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

Let $\mathcal{H}_{n,r}$ be the Ariki-Koike algebra associated to the complex reflection group $\mathfrak{S}_n\ltimes (\mathbb{Z}/r\mathbb{Z})^n$, and let $\mathcal{S}(\varLambda)$ be the cyclotomic $q$-Schur algebra associated to $\mathcal{H}_{n,r}$, introduced by Dipper, James and Mathas. For each $\mathbf{p} = (r_1, \dots, r_g) \in \mathbb{Z}_{>0}^g$ such that $r_1 +\cdots + r_g = r$, we define a subalgebra $\mathcal{S}^{\mathbf{p}}$ of $\mathcal{S}(\varLambda)$ and its quotient algebra $\overline{\mathcal{S}}^{\mathbf{p}}$. It is shown that $\mathcal{S}^{\mathbf{p}}$ is a standardly based algebra and $\overline{\mathcal{S}}^{\mathbf{p}}$ is a cellular algebra. By making use of these algebras, we prove a product formula for decomposition numbers of $\mathcal{S}(\varLambda)$, which asserts that certain decomposition numbers are expressed as a product of decomposition numbers for various cyclotomic $q$-Schur algebras associated to Ariki-Koike algebras $\mathcal{H}_{n_i,r_i}$ of smaller rank. This is a generalization of the result of N. Sawada. We also define a modified Ariki-Koike algebra $\overline{\mathcal{H}}^{\mathbf{p}}$ of type $\mathbf{p}$, and prove the Schur-Weyl duality between $\overline{\mathcal{H}}^{\mathbf{p}}$ and $\overline{\mathcal{S}}^{\mathbf{p}}$.