Journal of Algebraic Combinatorics

, Volume 13, Issue 2, pp 111–136

Kazhdan-Lusztig Polynomials for 321-Hexagon-Avoiding Permutations

Authors

  • Sara C. Billey
  • Gregory S. Warrington
Article

DOI: 10.1023/A:1011279130416

Cite this article as:
Billey, S.C. & Warrington, G.S. Journal of Algebraic Combinatorics (2001) 13: 111. doi:10.1023/A:1011279130416

Abstract

In (Deodhar, Geom. Dedicata, 36(1) (1990), 95–119), Deodhar proposes a combinatorial framework for determining the Kazhdan-Lusztig polynomials Px,w in the case where W is any Coxeter group. We explicitly describe the combinatorics in the case where \(W = \mathfrak{S}_n \) (the symmetric group on n letters) and the permutation w is 321-hexagon-avoiding. Our formula can be expressed in terms of a simple statistic on all subexpressions of any fixed reduced expression for w. As a consequence of our results on Kazhdan-Lusztig polynomials, we show that the Poincaré polynomial of the intersection cohomology of the Schubert variety corresponding to w is (1+q)l(w) if and only if w is 321-hexagon-avoiding. We also give a sufficient condition for the Schubert variety Xw to have a small resolution. We conclude with a simple method for completely determining the singular locus of Xw when w is 321-hexagon-avoiding. The results extend easily to those Weyl groups whose Coxeter graphs have no branch points (BCn, F4, G2).

321-hexagon-avoidingKazhdan-Lusztig polynomialsSchubert varietiessingular locusdefect graph

Copyright information

© Kluwer Academic Publishers 2001