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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Certain imprimitive reflection groups and their generic versions

Author: Jian-yi Shi
Journal: Trans. Amer. Math. Soc. 354 (2002), 2115-2129
MSC (2000): Primary 20F55
Published electronically: January 7, 2002
MathSciNet review: 1881032
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Abstract: The present paper is concerned with the connection between the imprimitive reflection groups $G(m,m,n)$, $m\in \mathbb{N} $, and the affine Weyl group $\widetilde {A}_{n-1}$. We show that $\widetilde {A}_{n-1}$ is a generic version of the groups $G(m,m,n)$, $m\in \mathbb{N} $. We introduce some new presentations of these groups which are shown to have some group-theoretic advantages. Then we define the Hecke algebras of these groups and of their braid versions, each in two ways according to two presentations. Finally we give a new description for the affine root system $\overline{\Phi }$ of $\widetilde {A}_{n-1}$ such that the action of $\widetilde {A}_{n-1}$ on $\overline{\Phi }$ is compatible with that of $G(m,m,n)$ on its root system in some sense.

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Jian-yi Shi
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556; Department of Mathematics, East China Normal University, Shanghai, 200062, P.R.C.

Received by editor(s): November 9, 1999
Received by editor(s) in revised form: May 24, 2001
Published electronically: January 7, 2002
Additional Notes: Supported partly by University of Notre Dame, and partly by the National Science Foundation of China, the Science Foundation of the University Doctorial Program of CNEC and the City Foundation for the Selected Academic Research of Shanghai.
Dedicated: Dedicated to Professor Cao Xi-hua on his 80th birthday
Article copyright: © Copyright 2002 American Mathematical Society