The Vector Invariants of Symmetric Groups

LetR be a commutative ring and let n,m be two positive integers. Let AR(n,m) := R[x11, . . . , x1m, . . . , xn1, . . . , xnm] be the polynomial ring in the commuting independent variables x11, . . . , x1m, . . . , xn1, . . . , xnm with coefficients in R. The symmetric group on n letters Sn acts on AR(n,m) by means of σ(xij) = xσ(i) j for all σ ∈ Sn and i = 1, . . . , n ; j = 1, . . . , m. Let us denote by AR(n,m) Sn the rings of invariants for this action. We give generators and relations of AR(n,m) Sn . Introduction Let R be a commutative ring and let n,m be two positive integers. Let AR(n,m) := R[x11, . . . , x1m, . . . , xn1, . . . , xnm] be the polynomial ring in the commuting independent variables x11, . . . , x1m, . . . , xn1, . . . , xnm with coefficients in R. The symmetric group on n letters Sn acts on AR(n,m) by means of σ(xij) = xσ(i) j for all σ ∈ Sn and i = 1, . . . , n ; j = 1, . . . , m. Let us denote by AR(n,m) Sn the rings of invariants for this action. Generators of AR(n,m) Sn are known when R is a characteristic zero field [3] and, when m = 1, AR(n, 1) Sn is a free polynomial ring for any ring R, [4]. Namely, let us introduce the elements en(α1,...,αm)(x1, . . . , xm) ∈ AR(n,m) Sn , with ∑m i=1 αi ≤ n, defined by n