A graphical description of $(D_n,A_{n-1})$ Kazhdan-Lusztig polynomials

TL;DR

This paper provides a diagrammatic approach to compute parabolic Kazhdan-Lusztig polynomials for type D Weyl groups, offering explicit formulas and categorification insights linked to representation theory and geometry.

Contribution

It introduces a simple diagrammatic method for these polynomials, yielding explicit counting formulas and categorifying certain modules, connecting combinatorics with geometric and algebraic structures.

Findings

Diagrammatic description of Kazhdan-Lusztig polynomials for type D

Explicit counting formula for morphism space dimensions

Categorification of irreducible modules

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 the morphism spaces between indecomposable projective objects in the corresponding category $\O_0^\p$. As a byproduct we categorify irreducible $W_n$-modules corresponding to pairs of one-line partitions. Finally we indicate the motivation for introducing the combinatorics by connections to Springer theory, the category of perverse sheaves on isotropic Grassmannians and to Brauer algebras which will be treated in \cite{ES}.