On a Theorem of Göbel on Permutation Invariants

Abstract

Let F be a field, let S = F[X 1,…, X n ] be a polynomial ring on variables X 1,…, X n , and let G be a group of permutations of {X 1,…, X n }. Göbel proved that for n ≥ 3 the ring of invariants S G is generated by homogeneous elements of degree at most . In this article, we obtain reductions in the set of generators introduced by Göbel and sharpen his bound for almost all permutation groups over any ground field.