Tree modules of the generalized Kronecker quiver

TL;DR

This paper studies indecomposable tree modules in the Kronecker quiver, showing their existence for all coprime dimension vectors and their role as stable fixed points, advancing understanding of quiver representations.

Contribution

It introduces a construction of indecomposable tree modules for all coprime dimension vectors using torus fixed points and reflection functors, extending known results.

Findings

Existence of indecomposable tree modules for all coprime dimension vectors

Construction of stable torus fixed points using these modules

Factor modules of fixed points include all roots

Abstract

For coprime dimension vectors certain torus fixed points of the Kronecker moduli space are indecomposable tree modules. They are indecomposable representations of the regular m-tree and can be glued in order to get stable torus fixed point for every coprime dimension vector. Using their stability and the reflection functor we show that for arbitrary roots there exist indecomposable tree modules of the Kronecker quiver as factor modules of these torus fixed points.