Specht Modules and Branching Rules for Ariki-koike Algebras

Specht modules for an Ariki-Koike algebra Hm have been investigated recently in the context of cellular algebras (see, e.g., [GL] and [DJM]). Thus, these modules are defined as quotient modules of certain “permutation” modules, that is, defined as “cell modules” via cellular bases. So cellular bases play a decisive rôle in these work. However, the classical theory [C] or the work in the case when m = 1, 2 (i.e., the case for type A and B Hecke algebras) suggest that a construction as submodules without using cellular bases should exist. Following our previous work [DR], 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 given in [JK] and [DJ] for type A. The second part of the paper moves on looking for modular branching rules for Specht and irreducible modules over an Ariki-Koike algebra. These rules for symmetric groups were recently established by Kleshchev [K], and were generalized to Hecke algebras of type A by Brundan [B]. We shall restrict to the case where Hm has a semi-simple bottom in the sense of [DR]. This is because the classification of irreducible modules is known in this case. Our Specht module theory and the Morita equivalence theorem established in [DR] are the main ingredients in this generalization. We point out that this realization as submodules is actually very important in the Specht/Young module theory for Ariki-Koike algebras and their associated endomorphism algebras. See [CPS] for more details. We emphasise that our method throughout the work is independent of the use of cellular bases. Moreover, in the proofs of the main results (2.2), (3.6), (4.2) and (4.10), one will see how the relevant level 1 results (i.e., the results for type A Hecke algebras) are “lifted” to the corresponding level m results.