On $G$-invariant Gorenstein ideals

Abstract

Let $k$ be a field and $G\subseteq G{l}_n(k)$ be a finite group with $|G{|}^{-1}\in k$. Let $G$ act linearly on $A=k[X_1,\dots ,X_n]$ and let $A^G$ be the ring of invariant's. Suppose there does not exist any non-trivial one-dimensional representation of $G$ over $k$. Then we show that if $Q$ is a $G$-invariant homogeneous ideal of $A$ such that $A/Q$ is a Gorenstein ring then $A^G/Q^G$ is also a Gorenstein ring.