Automorphisms and Twisted Forms of Rings of Invariants

Let \(G\subset \textrm{GL}_n(k)\) $G\subset {\text{GL}}_n(k)$ be a finite subgroup and \(k[x_1,\dots , x_n]^G\subset k[x_1,\dots , x_n]\) $k{[x_1,\cdots ,x_n]}^G\subset k[x_1,\cdots ,x_n]$ its ring of invariants. We show that, in many cases, the automorphism group of \(k[x_1,\dots , x_n]^G\) $k{[x_1,\cdots ,x_n]}^G$ is \(k^\times \) $k^{\times }$ .