Quivers with potentials and their representations II: Applications to cluster algebras
Harm Derksen, Jerzy Weyman, Andrei Zelevinsky
TL;DR
The paper develops a representation-theoretic framework for cluster algebras by interpreting g-vectors and F-polynomials through decorated quivers with potentials (QP) representations. It constructs indecomposable QP-representations that realize cluster data, proves mutation-invariance results, and introduces the E-invariant with a sharp lower bound, linking these to homological operations. This leads to a Caldero-Chapoton-type formula for coefficient-free cluster variables, sign-coherence of g-vectors, and a homological interpretation of mutation dynamics. The results provide a cohesive, category-theoretic and homological foundation for understanding cluster algebra structures in the skew-symmetric case, yielding both practical formulas and deep structural insights.
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.