, Volume 13, Issue 2, pp 111-136

Kazhdan-Lusztig Polynomials for 321-Hexagon-Avoiding Permutations

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In (Deodhar, Geom. Dedicata, 36(1) (1990), 95–119), Deodhar proposes a combinatorial framework for determining the Kazhdan-Lusztig polynomials P x , 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 X w to have a small resolution. We conclude with a simple method for completely determining the singular locus of X w when w is 321-hexagon-avoiding. The results extend easily to those Weyl groups whose Coxeter graphs have no branch points (B C n , F 4, G 2).