Vogan classes in type $B_n$

Edmund Howse

doi:10.5802/alco.74

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 Bonnafe--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 $(n-2,n-1)$.

Highlights

Lusztig has described how to partition a Coxeter group into left, right and twosided cells with respect to a weight function [20]. This is done via certain equivalence relations that are calculated in the corresponding Iwahori–Hecke algebra, and the resulting cells afford representations of both the group and the algebra
The left cells afford a complete list of irreducible representations of the corresponding Iwahori– Hecke algebra, a pair of left cells afford isomorphic representations if and only if they are contained in the same two-sided cell, and two elements of the group are in the same left cell if and only if they have the same recording tableaux under the Robinson–Schensted correspondence
We retain the notation for the Coxeter group of type Bn from § 1

Summary

Introduction

Lusztig has described how to partition a Coxeter group into left, right and twosided cells with respect to a weight function [20]. This is done via certain equivalence relations that are calculated in the corresponding Iwahori–Hecke algebra, and the resulting cells afford representations of both the group and the algebra. An important development in the study of the cells of Wn came in the form of a number of conjectures by Bonnafé, Geck, Iancu and Lam [7] These conjectures state conditions for two weight functions on Wn to be cell-equivalent, as well as a unified combinatorial description of the left, right and two-sided cells for each of these cases.

Kazhdan–Lusztig cells with unequal parameters

The Coxeter group of type Bn

A new descent set for finite Coxeter groups

An extension of the generalised τ -invariant in type Bn

Determining Ξ-orbits