The equivariant inverse Kazhdan-Lusztig polynomials of uniform matroids

Abstract

Motivated by the concepts of the inverse Kazhdan-Lusztig polynomial and the equivariant Kazhdan-Lusztig polynomial, Proudfoot defined the equivariant inverse Kazhdan-Lusztig polynomial for a matroid. In this paper, we show that the equivariant inverse Kazhdan-Lusztig polynomial of a matroid is very useful for determining its equivariant Kazhdan-Lusztig polynomials, and we determine the equivariant inverse Kazhdan-Lusztig polynomials for Boolean matroids and uniform matroids. As an application, we give a new proof of Gedeon, Proudfoot and Young's formula for the equivariant Kazhdan-Lusztig polynomials of uniform matroids. Inspired by Lee, Nasr and Radcliffe's combinatorial interpretation for the ordinary Kazhdan-Lusztig polynomials of uniform matroids, we further present a new formula for the corresponding equivariant Kazhdan-Lusztig polynomials.