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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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On decomposition numbers with Jantzen filtration of cyclotomic $q$-Schur algebras
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by Kentaro Wada PDF
Represent. Theory 14 (2010), 417-434 Request permission

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
  • 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