Leading Coefficients of Kazhdan–Lusztig Polynomials in Type D

Leading Coefficients of Kazhdan–Lusztig Polynomials in Type D

Kazhdan–Lusztig polynomials arise in the context of Hecke algebras associated to Coxeter groups. The computation of these polynomials is very difficult for examples of even moderate rank. In type A it is known that the leading coefficient, μ(x, w) of a Kazhdan–Lusztig polynomial P_(x,w) is either 0 or 1 when x is fully commutative and w is arbitrary. In type D Coxeter groups there are certain “bad” elements that make μ-value computation difficult.
The Robinson–Schensted correspondence between the symmetric group and pairs of standard Young tableaux gives rise to a way to compute cells of Coxeter groups of type A. A lesser known corre- spondence exists for signed permutations and pairs of so-called domino tableaux, which allows us to compute cells in Coxeter groups of types B and D. I will use this correspondence in type D to compute μ-values involving bad elements. I will conclude by showing that μ(x, w) is 0 or 1 when x is fully commutative in type D.

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Tyson Gern

March 21, 2013
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