The Le Bruyn-Procesi theorem following Lusztig

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.