# On Cluster Algebras Arising from Unpunctured Surfaces

@article{Schiffler2007OnCA,
title={On Cluster Algebras Arising from Unpunctured Surfaces},
author={Ralf Schiffler and Hugh Thomas},
journal={International Mathematics Research Notices},
year={2007},
volume={2009},
pages={3160-3189}
}
• Published 2007
• Mathematics
• International Mathematics Research Notices
We study cluster algebras that are associated to unpunctured surfaces, with coefficients arising from boundary arcs. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras. In the special case where the cluster algebra is acyclic, we also give a formula for the… Expand

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