Cluster expansion formulas and perfect matchings

Gregg Musiker, Ralf Schiffler

Research output: Contribution to journalArticle

32 Scopus citations

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 GT,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 GT,γ..

Original languageEnglish (US)
Pages (from-to)187-209
Number of pages23
JournalJournal of Algebraic Combinatorics
Volume32
Issue number2
DOIs
StatePublished - Sep 1 2010

Keywords

  • Cluster algebra
  • F-polynomial
  • Principal coefficients
  • Snake graph
  • Triangulated surface

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