Decomposition numbers for Rouquier blocks of Ariki-Koike algebras I

Abstract

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