A new degree bound for vector invariants of symmetric groups

Abstract:

Let $R$ be a commutative ring, $V$ a finitely generated free $R$-module and $G\le GL_R(V)$ a finite group acting naturally on the graded symmetric algebra $A=S(V)$. Let $\beta (V,G)$ denote the minimal number $m$, such that the ring $A^G$ of invariants can be generated by finitely many elements of degree at most $m$.

For $G=\Sigma _n$ and $V(n,k)$, the $k$-fold direct sum of the natural permutation module, one knows that $\beta (V(n,k),\Sigma _n) \le n$, provided that $n!$ is invertible in $R$. This was used by E. Noether to prove $\beta (V,G) \le |G|$ if $|G|! \in R^*$.

In this paper we prove $\beta (V(n,k),\Sigma _n) \le max\{n,k(n-1)\}$ for arbitrary commutative rings $R$ and show equality for $n=p^s$ a prime power and $R = \mathbb {Z}$ or any ring with $n\cdot 1_R=0$. Our results imply \begin{equation*} \beta (V,G)\le max\{|G|, \operatorname {rank}(V)(|G|-1)\}\end{equation*} for any ring with $|G| \in R^*$.