Polynomring/Gruppenoperation/Klein/F-Signatur/Fakt/en

Let ${}G\subseteq \operatorname {GL} _{n}\!{\left(K\right)}$ be a small finite group with its natural action on the polynomial ring ${}K[X_{1},\ldots ,X_{n}]$. Suppose that the characteristic of ${}K$ does not divide the order of the group.

Then the ${}F$-signature of the invariant ring

${}R=K[X_{1},\ldots ,X_{n}]^{G}\,$

is ${}{\frac {1}{\operatorname {ord} {\left(G\right)}}}$