Generalised canonical basic sets for Ariki-Koike algebras

TL;DR

This paper classifies when Ariki-Koike algebras admit a canonical basic set based on a generalized Lusztig's a-function, describing these sets via twisted multipartitions, and clarifies the algebra's decomposition matrix structure.

Contribution

It provides a complete classification of parameter values for which Ariki-Koike algebras have canonical basic sets and describes these sets explicitly.

Findings

Identifies parameter conditions for canonical basic sets.

Describes these sets using twisted Uglov and Kleshchev multipartitions.

Shows when the decomposition matrix is unitriangular.

Abstract

Let H be a non semi-simple Ariki-Koike algebra. According to [18] and [14], there is a generalisation of Lusztig's a-function which induces a natural order (parametrised by a tuple m) on Specht modules. In some cases, Geck and Jacon have proved that this order makes unitriangular the decomposition matrix of H. The algebra H is then said to admit a "canonical basic set". We fully classify which values of m afford a canonical basic set for H and which do not. When this is the case, we describe these sets in terms of "twisted Uglov" or "twisted Kleshchev" multipartitions.