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International Mathematics Research Notices Advance Access published online on April 11, 2009
International Mathematics Research Notices, doi:10.1093/imrn/rnp047
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© The Author 2009. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

On Cluster Algebras Arising from Unpunctured Surfaces

Ralf Schiffler1 and Hugh Thomas2

1 Department of Mathematics, University of Connecticut, Storrs, CT, 06269-3009, USA
2 Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, E3B 5A3, Canada

Correspondence: Correspondence to be sent to: schiffler{at}math.uconn.edu

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 expansion of cluster variables as a polynomial whose indeterminates are the cluster variables contained in the union of an arbitrary acyclic cluster and all its neighboring clusters in the mutation graph.


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