Abstract
We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph G T,γ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph G T,γ.
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The first author is supported by an NSF Mathematics Postdoctoral Fellowship, and the second author is supported by the NSF grant DMS-0908765 and by the University of Connecticut.
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Musiker, G., Schiffler, R. Cluster expansion formulas and perfect matchings. J Algebr Comb 32, 187–209 (2010). https://doi.org/10.1007/s10801-009-0210-3
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DOI: https://doi.org/10.1007/s10801-009-0210-3