The Le Bruyn-Procesi theorem following Lusztig

TL;DR

This paper provides a new proof of Lusztig's generalization of the Le Bruyn-Procesi theorem, describing the generators and relations of invariant rings for quiver representations with trivial group actions.

Contribution

It offers a simplified proof of Lusztig's theorem and explicitly determines relations among algebra generators for quivers with relations.

Findings

New proof of Lusztig's theorem on invariant rings

Explicit description of relations among generators

Applicable to quivers with relations

Abstract

For any quiver $Q$ and dimension vector $v$, Le Bruyn-Procesi proved that the invariant ring for the action of the change of basis group on the space of representations $\text{Rep}(Q,v)$ is generated by the traces of matrix products associated to cycles in the quiver. Lusztig generalised this to allow for vertices where the group acts trivially. Here we provide a simple new proof of Lusztig's theorem and determine the relations between his algebra generators for any quiver with relations.