#### Date of Award

Spring 1-1-2013

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Richard M. Green

#### Second Advisor

Nathaniel Thiem

#### Third Advisor

Martin Walter

#### Fourth Advisor

Stephen Doty

#### Fifth Advisor

James M. Douglass

#### Abstract

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 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.

#### Recommended Citation

Gern, Tyson Charles, "Leading Coefficients of Kazhdan–Lusztig Polynomials in Type D" (2013). *Mathematics Graduate Theses & Dissertations*. 26.

https://scholar.colorado.edu/math_gradetds/26