An elementary discussion of the representation and geometric invariant theory of equioriented quivers of type D with an application to quiver bundles

We first review a constructive proof of Gabriel's theorem on the representation finiteness of Dynkin quivers for the case of an equioriented quiver of type D. Then, we proceed to the analysis of polystable representations of such a quiver, give a direct proof of the characterisation of its semistable representations, and explain how we may recover a theorem of Abeasis and Koike from this characterisation . Finally, we illustrate how these results may be applied to the study of moduli spaces of quiver bundles.

Automated summary

This article reviews a constructive proof of Gabriel's theorem on the representation finiteness of Dynkin quivers, specifically focusing on the case of an equioriented quiver of type D. It then delves into the analysis of polystable representations of such a quiver, provides a direct proof of the characterisation of its semistable representations, and explains how a theorem of Abeasis and Koike can be derived from this characterisation. Finally, it demonstrates how these results can be applied to the study of moduli spaces of quiver bundles.