Tree normal forms for quiver representations

TL;DR

This paper introduces new methods for constructing normal forms of indecomposable quiver representations, combining homological and geometric tools to classify and parametrize these representations, especially for tree modules and certain roots.

Contribution

It develops recursive homological techniques and geometric approaches to explicitly construct and classify indecomposable quiver representations, including normal forms for all indecomposables of certain roots.

Findings

Constructed families of indecomposables as deformations of tree modules.

Developed geometric methods using torus actions on moduli spaces.

Provided normal forms for all indecomposables of specific roots.

Abstract

We explore methods for constructing normal forms of indecomposable quiver representations. The first part of the paper develops homological tools for recursively constructing families of indecomposable representations from indecomposables of smaller dimension vector. This is then specialized to the situation of tree modules, where the existence of a special basis simplifies computations and gives nicer normal forms. Motivated by a conjecture of Kac, we use this to construct cells of indecomposable representations as deformations of tree modules. The second part of the paper develops geometric tools for constructing cells of indecomposable representations from torus actions on moduli spaces of representations. As an application we combine these methods and construct families of indecomposables - grouped into affine spaces - which actually gives a normal form for all indecomposables of certain roots.