Canonical basic sets in type B

TL;DR

This paper investigates the canonical basic sets for Hecke algebras of type B across all parameter choices, extending previous combinatorial descriptions to new cases using Lusztig's a-function.

Contribution

It provides a comprehensive analysis of canonical basic sets in type B Hecke algebras for all parameter values, building on prior combinatorial and theoretical frameworks.

Findings

Extended the description of canonical basic sets to all parameter choices in type B

Connected Lusztig's a-function with combinatorial parametrizations

Unified understanding of irreducible representations in Hecke algebras

Abstract

More than 10 years ago, Dipper, James and Murphy developped the theory of Specht modules for Hecke algebras of type $B\_n$. More recently, using Lusztig's a-function, Geck and Rouquier showed how to obtain parametrisations of the irreducible representations of Hecke algebras (of any finite type) in terms of so-called canonical basic sets. For certain values of the parameters in type $B\_n$, combinatorial descriptions of these basic sets were found by Jacon, based on work of Ariki and Foda-Leclerc-Okado-Thibon-Welsh. Here, we consider the canonical basic sets for all the remaining choices of the parameters.