A Graph Theoretic Expansion Formula for Cluster Algebras of Classical Type

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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 An case while providing a novel interpretation for the Bn, Cn, and Dn cases.

Original languageEnglish (US)
Pages (from-to)147-184
Number of pages38
JournalAnnals of Combinatorics
Volume15
Issue number1
DOIs
StatePublished - Mar 1 2011

Keywords

  • classical type
  • cluster algebras
  • laurentness
  • perfect matchings

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