A Graph Theoretic Expansion Formula for Cluster Algebras of Classical Type
A Graph Theoretic Expansion Formula for Cluster Algebras of Classical Type
复制标题
经典型簇代数的图论展开式
DOI：
10.1007/s00026-011-0088-3
复制
发表时间：
2007
影响因子：
0.5
通讯作者：
Gregg Musiker
中科院分区：
文献类型：
--
作者：
Gregg Musiker
本文给出了大多数具有二部种子的有限型簇代数中出现的簇变量的图论组合解释。特别地，我们提供了一组图，使得它们完美匹配的加权枚举编码相关的Laurent多项式的分子，而图的分解对应于分母。这补充了Schiffler和Carroll-Price最近对An情况的簇展开公式的研究，同时为Bn、Cn和Dn情况提供了新的解释。
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.