Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations

Abstract

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.

24 pages, 18 figures, AMS-LaTeX

Cited by in corpus (5)

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• A combinatorial version of Sylvester's four-point problem
• Conjectures about certain parabolic Kazhdan--Lusztig polynomials
• Combinatorial results on (1,2,1,2)-avoiding $GL(p,\mathbb{C})\times GL(q,\mathbb{C})$-orbit closures on $GL(p+q,\mathbb{C})/B$
• 321-polygon-avoiding permutations and Chebyshev polynomials