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On tropical dualities in cluster algebras

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arxiv 1101.3736 v3 pith:FNJBCI5K submitted 2011-01-19 math.RA

Tomoki Nakanishi , Andrei Zelevinsky This is my paper · ORCID
classification math.RA
keywords vectorsclusteralgebrasparameterizingassumptioncoefficientscrucialdepend
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abstract

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.

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  1. From green mutation to $\mathrm{X}$-evolution: flows and foliations on cluster complexes

    math.RT 2025-01 conditional novelty 7.0 of 10

    Introduces a piecewise-linear flow on cluster complexes whose leaves generalize green mutation, proving these complexes are spheres for Dynkin quivers and contractible for Euclidean quivers.

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