Proof of Two Conjectures of Brenti and Simion on Kazhdan-Lusztig Polynomials

Abstract

We find an explicit formula for the Kazhdan-Lusztig polynomials P ui,a,v i of the symmetric group \(\mathfrak{S}\)(n) where, for a, i, n ∈ \(\mathbb{N}\) such that 1 ≤ a ≤ i ≤ n, we denote by u i,a = s a s a+1 ··· s i−1 and by v i the element of \(\mathfrak{S}\)(n) obtained by inserting n in position i in any permutation of \(\mathfrak{S}\)(n − 1) allowed to rise only in the first and in the last place. Our result implies, in particular, the validity of two conjectures of Brenti and Simion [4, Conjectures 4.2 and 4.3], and includes as a special case a result of Shapiro, Shapiro and Vainshtein [13, Theorem 1]. All the proofs are purely combinatorial and make no use of the geometry of the corresponding Schubert varieties.