Kazhdan-Lusztig Polynomials for 321-Hexagon-Avoiding Permutations

Abstract

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