The Kazhdan-Lusztig polynomials of uniform matroids

The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias et al. (2016) [4]. Let U-m,U-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 U-m,U-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 U-m,U-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 U-m,U-d's and prove the real-rootedness of the Z-polynomials of U(m,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 U-m,U-d's without using the equivariant Kazhdan-Lusztig polynomials. (C) 2020 Elsevier Inc. All rights reserved.