The representation theory of the Ariki-Koike and cyclotomic q-Schur algebras

The Ariki-Koike algebras were introduced by Ariki and Koike [8] who were interested in them because they are a natural generalization of the Iwahori-Hecke algebras of types A and B. At almost the same time, Broué and Malle [21] attached to each complex reflection group a cyclotomic Hecke algebra which, they conjectured, should play a role in the decomposition of the induced cuspidal representations of the finite groups of Lie type. The Ariki-Koike algebras are a special case of Broué and Malle's construction. The deepest conjectures of Broué and Malle (and others) concerning the Ariki-Koike algebras have not yet been proved; however, many of the consequences of these conjectures have been established. Further, the representation theory of these algebras is beginning to be well understood. For example, the simple modules of the Ariki-Koike algebras have been classified; the blocks are known; there are analogues of Kleshchev's modular branching rules; and, in principle, the decomposition matrices of the Ariki-Koike algebras are known in characteristic zero. In many respects this theory looks much like that of the symmetric groups; in particular, there is a rich combinatorial mosaic underpinning these results which involves familiar objects like standard tableaux (indexed by multipartitions), Specht modules and so on. The cyclotomic Schur algebras were introduced by Dipper, James and the author [41]; by definition these algebras are endomorphism algebras of a direct sum of " permutation modules " for an Ariki-Koike algebra. This is generalizes the Dipper-James definition [39] of the q-Schur algebras as endomorphism algebras of tensor space. We were interested in these algebras both as another tool for studying the Ariki-Koike algebras and because we hoped that there might be a cyclotomic analogue of the famous Dipper-James theory [27, 39].