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.