Decomposition numbers for Rouquier blocks of Ariki-Koike algebras I

Let $\mathcal{H}$ denote an Ariki-Koike algebra over a field of characteristic $p\ge 0$. For each $r$-multipartition $\mathbf{\lambda }$ of $n$, we define a $\mathcal{H}$-module $S^{\mathbf{\lambda }}$ and for each Kleshchev $r$-multipartition $\mathbf{\mu }$ of $n$, we define an irreducible $\mathcal{H}$-module $D^{\mathbf{\mu }}$. Given a multipartition $\mathbf{\lambda }$ and a Kleshchev multipartition $\mathbf{\mu }$ both lying in a Rouquier block and which have a common multicore, we give a closed formula for the graded decomposition number $[S^{\mathbf{\lambda }}:D^{\mathbf{\mu }}{]}_v$ when $p=0$.