Symmetric product as moduli space of linear representations

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.

10 pages, from the talk given at ICM 2006 Madrid

Cited by in corpus (3)

• Azumaya-type noncommutative spaces and morphisms therefrom: Polchinski's D-branes in string theory from Grothendieck's viewpoint
• Moduli of linear representations, symmetric products and the non commutative Hilbert scheme
• Hilbert-Chow morphism for non commutative Hilbert schemes and moduli spaces of linear representations