Decomposition numbers for Rouquier blocks of Ariki–Koike algebras I

Abstract

Let $\mathcal{H}={\mathcal{H}}_{r,n}(q,\mathbf{Q})$ denote an Ariki–Koike algebra over a field of characteristic $p\ge 0$. For each $r$-multipartition $\mathbf{\lambda }$ of $n$, there exists a $\mathcal{H}$-module $S^{\mathbf{\lambda }}$ and for each Kleshchev $r$-multipartition $\mathbf{\mu }$ of $n$, there exists 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 such that $\mathbf{\lambda }$ and $\mathbf{\mu }$ have the same multicore, we give a closed formula for the graded decomposition number $[S^{\mathbf{\lambda }}:D^{\mathbf{\mu }}{]}_v$ when $p=0$ or when each component of $\mathbf{\mu }$ has fewer than $p$ removable $e$-rim hooks.