The Le Bruyn-Procesi theorem and Hilbert schemes

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 \(\textrm{Rep}(Q,v)\) is generated by the traces of matrix products associated to cycles in the quiver. We generalise this to allow for vertices where the group acts trivially, and we give relations between the generators of the invariant algebra for quivers with relations. As a geometric application, we prove for \(n\geq 1\) that the Hilbert scheme of \(n\)-points on an ADE surface singularity is isomorphic to a Nakajima quiver variety. This allows us to generalise the well known theorem of [Fogarty] by showing that the Hilbert scheme of \(n\)-points on a normal surface with canonical singularities is a normal variety of dimension \(2n\) with canonical singularities. In addition, we show that if \(S\) has symplecic singularities over \(\mathbb{C}\), then so does the Hilbert scheme of \(n\)-points on \(S\), thereby generalising a result of Beauville.