Cyclotomic $q$-Schur algebras associated to the Ariki-Koike algebra

Let $S$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $H_{n,r}$ of rank $n$, introduced by Dipper-James-Mathas. For each $p = (r_1, …, r_g)$suchthat$r_1 + … + r_g = r$,wedefineasubalgebra$S^p$of$S$anditsquotientalgebra$\bar S^p$\mathrm{.}Itisshownthat$S^p$ is a standardly based algebra and ${\bar{S}}^p$ is a cellular algebra. By making use of these algebras, we show that certain decomposition numbers for $S$ can be expressed as a product of decomposition numbers for cyclotomic $q$-Schur algebras associated to smaller Ariki_koike algebras $H_{n_k,r_k}$.