A graphical description of (Dn,An-1) kazhdan-lusztig polynomials

Abstract

We give an easy diagrammatical description of the parabolic Kazhdan-Lusztig polynomials for the Weyl group W n of type D n with parabolic subgroup of type A n and consequently an explicit counting formula for the dimension of morphism spaces between indecomposable projective objects in the corresponding category . As a by-product we categorify irreducible W n-modules corresponding to the pairs of one-line partitions. Finally, we indicate the motivation for introducing the combinatorics by connections to the Springer theory, the category of perverse sheaves on isotropic Grassmannians, and to the Brauer algebras, which will be treated in two subsequent papers of the second author. © Glasgow Mathematical Journal Trust 2012.