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

Author: Kentaro Wada
Journal: Represent. Theory 14 (2010), 417-434
MSC (2010): Primary 20-XX, 16-XX
Published electronically: May 18, 2010
MathSciNet review: 2652073
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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.

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Additional 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

Received by editor(s): November 6, 2007
Published electronically: May 18, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.