Quivers with potentials and their representations II: Applications to cluster algebras
Abstract
We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the "Cluster algebras IV" paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called g-vectors, and a family of integer polynomials called F-polynomials. In the case of skew-symmetric exchange matrices we find an interpretation of these g-vectors and F-polynomials in terms of (decorated) representations of quivers with potentials. Using this interpretation, we prove most of the conjectures about g-vectors and F-polynomials made in loc. cit.
Cite
@article{arxiv.0904.0676,
title = {Quivers with potentials and their representations II: Applications to cluster algebras},
author = {Harm Derksen and Jerzy Weyman and Andrei Zelevinsky},
journal= {arXiv preprint arXiv:0904.0676},
year = {2010}
}
Comments
44 pages; version 2: revised according to the referee's suggestions; version 3: final version, typos corrected, references corrected and updated.