Quivers with potentials and their representations II: Applications to cluster algebras

Derksen, Harm; Weyman, Jerzy; Zelevinsky, Andrei

doi:10.1090/S0894-0347-10-00662-4

Quivers with potentials and their representations II: Applications to cluster algebras
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by Harm Derksen, Jerzy Weyman and Andrei Zelevinsky;

J. Amer. Math. Soc. 23 (2010), 749-790

DOI: https://doi.org/10.1090/S0894-0347-10-00662-4

Published electronically: February 8, 2010

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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 $\mathbf {g}$-vectors, and a family of integer polynomials called $F$-polynomials. In the case of skew-symmetric exchange matrices we find an interpretation of these $\mathbf {g}$-vectors and $F$-polynomials in terms of (decorated) representations of quivers with potentials. Using this interpretation, we prove most of the conjectures about $\mathbf {g}$-vectors and $F$-polynomials made in loc. cit.

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