Q-system Cluster Algebras, Paths and Total Positivity

Abstract

In the first part of this paper, we provide a concise review of our method of solution of the $A_r$ 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.

36 pages, 16 Postscript figures, typos corrected and one reference added

References in corpus (7)

• Q-systems as cluster algebras II: Cartan matrix of finite type and the polynomial property
• Q-systems, Heaps, Paths and Cluster Positivity
• Preprojective algebras and cluster algebras
• Q-systems as cluster algebras
• Positivity for cluster algebras from surfaces
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Cited by in corpus (6)

• Q-systems, Heaps, Paths and Cluster Positivity
• Noncommutative integrability, paths and quasi-determinants
• A compendium on the cluster algebra and quiver package in sage
• The solution of the $A_r$ T-system for arbitrary boundary
• Generalized Bäcklund-Darboux transformations for Coxeter-Toda flows from a cluster algebra perspective
• Linear recurrence relations in $Q$-systems and difference $L$-operators