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Equivariant Kazhdan--Lusztig Polynomials of Thagomizer Matroids with a Hyperoctahedral Group Action

Matthew H. Y. Xie, Philip B. Zhang, Michael X. X. Zhong

TL;DR

The paper addresses the problem of computing equivariant Kazhdan–Lusztig and inverse Kazhdan–Lusztig polynomials for thagomizer matroids under the full hyperoctahedral action. It introduces a palindromic $Z$-polynomial strategy to reduce to the known $\mathfrak{S}_n$-equivariant theory for cycle-graph matroids and then applies Proudfoot’s Kazhdan–Lusztig–Stanley inversion to obtain the inverse polynomial. The authors prove that each coefficient is an honest $\mathcal{B}_n$-representation with multiplicity-free irreducible decompositions, and they provide explicit, wreath-product-Frobenius-characteristic expressions for the coefficients. Passing to dimensions recovers the classical nonequivariant polynomials, and the results unify type-B representation-theoretic data with the cycle-graph KL framework. The paper also formulates induced log-concavity conjectures for the coefficient sequences, suggesting positivity phenomena in the equivariant setting.

Abstract

The thagomizer matroid, realized as the graphic matroid of the complete tripartite graph $K_{1,1,n}$, has full automorphism group isomorphic to the hyperoctahedral group whenever $n \ge 2$. In the equivariant setting for this action, we compute both the Kazhdan--Lusztig polynomial and the inverse Kazhdan--Lusztig polynomial in the sense of Proudfoot's Kazhdan--Lusztig--Stanley theory, and we show that each coefficient is an honest representation with a multiplicity-free irreducible decomposition. Our main idea is to exploit the palindromicity of the equivariant $Z$-polynomial, reducing the computation to the already established symmetric-group equivariant Kazhdan--Lusztig theory for the graphic matroids of cycle graphs, and then to apply Proudfoot's equivariant Kazhdan--Lusztig--Stanley inversion identity to obtain the inverse polynomial. Passing to dimensions recovers the previously known nonequivariant thagomizer polynomials, while the coefficient formulas admit a natural expression in terms of the wreath product Frobenius characteristic for the hyperoctahedral group.

Equivariant Kazhdan--Lusztig Polynomials of Thagomizer Matroids with a Hyperoctahedral Group Action

TL;DR

The paper addresses the problem of computing equivariant Kazhdan–Lusztig and inverse Kazhdan–Lusztig polynomials for thagomizer matroids under the full hyperoctahedral action. It introduces a palindromic -polynomial strategy to reduce to the known -equivariant theory for cycle-graph matroids and then applies Proudfoot’s Kazhdan–Lusztig–Stanley inversion to obtain the inverse polynomial. The authors prove that each coefficient is an honest -representation with multiplicity-free irreducible decompositions, and they provide explicit, wreath-product-Frobenius-characteristic expressions for the coefficients. Passing to dimensions recovers the classical nonequivariant polynomials, and the results unify type-B representation-theoretic data with the cycle-graph KL framework. The paper also formulates induced log-concavity conjectures for the coefficient sequences, suggesting positivity phenomena in the equivariant setting.

Abstract

The thagomizer matroid, realized as the graphic matroid of the complete tripartite graph , has full automorphism group isomorphic to the hyperoctahedral group whenever . In the equivariant setting for this action, we compute both the Kazhdan--Lusztig polynomial and the inverse Kazhdan--Lusztig polynomial in the sense of Proudfoot's Kazhdan--Lusztig--Stanley theory, and we show that each coefficient is an honest representation with a multiplicity-free irreducible decomposition. Our main idea is to exploit the palindromicity of the equivariant -polynomial, reducing the computation to the already established symmetric-group equivariant Kazhdan--Lusztig theory for the graphic matroids of cycle graphs, and then to apply Proudfoot's equivariant Kazhdan--Lusztig--Stanley inversion identity to obtain the inverse polynomial. Passing to dimensions recovers the previously known nonequivariant thagomizer polynomials, while the coefficient formulas admit a natural expression in terms of the wreath product Frobenius characteristic for the hyperoctahedral group.
Paper Structure (11 sections, 21 theorems, 87 equations, 1 figure)

This paper contains 11 sections, 21 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Frobenius characteristic and plethystic notation
  4. Equivariant Kazhdan--Lusztig and Z polynomials
  5. The thagomizer matroid under the Bn-action
  6. The automorphism group
  7. Flats of Tn and their Bn-orbits
  8. Polynomials and generating series of the graphic matroids of cycle graphs
  9. Proof of Theorem \ref{['thm:main']} via a Z-polynomial comparison
  10. Equivariant inverse Kazhdan--Lusztig polynomials
  11. Induced log-concavity

Key Result

Theorem 1

For all $n\ge 0$, we have in $\operatorname{grVRep}(\mathcal{B}_n)$ where $(0)=\varnothing$.

Figures (1)

  • Figure 1: A picture of $K_{1,1,n}$ (and hence of the thagomizer matroid $T_n$): the spine edge is $e_*=AB$, and the $i$th spike is the pair of edges $\{a_i,b_i\}$ joining the vertex $i$ to $A$ and $B$.

Theorems & Definitions (40)

  • Theorem 1
  • Theorem 2
  • Lemma 3: haglund2008q
  • Lemma 4
  • proof
  • Definition 5: ferroni2025deletion
  • Lemma 6: gedeon2017equivariant
  • Proposition 7: xie2019thagomizer
  • Remark 8
  • Proposition 9
  • ...and 30 more