Kazhdan-Lusztig Polynomials for 321-Hexagon-Avoiding Permutations
نویسنده
چکیده
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 = Sn (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 (Bn , F4, G2).
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