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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

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.