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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

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Cyclotomic $q$-Schur algebras associated to the Ariki-Koike algebra
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by Toshiaki Shoji and Kentaro Wada
Represent. Theory 14 (2010), 379-416
Published electronically: May 6, 2010


Let $\mathcal {H}_{n,r}$ be the Ariki-Koike algebra associated to the complex reflection group $\mathfrak {S}_n\ltimes (\mathbb {Z}/r\mathbb {Z})^n$, and let $\mathcal {S}(\varLambda )$ be the cyclotomic $q$-Schur algebra associated to $\mathcal {H}_{n,r}$, introduced by Dipper, James and Mathas. For each $\mathbf {p} = (r_1, \dots , r_g) \in \mathbb {Z}_{>0}^g$ such that $r_1 +\cdots + r_g = r$, we define a subalgebra $\mathcal {S}^{\mathbf {p}}$ of $\mathcal {S}(\varLambda )$ and its quotient algebra $\overline {\mathcal {S}}^{\mathbf {p}}$. It is shown that $\mathcal {S}^{\mathbf {p}}$ is a standardly based algebra and $\overline {\mathcal {S}}^{\mathbf {p}}$ is a cellular algebra. By making use of these algebras, we prove a product formula for decomposition numbers of $\mathcal {S}(\varLambda )$, which asserts that certain decomposition numbers are expressed as a product of decomposition numbers for various cyclotomic $q$-Schur algebras associated to Ariki-Koike algebras $\mathcal {H}_{n_i,r_i}$ of smaller rank. This is a generalization of the result of N. Sawada. We also define a modified Ariki-Koike algebra $\overline {\mathcal {H}}^{\mathbf {p}}$ of type $\mathbf {p}$, and prove the Schur-Weyl duality between $\overline {\mathcal {H}}^{\mathbf {p}}$ and $\overline {\mathcal {S}}^{\mathbf {p}}$.
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Bibliographic Information
  • Toshiaki Shoji
  • Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan
  • 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 1, 2007
  • Received by editor(s) in revised form: February 6, 2010
  • Published electronically: May 6, 2010
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 14 (2010), 379-416
  • MSC (2010): Primary 20C08, 20G43
  • DOI:
  • MathSciNet review: 2644457