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 language | English (US) |
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Pages (from-to) | 187-209 |
Number of pages | 23 |
Journal | Journal of Algebraic Combinatorics |
Volume | 32 |
Issue number | 2 |
DOIs | |
State | Published - 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.
Keywords
- Cluster algebra
- F-polynomial
- Principal coefficients
- Snake graph
- Triangulated surface