Equivariant inverse Kazhdan--Lusztig polynomials of thagomizer matroids

TL;DR

The paper computes explicit $\,\mathfrak{S}_n$-equivariant inverse Kazhdan–Lusztig polynomials for thagomizer matroids $T_n$ and, via generating functions, the non-equivariant polynomials. It leverages symmetric-function techniques, including the Frobenius characteristic map, plethystic substitution, and the Pieri rule, to obtain closed forms and to relate the equivariant and ordinary cases. It also proves that the inverse Kazhdan–Lusztig polynomials $Q_{T_n}(t)$ are log-concave and derives a generating-function expression for $Q_{T_n}(t)$, with a corollary for the graphic matroids of complete bipartite graphs $K_{2,n}$. The results deepen understanding of valuative and functorial properties of inverse KL polynomials in graphic matroids and provide concrete, verifiable formulas for both equivariant and ordinary settings.

Abstract

In this paper, we focus on the equivariant inverse Kazhdan--Lusztig polynomials of thagomizer matroids, a natural family of graphic matroids associated with the complete tripartite graphs $K_{1,1,n}$. These polynomials were introduced by Proudfoot as an extension of the Kazhdan--Lusztig theory for matroids. We derive closed-form expressions for the $\mathfrak{S}_n$-equivariant inverse Kazhdan--Lusztig polynomials of thagomizer matroids and present them explicitly in terms of the irreducible representations of $\mathfrak{S}_n$. As an application, we also provide explicit formulas for the non-equivariant inverse Kazhdan--Lusztig polynomials, originally defined by Gao and Xie, and give an alternative proof using generating functions. Furthermore, we prove that the inverse Kazhdan--Lusztig polynomials of thagomizer matroids are log-concave.

Figures (1)

• Figure 1: The hook set of $(1,2)$ composed of gray cells.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: haglund2008q
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• proof