Q-system cluster algebras, paths and total positivity

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Abstract

In the first part of this paper, we provide a concise review of our method of solution of the Ar Q-systems in terms of the partition function of paths on a weighted graph. In the second part, we show that it is possible to modify the graphs and transfer matrices so as to provide an explicit connection to the theory of planar networks introduced in the context of totally positive matrices by Fomin and Zelevinsky. As an illustration of the further generality of our method, we apply it to give a simple solution for the rank 2 affine cluster algebras studied by Caldero and Zelevinsky.

Original languageEnglish (US)
Article number014
JournalSymmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume6
DOIs
StatePublished - Jan 1 2010

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Total Positivity
Cluster Algebra
Totally Positive Matrices
Path
Weighted Graph
Transfer Matrix
Partition Function
Graph in graph theory
Review
Context

Keywords

  • Cluster algebras
  • Total positivity

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics
  • Geometry and Topology

Cite this

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abstract = "In the first part of this paper, we provide a concise review of our method of solution of the Ar Q-systems in terms of the partition function of paths on a weighted graph. In the second part, we show that it is possible to modify the graphs and transfer matrices so as to provide an explicit connection to the theory of planar networks introduced in the context of totally positive matrices by Fomin and Zelevinsky. As an illustration of the further generality of our method, we apply it to give a simple solution for the rank 2 affine cluster algebras studied by Caldero and Zelevinsky.",
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