Specht Modules and Branching Rules for Ariki-Koike Algebras

Abstract

Specht modules for an Ariki-Koike algebra have been investigated recently in the context of cellular algebras. Thus, these modules are defined as quotient modules of certain ``permutation'' modules, that is, defined as ``cell modules'' via cellular bases. We shall introduce in this paper Specht modules for an Ariki-Koike algebra as submodules of those ``permutation'' modules and investigate their basic properties such as Standard Basis Theorem and the ordinary Branching Theorem, generalizing several classical constructions for type $A$. The second part of the paper moves on looking for Kleshchev's branching rules for Specht and irreducible modules over an Ariki-Koike algebra. We shall restrict to the case where the Ariki-Koike algebra has a semi-simple bottom. With a recent work of Ariki on the classification of irreducible modules, we conjecture that the results should be true in general if $l$-regular multipartitions are replaced by Kleshchev's multipartitions. We point out that our approach is independent of the use of cellular bases.