform and q-uniform matroids, and conjectures unimodality and log-concavity.

Findings

Coefficients are honest representations.

Polynomials are palindromic.

Polynomials are equivariantly unimodal and log-concave.

Abstract

Motivated by the notion of the inverse $Z$-polynomial introduced by Ferroni, Matherne, Stevens, and Vecchi, we study the equivariant inverse $Z$-polynomial of a matroid equipped with a finite group. We prove that the coefficients of the equivariant inverse $Z$-polynomials are honest representations and that these polynomials are palindromic. Explicit formulas are obtained for uniform matroids equipped with the symmetric group. The corresponding formulas for $q$-niform matroids are derived using the Comparison Theorem for unipotent representations. For arbitrary equivariant paving matroids, explicit expressions are obtained by relating the polynomials of a matroid to those of its relaxation. We show that these polynomials are equivariantly unimodal and strongly inductively log-concave for both uniform and $q$-niform matroids. Motivated by the properties of equivariant $Z$-polynomials, we conjecture that the coefficients of the equivariant inverse $Z$-polynomials are equivariantly unimodal and strongly equivariantly log-concave.