Poincaré duality algebras and rings of coinvariants
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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 $|G|\in \mathbb {F} ^{\times }$ (i.e. the order $|G|$ of $G$ is relatively prime to the characteristic of $\mathbb {F}$ ).References
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Additional Information
- 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
- Email: lintc@fcu.edu.tw
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 1599-1604
- MSC (2000): Primary 13A50; Secondary 20F55
- DOI: https://doi.org/10.1090/S0002-9939-05-08170-0
- MathSciNet review: 2204269