Leading Coefficients of KazhdanLusztig Polynomials in Type $D$
Abstract
KazhdanLusztig 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, $\mu(x,w)$ of a KazhdanLusztig 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 $\mu$value computation difficult. The RobinsonSchensted 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 socalled 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 $\mu$values involving bad elements. I will conclude by showing that $\mu(x,w)$ is 0 or 1 when $x$ is fully commutative in type $D$.
 Publication:

arXiv eprints
 Pub Date:
 April 2013
 arXiv:
 arXiv:1304.6074
 Bibcode:
 2013arXiv1304.6074G
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Quantum Algebra
 EPrint:
 Author's Ph.D. Thesis (2013) directed by R.M. Green at the University of Colorado Boulder. 68 pages