Matrix Units and Generic Degrees for the Ariki–koike Algebras

The cyclotomic Hecke algebras were introduced by Ariki and Koike [2,4] and Broué and Malle [7]. It is conjectured [7] that these algebras play a rôle in the representation theory of reductive groups similar to (but more complicated than) that played by the Iwahori–Hecke algebras (see, for example, [8]). In particular, it should be possible to use these algebras to compute the degrees (and more generally characters) of certain representations of reductive groups; more precisely, we can attach a polynomial to each irreducible representation H , called its generic degree, and appropriate specialisations of this polynomial should compute dimensions of corresponding irreducible representations of the finite groups of Lie type. The purpose of this paper is to compute the generic degrees for the cyclotomic Hecke algebras of type G(r, 1, n); the generic degrees of the Ariki–Koike algebras have also been computed independently by Geck, Iancu and Malle [15]. Further results of Malle [19] and Malle and the author [21] give the generic degrees for all of the cyclotomic Hecke algebras corresponding to imprimitive complex reflection groups. Malle [20] has recently computed the generic degrees for cyclotomic algebras for the primitive complex reflection groups (modulo the assumption that the corresponding cyclotomic Hecke algebras are symmetric), so this completes the calculation of the generic degrees of the cyclotomic Hecke algebras associated with complex reflection groups. Two important special cases of the Ariki–Koike algebras are the Iwahori–Hecke algebras of types A and B; the generic degrees of these algebras are well–known and were first computed by Hoefsmit [16]. Later Murphy [25] gave an easier derivation of Hoefsmit’s formulae for the generic degrees of the Iwahori–Hecke algebras of type A using different, but related, techniques. This article is largely inspired by Murphy’s paper; however, with hindsight we are able to take quite a few shortcuts. Along the way we give a new, and quite elegant, treatment of the representation theory of the semisimple Ariki–Koike algebras. In particular, we explicitly construct the primitive idempotents and the matrix units in the Wedderburn decomposition of H . One of the nice features of our approach is that we use the modular theory (more accurately, the cellular theory) to understand the semisimple case. Another advantage of our approach is that our calculation of the generic degrees is almost entirely representation theoretic and it is only near of the end of the paper that we