The Uglov map sends a multipartition (with an associated multicharge) to a partition. Using this Uglov map, I will show how one can use the e-abacus to define the e-core (which is a partition) and the e-weight (which is a non-negative integer) of a multipartition associated to a multi-e-residue.
This combinatorial definition of $e$-weight coincides with the definition first introduced by Fayers. Furthermore, two Specht modules of an Ariki-Koike algebra lie in the same block if and only if they are labelled by multipartitions with the same e-core and the same e-weight. This thus provides a characterisation of the blocks of Ariki-Koike algebras that is analogous to that of Iwahori-Hecke algebras. If time allows, I will discuss the implications of these results for Scopes's equivalences for the blocks of Ariki-Koike algebras, as well as suggest a definition of Rouquier blocks of Ariki-Koike algebras that is different from Lyle's, but is perhaps more natural.
