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# Invariant Theory of Abelian Transvection Groups

Published:2010-05-11
Printed: Sep 2010
• Abraham Broer,
Département de mathématiques et de statistique, Université de Montréal, Montréal, QC, Canada
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## Abstract

Let $G$ be a finite group acting linearly on the vector space $V$ over a field of arbitrary characteristic. The action is called \emph{coregular} if the invariant ring is generated by algebraically independent homogeneous invariants, and the \emph{direct summand property} holds if there is a surjective $k[V]^G$-linear map $\pi\colon k[V]\to k[V]^G$. The following Chevalley--Shephard--Todd type theorem is proved. Suppose $G$ is abelian. Then the action is coregular if and only if $G$ is generated by pseudo-reflections and the direct summand property holds.
 MSC Classifications: 13A50 - Actions of groups on commutative rings; invariant theory [See also 14L24]

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