## On tropical dualities in cluster algebras.(English)Zbl 1317.13054

Ariki, Susumu (ed.) et al., Algebraic groups and quantum groups. International conference on representation theory of algebraic groups and quantum groups ’10, Graduate School of Mathematics, Nagoya University, Nagoya, Japan, August 2–6, 2010. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-5317-7/pbk; 978-0-8218-8536-9/ebook). Contemporary Mathematics 565, 217-226 (2012).
Summary: We study two families of integer vectors playing a crucial part in the structural theory of cluster algebras: the $$\mathbf g$$-vectors parameterizing cluster variables, and the $$\mathbf c$$-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.
For the entire collection see [Zbl 1239.20002].

### MSC:

 13F60 Cluster algebras 15A80 Max-plus and related algebras 14T05 Tropical geometry (MSC2010)
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