Vector Invariants of Symmetric Groups

Abstract

Let M be a free module of rank n over a commutative ring R with unit and let Σ_n denote the symmetric group acting on a fixed basis of M in the usual way. Let M^m denote the direct sum of m copies of M and let S be the symmetric ring of M^m over R. Then each element of Σ_n acts diagonally on M^m and consequently on S; denote by Xn the subgroup of Gl(M^m) so defined. The ring of invariants, S^Σ_n, is called the ring of vector invariants by H. Weyl [ 3, Chapter II, p. 27] when R = Q. In this paper a set of generators valid over any ring R is given. This set of generators is somewhat larger than Weyl's. It is interesting to note that, over the integers, his algebra and S^Σ_n have the same Hilbert-Poincaré series, are equal after tensoring with the rationals, and have the same fraction fields, although they are not equal.