The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias et al. (2016) [4]. Let Um,d denote the uniform rank d on set m+d elements. Gedeon (2017) [7] pointed out that they can derive an explicit formula polynomials using equivariant polynomials. In this paper we give alternative formula, which allows us to prove real-rootedness for 2?m?15 and all d's. case m=1 previously proved [8]. ...

We prove the relative hard Lefschetz theorem for Soergel bimodules. It follows that structure constants of Kazhdan–Lusztig basis are unimodal. explain why implies tensor category associated by Lusztig to any two-sided cell in a Coxeter group is rigid and pivotal.

We consider the Kazhdan-Lusztig polynomials Pu,v(q) indexed by permutations u, v having particular forms with regard to their monotonicity patterns. The main results are the following. First we obtain a simplified recurrence relation satisfied by Pu,v(q) when the maximum value of v ∈ Sn occurs in position n − 2 or n − 1. As a corollary we obtain the explicit expression for Pe,3 4 ... n 1 2(q) (...