Moving vectors and core blocks of Ariki-Koike algebras

We classify the core blocks of Ariki-Koike algebras by their moving vectors. Using this classification, we obtain a necessary and sufficient condition for Scopes equivalence between two core blocks, and express the number of simple modules lying in a core block as a classical Kostka number. Under certain conditions on the multicharge and moving vector, we further relate the graded decomposition numbers of these blocks in characteristic zero with the graded decomposition numbers of the Iwahori-Hecke algebras of type $A$.