Cyclotomic -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 | References | Similar Articles | Additional Information

Abstract: Let be the Ariki-Koike algebra associated to the complex reflection group , and let be the cyclotomic -Schur algebra associated to , introduced by Dipper, James and Mathas. For each such that , we define a subalgebra of and its quotient algebra . It is shown that is a standardly based algebra and is a cellular algebra. By making use of these algebras, we prove a product formula for decomposition numbers of , which asserts that certain decomposition numbers are expressed as a product of decomposition numbers for various cyclotomic -Schur algebras associated to Ariki-Koike algebras of smaller rank. This is a generalization of the result of N. Sawada. We also define a modified Ariki-Koike algebra of type , and prove the Schur-Weyl duality between and .

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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.