Vector invariants of permutation groups in characteristic zero

We consider a finite permutation group acting naturally on a vector space V over a field k. A well-known theorem of G¨obel asserts that the corresponding ring of invariants k[V ] G is generated by the invariants of degree at most dim V 2 ´ . 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 dim V 2 ´ , which implies the degree bound `dim V 2 ´ + 1 for the ring of vector invariants k[V m] G. So, G¨obel’s bound almost holds for vector invariants in characteristic zero as well.