Journal of Algebraic Combinatorics

, Volume 32, Issue 2, pp 187–209

Cluster expansion formulas and perfect matchings


DOI: 10.1007/s10801-009-0210-3

Cite this article as:
Musiker, G. & Schiffler, R. J Algebr Comb (2010) 32: 187. doi:10.1007/s10801-009-0210-3


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.

Cluster algebra Triangulated surface Principal coefficients F-polynomial Snake graph 

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Mathematics, Room 2-336Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.Department of MathematicsUniversity of ConnecticutStorrsUSA

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