Q-system Cluster Algebras, Paths and Total Positivity

Details Q-system Cluster Algebras, Paths and Total Positivity Q-system Cluster Algebras, Paths and Total Positivity

2009-06-18

arxiv.org/abs/0906.3421v3

SIGMA 6 (2010), 014, 36 pages

Mathematics

Combinatorics

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

Scientific paper

10.3842/SIGMA.2010.014

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.

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