Moving vectors I: Representation type of blocks of Ariki-Koike algebras

We introduce a new invariant for blocks of Ariki-Koike algebras, called block moving vector, which is a vector of non-negative integers summing up to the weight of the block. In this paper, we use moving vectors to classify representation-finite blocks of Ariki-Koike algebras. As applications, we obtain examples of blocks with the same weight associated with the same multicharge that are not derived equivalent and examples of derived equivalent blocks being in different orbits under the adjoint action of the affine Weyl group. We also determine the representation type for blocks of cyclotomic $q$-Schur algebras.