Kazhdan-Lusztig Polynomials and 321-Hexagon Avoiding Permutations

Gregory Warrington

Harvard University

November 17,
refreshments at 3:45pm


The Kazhdan-Lusztig polynomials $P_{x,w}$ are associated to a pair of elements in any finite reflection group. These polynomials have important interpretations in representation theory. The $P_{x,w}$ have a simple recursive definition that has not been conducive to expression in closed form. Deodhar proposes a non-constructive combinatorial framework for determining the $P_{x,w}$. Using this framework, we explicitly describe the combinatorics of the $P_{x,w}$ in the case where our group is the symmetric group and the permutation $w$ is 321-hexagon-avoiding. Our formula can be expressed in terms of a statistic on all subexpressions of any fixed reduced word for $w$. These results give a simple criterion for the Schubert variety $X_w$ to have a small resolution. We conclude with an explicit 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 ($B_n$, $F_4$, $G_2$).

Speaker's Contact Info: gwar(at-sign)math.harvard.edu

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