Vector invariants of permutation groups in characteristic zero

Abstract

We consider a finite permutation group acting naturally on a vector space [Formula: see text] over a field [Formula: see text]. A well-known theorem of Göbel asserts that the corresponding ring of invariants [Formula: see text] is generated by the invariants of degree at most [Formula: see text]. In this paper, we show that if the characteristic of [Formula: see text] is zero, then the top degree of vector coinvariants [Formula: see text] is also bounded above by [Formula: see text], which implies the degree bound [Formula: see text] for the ring of vector invariants [Formula: see text]. So, Göbel’s bound almost holds for vector invariants in characteristic zero as well.