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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

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Cyclotomic $q$-Schur algebras associated to the Ariki-Koike algebra
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by Toshiaki Shoji and Kentaro Wada PDF
Represent. Theory 14 (2010), 379-416 Request permission

Abstract:

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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Additional 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: https://doi.org/10.1090/S1088-4165-10-00375-4
  • MathSciNet review: 2644457