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Representations of Ariki–Koike algebras and Gröbner–Shirshov bases

Published online by Cambridge University Press:  30 June 2004

Seok-Jin Kang ,
In-Sok Lee ,
Kyu-Hwan Lee  and
Hyekyung Oh
Seok-Jin Kang
Affiliation:
School of Mathematics, Korea Institute for Advanced Study, 207-43 Cheongryangri-Dong, Dongdaemun-Gu, Seoul 130-722, Korea. E-mail: sjkang@kias.re.kr
In-Sok Lee
Affiliation:
Department of Mathematics, Seoul National University, Seoul 151-747, Korea. E-mail: islee@math.snu.ac.kr
Kyu-Hwan Lee
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, Canada. E-mail: khlee@math.toronto.edu
Hyekyung Oh
Affiliation:
Department of Mathematics, Seoul National University, Seoul 151-747, Korea. E-mail: hyekyung@math.snu.ac.kr
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Abstract

In this paper, we investigate the structure of Ariki–Koike algebras and their Specht modules using Gröbner–Shirshov basis theory and combinatorics of Young tableaux. For a multipartition $\lambda$, we find a presentation of the Specht module $S^{\lambda}$ given by generators and relations, and determine its Gröbner–Shirshov pair. As a consequence, we obtain a linear basis of $S^{\lambda}$ consisting of standard monomials with respect to the Gröbner–Shirshov pair. We show that this monomial basis can be canonically identified with the set of cozy tableaux of shape $\lambda$.

Keywords

Ariki–Koike algebrasSpecht modulesGröbner–Shirshov basistableaux16Gxx05Exx
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 89 , Issue 1 , July 2004 , pp. 54 - 70
Copyright
2004 London Mathematical Society

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Footnotes

The research of the first author was supported by KOSEF Grant # 98-0701-01-5-L and the Young Scientist Award, Korean Academy of Science and Technology.
The research of the other three authors was supported by KOSEF Grant # 98-0701-01-5-L and BK21 Mathematical Sciences Division, Seoul National University.