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