On tropical dualities in cluster algebras

Tomoki Nakanishi; Andrei Zelevinsky

On tropical dualities in cluster algebras

Rings and Algebras 2012-03-15 v3

Authors: Tomoki Nakanishi , Andrei Zelevinsky

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.

Keywords

cluster algebras tropical dualities g-vectors c-vectors sign-coherence

Cite

@article{arxiv.1101.3736,
  title  = {On tropical dualities in cluster algebras},
  author = {Tomoki Nakanishi and Andrei Zelevinsky},
  journal= {arXiv preprint arXiv:1101.3736},
  year   = {2012}
}

Comments

10 pages; v2: expanded a little, an acknowledgment and reference added, minor editorial changes; v3: minor changes, final version, to appear in Proceedings of Representation Theory of Algebraic groups and Quantum groups, 10, Contemp. Math., AMS

On tropical dualities in cluster algebras

Abstract

Keywords

Cite

Comments

Related papers