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Annals of Combinatorics

, Volume 15, Issue 1, pp 147–184 | Cite as

A Graph Theoretic Expansion Formula for Cluster Algebras of Classical Type

  • Gregg MusikerEmail author
Article

Abstract

In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type with bipartite seed. In particular, we provide a family of graphs such that a weighted enumeration of their perfect matchings encodes the numerator of the associated Laurent polynomial while decompositions of the graphs correspond to the denominator. This complements recent work by Schiffler and Carroll-Price for a cluster expansion formula for the A n case while providing a novel interpretation for the B n , C n , and D n cases.

Mathematics Subject Classification

05E15 16S99 

Keywords

cluster algebras classical type perfect matchings laurentness 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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