Journal of Algebraic Combinatorics

, Volume 11, Issue 3, pp 187-196

First online:

Explicit Formulae for Some Kazhdan-Lusztig Polynomials

  • Francesco BrentiAffiliated withDipartimento di Matematica, Universitá di Roma “Tor Vergata”
  • , Rodica SimionAffiliated withDepartment of Mathematics, The George Washington University

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We consider the Kazhdan-Lusztig polynomials P u,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 P u,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) (where e denotes the identity permutation), as a q-analogue of the Fibonacci number. This establishes a conjecture due to M. Haiman. Second, we obtain an explicit expression for Pe, 3 4 ... (n − 2) n (n − 1) 1 2(q). Our proofs rely on the recurrence relation satisfied by the Kazhdan-Lusztig polynomials when the indexing permutations are of the form under consideration, and on the fact that these classes of permutations lend themselves to the use of induction. We present several conjectures regarding the expression for P u,v (q) under hypotheses similar to those of the main results.

Kazhdan-Lusztig polynomial q-Fibonacci number Bruhat order