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

Let S (Λ) be the cyclotomic q-Schur algebra associated to the ArikiKoike algebra Hn,r, introduced by Dipper-James-Mathas. In this paper, we consider v-decomposition numbers of S (Λ), 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 S (Λ), 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 Hni,ri of smaller rank. Moreover we prove a similar formula for v-decomposition numbers of Hn,r by using a Schur functor.