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Cyclotomic $ q$-Schur algebras associated to the Ariki-Koike algebra


Authors: Toshiaki Shoji and Kentaro Wada
Journal: Represent. Theory 14 (2010), 379-416
MSC (2010): Primary 20C08, 20G43
DOI: https://doi.org/10.1090/S1088-4165-10-00375-4
Published electronically: May 6, 2010
MathSciNet review: 2644457
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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

DOI: https://doi.org/10.1090/S1088-4165-10-00375-4
Received by editor(s): November 1, 2007
Received by editor(s) in revised form: February 6, 2010
Published electronically: May 6, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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