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

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

2007-07-12

arxiv.org/abs/0707.1753v1

Mathematics

Representation Theory

20 pages

Scientific paper

Let $\Sc(\vL)$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $\He_{n,r}$, introduced by Dipper-James-Mathas. In this paper, we consider $v$-decomposition numbers of $\Sc(\vL)$, 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 $\Sc(\vL)$, 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 $\He_{n_i,r_i}$ of smaller rank. Moreover we prove a similar formula for $v$-decomposition numbers of $\He_{n,r}$ by using a Schur functor.

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