Gregg Musiker ; Ralf Schiffler - Cluster algebras of unpunctured surfaces and snake graphs

dmtcs:2685 - Discrete Mathematics & Theoretical Computer Science, January 1, 2009, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) - https://doi.org/10.46298/dmtcs.2685
Cluster algebras of unpunctured surfaces and snake graphs

Authors: Gregg Musiker ; Ralf Schiffler

    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 $G_{T,\gamma}$ .


    Volume: DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)
    Section: Proceedings
    Published on: January 1, 2009
    Imported on: January 31, 2017
    Keywords: cluster algebra,triangulated surface,principal coefficients,F-polynomial,height function,snake graphs,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
    Fundings :
      Source : OpenAIRE Research Graph
    • Cluster algebras and tilting theory; Funder: National Science Foundation; Code: 0700358
    • PostDoctoral Research Fellowship; Funder: National Science Foundation; Code: 0703691

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