Specht Modules and Branching Rules for Ariki-Koike Algebras

TL;DR

This paper introduces Specht modules for Ariki-Koike algebras as submodules of permutation modules, explores their properties, and investigates branching rules, extending classical results and proposing conjectures for more general cases.

Contribution

It defines Specht modules as submodules, generalizes classical constructions, and examines branching rules without relying on cellular bases, advancing the understanding of Ariki-Koike algebras.

Findings

Standard Basis Theorem established for Specht modules

Branching Theorem extended to Ariki-Koike algebras

Conjecture on Kleshchev's branching rules for general cases

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.