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Cluster algebras of unpunctured surfaces and snake graphs

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 $G_{T,\gamma}$ 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 $G_{T,\gamma}$ .
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Gregg Musiker, Ralf Schiffler. Cluster algebras of unpunctured surfaces and snake graphs. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. pp.673-684, ⟨10.46298/dmtcs.2685⟩. ⟨hal-01185377⟩

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