Symmetric product as moduli space of linear representations

TL;DR

This paper establishes a deep connection between symmetric products of affine schemes and moduli spaces of linear representations, providing explicit descriptions of invariants and equations, with extensions to positive characteristic cases.

Contribution

It demonstrates that symmetric products can be realized as quotients of representation schemes, offering explicit generators, relations, and trace-based equations, extending to positive characteristic.

Findings

Symmetric products are isomorphic to quotient schemes of representation spaces.

Explicit generators and relations for invariant rings are provided.

Trace functions describe equations of symmetric products.

Abstract

We show that the $n-$th symmetric product of an affine scheme $X=\mathrm{Spec} A$ over a characteristic zero field is isomorphic as a scheme to the quotient by the general linear group of the scheme parameterizing $n-$dimensional linear representations of $A$. As a consequence we give generators and relations of the related rings of invariants as well as the equations of any symmetric products in term of traces. In positive characteristic we prove an analogous result for the associated varieties.