Leading coefficients of Kazhdan---Lusztig polynomials for Deodhar elements

Authors:
Brant C. Jones
Affiliations:
Department of Mathematics, University of California, Davis, USA 95616
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
2009

Citing 5
Cited 1

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The Enumeration of Fully Commutative Elements of Coxeter Groups

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Kazhdan-Lusztig Polynomials for 321-Hexagon-Avoiding Permutations

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On rank functions for heaps

Journal of Combinatorial Theory Series A
Acyclic Heaps of Pieces, I

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Leading coefficients of Kazhdan---Lusztig polynomials and fully commutative elements

Journal of Algebraic Combinatorics: An International Journal

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Abstract

We show that the leading coefficient of the Kazhdan---Lusztig polynomial P x,w (q) known as μ(x,w) is always either 0 or 1 when w is a Deodhar element of a finite Weyl group. The Deodhar elements have previously been characterized using pattern avoidance in Billey and Warrington (J. Algebraic Combin. 13(2):111---136, [2001]) and Billey and Jones (Ann. Comb. [2008], to appear). In type A, these elements are precisely the 321-hexagon avoiding permutations. Using Deodhar's algorithm (Deodhar in Geom. Dedicata 63(1):95---119, [1990]), we provide some combinatorial criteria to determine when μ(x,w)=1 for such permutations w.