Journal of Algebraic Combinatorics

, Volume 11, Issue 3, pp 187–196

Explicit Formulae for Some Kazhdan-Lusztig Polynomials

  • Francesco Brenti
  • Rodica Simion

DOI: 10.1023/A:1008741113381

Cite this article as:
Brenti, F. & Simion, R. Journal of Algebraic Combinatorics (2000) 11: 187. doi:10.1023/A:1008741113381


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) (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 Pu,v(q) under hypotheses similar to those of the main results.

Kazhdan-Lusztig polynomialq-Fibonacci numberBruhat order

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Francesco Brenti
    • 1
  • Rodica Simion
    • 2
  1. 1.Dipartimento di MatematicaUniversitá di Roma “Tor Vergata”RomaItaly
  2. 2.Department of MathematicsThe George Washington UniversityWashingtonU.S.A.