A graph theoretic expansion formula for cluster algebras of classical type

2007/10/18 by Gregg Musiker, Musiker, Gregg
Mathematics · #05E15 #16S99 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.0710.3574

openalex publication_date 2007/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type. 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.

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