LAUSR

Generic (canonical) decomposition of dimension vector for a quiver was introduced by Victor Kac as characterizing the generic module indecomposable summands dimensions, hence, the generic orbit. Derksen and Weyman proposed an elegant algorithm to compute that decomposition, extensively using Schofield’s results. We consider generic semi-simple decomposition, which corresponds to generic closed orbit and provide a simple and fast algorithm to compute this decomposition. Generic semi-simple decomposition has two useful application. First, it reduces the computation of generic decomposition to the case of quiver without oriented cycles in a geometric way. Second, it provides a nice novel presentation of the algebra of invariants of quiver representations as a tensor product of similar algebras for the summands.