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 $𝕜$. A well-known theorem of Göbel asserts that the corresponding ring of invariants $𝕜{[V]}^G$ is generated by the invariants of degree at most $\left(\genfrac{}{}{0.0pt}{}{dimV}{2}\right)$. In this paper, we show that if the characteristic of $𝕜$ is zero, then the top degree of vector coinvariants $𝕜{[V^m]}_G$ is also bounded above by $\left(\genfrac{}{}{0.0pt}{}{\mathrm{\text{dim}}V}{2}\right)$, which implies the degree bound $\left(\genfrac{}{}{0.0pt}{}{dimV}{2}\right)+1$ for the ring of vector invariants $𝕜{[V^m]}^G$. So, Göbel’s bound almost holds for vector invariants in characteristic zero as well.