Cluster expansion formulas and perfect matchings

Gregg Musiker, Ralf Schiffler

Research output: Contribution to journalArticlepeer-review

45 Scopus citations


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
Issue number2
StatePublished - Sep 2010

Bibliographical note

Funding Information:
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.


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


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