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

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

We find an explicit formula for the Kazhdan-Lusztig polynomials Pui,a ,vi of the symmetric group S(n) where, for a, i, n ∈ N such that 1 ≤ a ≤ i ≤ n, we denote by ui,a = sasa+1 · · · si−1 and by vi the element ofS(n) obtained by inserting n in position i in any permutation ofS(n −1) allowed to rise only in the first and in the last place. Our result implies, in particular, the validity of two c...

Perverse Sheaves and the Kazhdan-lusztig Conjectures

Brundan-kazhdan-lusztig and Super Duality Conjectures

We formulate a general super duality conjecture on connections between parabolic categories O of modules over Lie superalgebras and Lie algebras of type A, based on a Fock space formalism of their Kazhdan-Lusztig theories which was initiated by Brundan. We show that the Brundan-Kazhdan-Lusztig (BKL) polynomials for gl(m|n) in our parabolic setup can be identified with the usual parabolic Kazhda...

Moment graphs and Kazhdan-Lusztig polynomials

Motivated by a result of Fiebig (2007), we categorify some properties of Kazhdan-Lusztig polynomials via sheaves on Bruhat moment graphs. In order to do this, we develop new techniques and apply them to the combinatorial data encoded in these moment graphs. Résumé. Motivés par un resultat de Fiebig (2007), nous categorifions certaines propriétés des polynômes de KazhdanLusztig en utilisant fais...

Fock Space and Kazhdan-lusztig Polynomials

1. Lecture 1: Affine Lie algebras and the Fock representation of ĝln. 1.1. The loop algebra construction. Let g be a complex reductive Lie algebra and let L denote the algebra of Laurent polynomials in one variable L = C[t, t−1]. The loop algebra over g is L(g) = L ⊗ g, which is a Lie algebra with the bracket [t ⊗ x, t ⊗ y]0 = t[x, y]. (1.1) The elements of the loop algebra may be regarded as r...