Equivariant inverse Kazhdan--Lusztig polynomials of thagomizer matroids

2025/10/13 by Gao, Alice L. L., Li, Yun, Xie, Matthew H. Y.
#05B35 #05E05 #20C30 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.11322

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 \mathfrakS_n-equivariant inverse Kazhdan--Lusztig polynomials of thagomizer matroids and present them explicitly in terms of the irreducible representations of \mathfrakS_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.

Citations

• Inverse Kazhdan-Lusztig polynomials of matroids under deletion
• Log-concavity of inverse Kazhdan-Lusztig polynomials of paving matroids
• Equivariant Kazhdan-Lusztig theory of paving matroids
• The equivariant inverse Kazhdan-Lusztig polynomials of uniform matroids
• Valuations and the Hopf Monoid of Generalized Permutahedra
• Singular Hodge theory for combinatorial geometries
• Equivariant incidence algebras and equivariant Kazhdan-Lusztig-Stanley theory
• Equivariant Kazhdan-Lusztig polynomials of thagomizer matroids
• The equivariant Kazhdan-Lusztig polynomial of a matroid
• The Kazhdan-Lusztig polynomial of a matroid

Related

• The equivariant inverse Kazhdan-Lusztig polynomials of uniform matroids
• Equivariant Kazhdan-Lusztig polynomials of thagomizer matroids
• Kazhdan-Lusztig polynomials of thagomizer matroids
• The equivariant Kazhdan-Lusztig polynomial of a matroid
• Log-concavity of inverse Kazhdan-Lusztig polynomials of paving matroids