Vector invariants of permutation groups in characteristic zero

Abstract

We consider a finite permutation group acting naturally on a vector space $V$ over a field $\Bbbk$. A well known theorem of G\"obel asserts that the corresponding ring of invariants $\Bbbk[V]^G$ is generated by invariants of degree at most $\binom{\dim V}{2}$. In this note we show that if the characteristic of $\Bbbk$ is zero then the top degree of vector coinvariants $\Bbbk[V^m]_G$ is also bounded above by $\binom{\dim V}{2}$, which implies the degree bound $\binom{\dim V}{2}+ 1$ for the ring of vector invariants $\Bbbk[V^m]^G$. So G\"obel's bound almost holds for vector invariants in characteristic zero as well.