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For any quiver $Q$ and dimension vector $v$, Le Bruyn-Procesi proved that theinvariant ring for the action of the change of basis group on the space ofrepresentations $\operatorname{Rep}(Q,v)$ is generated by the traces of matrixproducts associated to cycles in the quiver. We generalise this to allow forvertices where the group acts trivially, and we give relations between thegenerators of the invariant algebra for quivers with relations. As a geometricapplication, we prove for $n\geq 1$ that the Hilbert scheme of $n$-points on anADE surface singularity is isomorphic to a Nakajima quiver variety. This allowsus to generalise the well known theorem of [Fogarty] by showing that theHilbert scheme of $n$-points on a normal surface with canonical singularitiesis a normal variety of dimension $2n$. In doing so, we also provide a new proofof Fogarty's result.

The Le Bruyn-Procesi theorem and Hilbert schemes