On decomposition numbers with Jantzen filtration of cyclotomic $q$-Schur algebras

Let $\mathscr{S}(\varLambda)$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $\mathscr{H}_{n,r}$, introduced by Dipper, James and Mathas. In this paper, we consider $v$-decomposition numbers of $\mathscr{S}(\varLambda)$, namely decomposition numbers with respect to the Jantzen filtrations of Weyl modules. We prove, as a $v$-analogue of the result obtained by Shoji and Wada, a product formula for $v$-decomposition numbers of $\mathscr{S}(\varLambda)$, which asserts that certain $v$-decomposition numbers are expressed as a product of $v$-decomposition numbers for various cyclotomic $q$-Schur algebras associated to Ariki-koike algebras $\mathscr{H}_{n_i,r_i}$ of smaller rank. Moreover, we prove a similar formula for $v$-decomposition numbers of $\mathscr{H}_{n,r}$ by using a Schur functor.