Cluster expansion formulas and perfect matchings

Gregg Musiker, Ralf Schiffler

Research output: Contribution to journalArticle

32 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 1 2010


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

Fingerprint Dive into the research topics of 'Cluster expansion formulas and perfect matchings'. Together they form a unique fingerprint.

  • Cite this