The Kazhdan-Lusztig polynomials of uniform matroids

Abstract

The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias et al. (2016) [4]. Let Um,d denote the uniform matroid of rank d on a set of m+d elements. Gedeon et al. (2017) [7] pointed out that they can derive an explicit formula of the Kazhdan-Lusztig polynomials of Um,d using equivariant Kazhdan-Lusztig polynomials. In this paper we give an alternative explicit formula, which allows us to prove the real-rootedness of the Kazhdan-Lusztig polynomials of Um,d for 2≤m≤15 and all d's. The case m=1 was previously proved by Gedeon et al. (2017) [8]. We further determine the Z-polynomials of all Um,d's and prove the real-rootedness of the Z-polynomials of Um,d for 2≤m≤15 and all d's. Our formula also enables us to give an alternative proof of Gedeon, Proudfoot, and Young's formula for the Kazhdan-Lusztig polynomials of Um,d's without using the equivariant Kazhdan-Lusztig polynomials.