  We consider the representation theory of the Ariki-Koike algebra, a
$q$-deformation of the group algebra of the complex reflection group $C_r \wr
\mathfrak{S}_n$. We examine blocks of the Ariki-Koike algebra. In particular,
we prove a sufficient condition such that restriction of modules leads to a
natural correspondence between the multipartitions of $n$ whose Specht modules
belong to a block $B$ and those of $n-\delta_i(B)$ whose Specht modules belong
to the block $B'$, obtained from $B$ applying a Scopes' equivalence. This
bijection gives us an equivalence for the $v$-decomposition numbers of the
Ariki-Koike algebras.
