Core blocks of Ariki–Koike algebras II: the weight of a core block

The study of the representation theory of the symmetric groups (and, more recently, the Iwahori–Hecke algebras of type A) has always been inextricably linked with the combinatorics of partitions. More recently, the complex reflection group of type G(r, 1, n) and its Hecke algebras (the Ariki–Koike algebras or cyclotomic Hecke algebras) have been studied, and it is clear that there is a similar link to algebraic combinatorics, but with multipartitions playing the rôle of partitions. This paper is intended as a contribution to the study of the combinatorics of multipartitions, as it relates to the Ariki–Koike algebra. An important manifestation of multipartition combinatorics is in the block classification for Ariki–Koike algebras. Given an Ariki–Koike algebra Hn of G(r, 1, n) and a multipartition λ of n with r components, there is an important Hn-module S λ called a Specht module. Each Specht module lies in one block of Hn and each block contains at least one Specht module, so the block classification for Hn amounts to deciding when two Specht modules lie in the same block. Graham and Lehrer [4] gave a combinatorially-defined equivalence relation ∼ on the set of multipartitions, and conjectured that Specht modules S λ and S μ lie in the same block