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

TL;DR

This paper introduces the concept of maximal rank type for quiver representations, shows real root representations possess this property, and uses it to construct all such representations for a specific wild quiver with three vertices.

Contribution

It defines maximal rank type for quiver representations and applies it to explicitly construct all real root representations of a particular wild quiver.

Findings

Real root representations are of maximal rank type.

All real root representations of the specific wild quiver are constructed.

Real root representations are shown to be tree modules.

Abstract

We introduce the notion of ''maximal rank type'' for representations of quivers, which requires certain collections of maps involved in the representation to be of maximal rank. We show that real root representations of quivers are of maximal rank type. By using the maximal rank type property and universal extension functors we construct all real root representations of a particular wild quiver with three vertices. From this construction it follows that real root representations of this quiver are tree modules. Moreover, formulae given by Ringel can be applied to compute the dimension of the endomorphism ring of a given real root representation.