Invariant Theory of Abelian Transvection Groups

http://dx.doi.org/10.4153/CMB-2010-044-6
Canad. Math. Bull. 53(2010), 404-411

  Published:2010-05-11
 Printed: Sep 2010

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.