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Poincaré duality algebras and rings of coinvariants

Author: Tzu-Chun Lin
Journal: Proc. Amer. Math. Soc. 134 (2006), 1599-1604
MSC (2000): Primary 13A50; Secondary 20F55
Published electronically: December 2, 2005
MathSciNet review: 2204269
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Abstract: Let $ \varrho :G\hookrightarrow GL(n, \mathbb{F})$ be a faithful representation of a finite group $ G$ over the field $ \mathbb{F}$. Via $ \varrho$ the group $ G$ acts on $ V=\mathbb{F} ^n$ and hence on the algebra $ {\mathbb{F}}[V]$ of homogenous polynomial functions on the vector space $ V$. R. Kane (1994) formulated the following result based on the work of R. Steinberg (1964): If the field $ \mathbb{F}$ has characteristic 0, then $ {\mathbb{F}}[V] _G$ is a Poincaré duality algebra if and only if $ G$ is a pseudoreflection group. The purpose of this note is to extend this result to the case $ \vert G\vert\in \mathbb{F} ^{\times}$ (i.e. the order $ \vert G\vert$ of $ G$ is relatively prime to the characteristic of $ \mathbb{F}$ ).

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Tzu-Chun Lin
Affiliation: Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstraße 3-5, D-37073 Göttingen, Germany – and – Department of Applied Mathematics, Feng Chia University, 100 Wenhwa Road, Taichung 407, Taiwan, Republic of China

Keywords: Invariant theory, pseudoreflection groups
Received by editor(s): June 23, 2003
Received by editor(s) in revised form: January 7, 2005
Published electronically: December 2, 2005
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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