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..
Bibliographical noteFunding 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
- Principal coefficients
- Snake graph
- Triangulated surface