The equivariant inverse Kazhdan-Lusztig polynomials of uniform matroids

TL;DR

This paper explores the equivariant inverse Kazhdan-Lusztig polynomials of uniform matroids, providing explicit formulas and demonstrating their utility in calculating equivariant Kazhdan-Lusztig polynomials, with applications to known formulas.

Contribution

It introduces the equivariant inverse Kazhdan-Lusztig polynomial for matroids and derives explicit formulas for Boolean and uniform matroids, offering new proofs and formulas for related polynomials.

Findings

Explicit formulas for equivariant inverse Kazhdan-Lusztig polynomials of uniform matroids

New proof of existing formulas for equivariant Kazhdan-Lusztig polynomials

A novel combinatorial formula for equivariant Kazhdan-Lusztig polynomials

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.