A New Degree Bound for Vector Invariants of Symmetric Groups

Let R be a commutative ring, V a nitely generated free R-module and G leGL R (V) a nite group acting naturally on the graded symmetric algebra A = S(V). Let (V; G) denote the minimal number m, such that the ring A G of inva riants can be generated by nitely many elements of degree at most m. For G = n and V (n; k), the k-fold direct sum of the natural permutat ion module, one knows that (V (n; k); n) n, provided that n! is invertible in R. This was used by E.Noether to prove (V; G) jGj if jGj! 2 R. In this paper we prove (V (n; k); n) maxfn; k(n ? 1)g for arbitrary com-mutative rings R and show equality for n = p s a prime power and R = Zor any ring with n 1 R = 0. Our results imply (V; G) maxfjGj; rank(V)(jGj ? 1)g for any ring with jGj 2 R .