KazhdanLusztig Polynomials and 321Hexagon Avoiding Permutations
Gregory Warrington
Harvard University
November 17,
4:15pm
refreshments at 3:45pm
2338
ABSTRACT

The KazhdanLusztig 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 nonconstructive
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 321hexagonavoiding. 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
321hexagonavoiding. 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(atsign)math.harvard.edu
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