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Equivariant inverse $Z$-polynomials of matroids

Alice L. L. Gao, Yun Li, Matthew H. Y. Xie

TL;DR

The paper develops the theory of equivariant inverse $Z$-polynomials for matroids under finite group actions. It introduces a representation-theoretic framework and proves that the coefficients are honest representations and that these polynomials are palindromic, then provides explicit decompositions for uniform and $q$-niform matroids, along with a relaxation-based method to handle paving matroids. The authors establish equivariant unimodality and strong inductive log-concavity for the uniform and $q$-niform cases and conjecture the same for general matroids. The work connects to broader equivariant Kazhdan–Lusztig–Stanley theory and yields tools for analyzing symmetry-protected matroid invariants with potential geometric and combinatorial implications.

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.

Equivariant inverse $Z$-polynomials of matroids

TL;DR

The paper develops the theory of equivariant inverse -polynomials for matroids under finite group actions. It introduces a representation-theoretic framework and proves that the coefficients are honest representations and that these polynomials are palindromic, then provides explicit decompositions for uniform and -niform matroids, along with a relaxation-based method to handle paving matroids. The authors establish equivariant unimodality and strong inductive log-concavity for the uniform and -niform cases and conjecture the same for general matroids. The work connects to broader equivariant Kazhdan–Lusztig–Stanley theory and yields tools for analyzing symmetry-protected matroid invariants with potential geometric and combinatorial implications.

Abstract

Motivated by the notion of the inverse -polynomial introduced by Ferroni, Matherne, Stevens, and Vecchi, we study the equivariant inverse -polynomial of a matroid equipped with a finite group. We prove that the coefficients of the equivariant inverse -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 -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 -niform matroids. Motivated by the properties of equivariant -polynomials, we conjecture that the coefficients of the equivariant inverse -polynomials are equivariantly unimodal and strongly equivariantly log-concave.
Paper Structure (7 sections, 19 theorems, 116 equations)

This paper contains 7 sections, 19 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Equivariant inverse Z-polynomials
  3. Uniform and q-niform matroids
  4. Equivariant inverse Z-polynomials of uniform matroids
  5. Equivariant unimodality and strongly induced log-concavity
  6. Q-niform matroids
  7. Paving matroids

Key Result

Theorem 1.1

For the equivariant uniform matroid $\mathfrak{S}_n \curvearrowright \mathrm{U}_{k,n}$ with $n\geq k \geq 1$, where $V_{\lambda}$ denotes the irreducible representation of $\mathfrak{S}_{|\lambda|}$ indexed by $\lambda$, and we set $V_{\lambda} = 0$ if $\lambda$ is not a valid partition.

Theorems & Definitions (38)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Proposition 2.1
  • proof
  • ...and 28 more