  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.
