A Gabriel-type theorem for cluster tilting

TL;DR

This paper generalizes Gabriel's theorem to cluster tilting theory, linking indecomposable modules' dimension vectors to roots of Euler forms, and explores properties of these roots and related quiver transformations.

Contribution

It introduces the concept of cluster-roots, generalizes BGP reflection functors, and describes quivers with relations for n-APR tilts, extending classical representation theory results.

Findings

Dimension vectors of cluster-indecomposable modules are roots of the Euler form.

Cluster-indecomposable modules are uniquely determined by their dimension vectors.

Provides a description of quivers with relations for n-APR tilts and generalizes BGP reflection functors.

Abstract

We study the relationship between $n$-cluster tilting modules over $n$ representation finite algebras and the Euler forms. We show that the dimension vectors of cluster-indecomposable modules give the roots of the Euler form. Moreover, we show that cluster-indecomposable modules are uniquely determined by their dimension vectors. This is a generalization of Gabriel's theorem by cluster tilting theory. We call the above roots cluster-roots and investigate their properties. Furthermore, we provide the description of quivers with relations of $n$-APR tilts. Using this, we provide a generalization of BGP reflection functors.