Leading Coefficients of Kazhdan--Lusztig Polynomials in Type D

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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 Pₓ,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 correspondence 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.