Equivariant inverse Kazhdan--Lusztig polynomials of thagomizer matroids

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.