Decomposition numbers for Rouquier blocks of Ariki–Koike algebras I

Abstract

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