Kazhdan-Lusztig Polynomials for 321-Hexagon-Avoiding Permutations

  • Authors:

  • Sara C. Billey;Gregory S. Warrington

  • Affiliations:

  • Department of Mathematics, 2-363c, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. billey@math.mit.edu;Department of Mathematics, Harvard University, Cambridge, MA 02138, USA. gwar@math.harvard.edu

  • Venue:
  • Journal of Algebraic Combinatorics: An International Journal
  • Year:
  • 2001

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Abstract

In (Deodhar, iGeom. Dedicata, 36(1) (1990), 95–119), Deodhar proposes a combinatorial framework for determining the Kazhdan-Lusztig polynomials iPx,w in the case where iW is any Coxeter group. We explicitly describe the combinatorics in the case where W=\hbox{\ca}_n (the symmetric group on in letters) and the permutation iw is 321-hexagon-avoiding. Our formula can be expressed in terms of a simple statistic on all subexpressions of any fixed reduced expression for iw. 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 iw is (1+q)^{l(w)} if and only if iw is 321-hexagon-avoiding. We also give a sufficient condition for the Schubert variety iXw to have a small resolution. We conclude with a simple method for completely determining the singular locus of iXw when iw is 321-hexagon-avoiding. The results extend easily to those Weyl groups whose Coxeter graphs have no branch points (iBn, iF4, iG2).