Quiver representations of maximal rank type and an application to representations of a quiver with three vertices

We introduce the notion of ‘maximal rank type’ forrepresentations of quivers, which requires certain collectionsof maps involved in the representation to be of maximal rank.We show that real root representations of quivers are of maximalrank type. By using the maximal rank type property and universalextension functors we construct all real root representationsof a particular wild quiver with three vertices. From this constructionit follows that real root representations of this quiver aretree modules. Moreover, formulae given by Ringel can be appliedto compute the dimension of the endomorphism ring of a givenreal root representation.