Vogan classes in type $B_n$

TL;DR

This paper provides a combinatorial description of Kazhdan-Lusztig left cells in type B_n Coxeter systems using Vogan classes, especially when the weight ratio is n-1 or in (n-2,n-1).

Contribution

It introduces a new combinatorial approach to describe left cells in type B_n Coxeter systems via Vogan classes, extending previous understanding.

Findings

Describes left cells for ratio n-1 using Vogan classes.

Provides partial information on cells for ratios in (n-2,n-1).

Enhances combinatorial understanding of Kazhdan-Lusztig cells.

Abstract

Consider a weighted Coxeter system $(W,S,\mathscr{L})$. Via its associated Iwahori-Hecke algebra, we may determine the partition of $W$ into Kazhdan-Lusztig cells. In this paper, we use the theory of Vogan classes introduced by Bonnaf\'e--Geck to obtain a combinatorial description of the left cells of type $B_n$ when the ratio of the weights of the first to second generator is $n-1$. We further give information on the left cells when this ratio lies in the interval $(n-2,n-1)$.