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

Let $\text{Sc}(\text{vL})$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra ${\text{He}}_{n,r}$, introduced by Dipper-James-Mathas. In this paper, we consider $v$-decomposition numbers of $\text{Sc}(\text{vL})$, namely decomposition numbers with respect to the Jantzen filtrations of Weyl modules. We prove, as a $v$-analogue of the result obtained by Shoji-Wada, a product formula for $v$-decomposition numbers of $\text{Sc}(\text{vL})$, 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 ${\text{He}}_{n_i,r_i}$ of smaller rank. Moreover we prove a similar formula for $v$-decomposition numbers of ${\text{He}}_{n,r}$ by using a Schur functor.