Applications of multisymmetric syzygies in invariant theory

TL;DR

This paper explores multisymmetric syzygies to describe the structure of symmetric powers of algebras, providing new generators and relations, and applies these results to invariant theory and classification problems.

Contribution

It introduces a presentation of symmetric powers using generators and relations, and applies this to invariant theory, including specific cases like the algebra of invariants of the symmetric group.

Findings

Presented generators and relations for the nth symmetric power of a commutative algebra.

Revealed the isomorphism between the algebra and coordinate rings of representation schemes in characteristic zero.

Derived generators and relations for invariants of the symmetric group in a specific case.

Abstract

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.