A remark on the constructibility of real root representations of quivers using universal extension functors

TL;DR

This paper investigates whether all real root representations of a quiver can be constructed from a real Schur representation using universal extension functors, providing a counterexample that shows this is not always possible.

Contribution

The paper presents a concrete example demonstrating that not all real root representations can be obtained via universal extension functors from a real Schur representation.

Findings

Counterexample showing limitations of universal extension functors

Not all real root representations are constructible from a real Schur representation

Highlights the need for alternative methods in representation construction

Abstract

In this paper we consider the following question: Is it possible to construct all real root representations of a given quiver Q by using universal extension functors, starting with a real Schur representation? We give a concrete example answering this question negatively.