Cluster algebras of unpunctured surfaces and snake graphs

Gregg Musiker, Ralf Schiffler

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 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 .

Original languageEnglish (US)
Title of host publicationFPSAC'09 - 21st International Conference on Formal Power Series and Algebraic Combinatorics
Pages673-684
Number of pages12
StatePublished - Dec 1 2009
Event21st International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'09 - Linz, Austria
Duration: Jul 20 2009Jul 24 2009

Other

Other21st International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'09
CountryAustria
CityLinz
Period7/20/097/24/09

Keywords

  • Cluster algebra
  • F-polynomial
  • Height function
  • Principal coefficients
  • Snake graphs
  • Triangulated surface

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