Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations

Details Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations

2000-05-05

arxiv.org/abs/math/0005052v1

Mathematics

Combinatorics

24 pages, 18 figures, AMS-LaTeX

Scientific paper

We give a combinatorial formula for the Kazhdan-Lusztig polynomials $P_{x,w}$ in the symmetric group when $w$ is a 321-hexagon-avoiding permutation. Our formula, which depends on a combinatorial framework developed by Deodhar, can be expressed in terms of a simple statistic on all subexpressions of any fixed reduced expression for $w$. We also show that $w$ being 321-hexagon-avoiding is equivalent to several other conditions, such as the Bott-Samelson resolution of the Schubert variety $X_w$ being small. We conclude with a simple method for completely determining the singular locus of $X_w$ when $w$ is 321-hexagon-avoiding.

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