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2011

On tropical dualities in cluster algebras

Nakanishi, Tomoki, Zelevinsky, Andrei

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We study two families of integer vectors playing a crucial part in the structural theory of cluster algebras: the $\gg$-vectors parameterizing cluster variables, and the $\cc$-vectors parameterizing the coefficients.

We prove two identities relating these vectors to each other.
The proofs depend on the sign-coherence assumption for $\cc$-vectors that still remains unproved in full generality.

On tropical dualities in cluster algebras · Around

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S. Fomin, A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc
2002
Earlier work this paper cites.
A. Berenstein, S. Fomin, A. Zelevinsky, Cluster algebras III: Upper bounds and double Bruhat cells, Duke Math. J
2005
Earlier work this paper cites.

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S. Fomin, A. Zelevinsky, Cluster algebras IV: Coefficients, Comp. Math
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V. V. Fock, A. B. Goncharov, Cluster ensembles, quantization and the dilogarithm, Annales Sci. de l’École Norm. Sup. (4)
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K. Nagao, Donaldson-Thomas theory and cluster algebras, arXiv:1002.4884
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T. Nakanishi, Periodicities in cluster algebras and dilogarithm identities, arXiv:1006.0632
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P. Plamondon, Cluster algebras via cluster categories with infinite-dimensional morphism spaces, arXiv:1004.0830
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Then

H. Derksen, J. Weyman, A. Zelevinsky, Quivers with potentials and their representations II: Applications to cluster algebras, J. Amer. Math. Soc
2010
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