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.
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The author received support from NSF grants DMS-9983797 and DMS-0636297.
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Jones, B.C. Leading coefficients of Kazhdan–Lusztig polynomials for Deodhar elements. J Algebr Comb 29, 229–260 (2009). https://doi.org/10.1007/s10801-008-0131-6
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DOI: https://doi.org/10.1007/s10801-008-0131-6