Applications of multisymmetric syzygies in invariant theory

A presentation by generators and relations of the $n$th symmetric power $B$ of a commutative algebra $A$ over a field of characteristic zero or greater than $n$ is given. This is applied to get information on a minimal homogeneous generating system of $B$ (in the graded case). The known result that in characteristic zero the algebra $B$ is isomorphic to the coordinate ring of the scheme of $n$-dimensional representations of $A$ is also recovered. The special case when $A$ is the two-variable polynomial algebra and $n=3$ is applied to find generators and relations of an algebra of invariants of the symmetric group of degree four that was studied in connection with the problem of classifying sets of four unit vectors in the Euclidean space.