Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 014, 36 pages      arXiv:0906.3421
Contribution to the Proceedings of the Workshop “Geometric Aspects of Discrete and Ultra-Discrete Integrable Systems”

Q-system Cluster Algebras, Paths and Total Positivity

Philippe Di Francesco a and Rinat Kedem b
a) Institut de Physique Théorique du Commissariat à l'Energie Atomique, Unité de Recherche associée du CNRS, CEA Saclay/IPhT/Bat 774, F-91191 Gif sur Yvette Cedex, France
b) Department of Mathematics, University of Illinois Urbana, IL 61801, USA

Received October 15, 2009, in final form January 15, 2010; Published online February 02, 2010

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.

Key words: cluster algebras; total positivity.

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