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 $\mathbb{k}$. A well known theorem of Göbel asserts that the corresponding ring of invariants $\mathbb{k}[V{]}^G$ is generated by invariants of degree at most $\left(\genfrac{}{}{0ex}{}{\dim V}{2}\right)$. In this note we show that if the characteristic of $\mathbb{k}$ is zero then the top degree of vector coinvariants $\mathbb{k}[V^m{]}_G$ is also bounded above by $\left(\genfrac{}{}{0ex}{}{\dim V}{2}\right)$, which implies the degree bound $\left(\genfrac{}{}{0ex}{}{\dim V}{2}\right)+1$ for the ring of vector invariants $\mathbb{k}[V^m{]}^G$. So Göbel's bound almost holds for vector invariants in characteristic zero as well.

preliminary version