Decomposition numbers for Rouquier blocks of Ariki–Koike algebras I

Let H=Hr,n(q,Q) denote an Ariki–Koike algebra over a field of characteristic p≥ 0 . For each r -multipartition λ of n , there exists a H -module Sλ and for each Kleshchev r -multipartition μ of n , there exists an irreducible H -module Dμ . Given a multipartition λ and a Kleshchev multipartition μ both lying in a Rouquier block such that λ and μ have the same multicore, we give a closed formula for the graded decomposition number [Sλ Dμ]ᵥ when p=0 or when each component of μ has fewer than p removable e -rim hooks.