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

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- by Kentaro Wada
- Represent. Theory
**14**(2010), 417-434 - DOI: https://doi.org/10.1090/S1088-4165-2010-00376-3
- Published electronically: May 18, 2010
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## Abstract:

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.## References

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## Bibliographic Information

**Kentaro Wada**- Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan
- Address at time of publication: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
- Email: wada@kurims.kyoto-u.ac.jp
- Received by editor(s): November 6, 2007
- Published electronically: May 18, 2010
- © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**14**(2010), 417-434 - MSC (2010): Primary 20-XX, 16-XX
- DOI: https://doi.org/10.1090/S1088-4165-2010-00376-3
- MathSciNet review: 2652073