Hone, Andrew N.W. and Inoue, Rei (2014) Discrete Painlevé equations from Ysystems. Journal of Physics A: Mathematical and Theoretical, 47 . ISSN 17518113. (doi:https://doi.org/10.1088/17518113/47/47/474007) (Full text available)
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Official URL http://iopscience.iop.org/article/10.1088/1751811... 
Abstract
We consider Tsystems and Ysystems arising from cluster mutations applied to quivers that have the property of being periodic under a sequence of mutations. The corresponding nonlinear recurrences for cluster variables (coefficientfree Tsystems) were described in the work of Fordy and Marsh, who completely classified all such quivers in the case of period 1, and characterized them in terms of the skewsymmetric exchange matrix B that defines the quiver. A broader notion of periodicity in general cluster algebras was introduced by Nakanishi, who also described the corresponding Ysystems, and Tsystems with coefficients. A result of Fomin and Zelevinsky says that the coefficientfree Tsystem provides a solution of the Ysystem. In this paper, we show that in general there is a discrepancy between these two systems, in the sense that the solution of the former does not correspond to the general solution of the latter. This discrepancy is removed by introducing additional nonautonomous coefficients into the Tsystem. In particular, we focus on the period 1 case and show that, when the exchange matrix B is degenerate, discrete Painlev\'e equations can arise from this construction.
Item Type:  Article 

Subjects: 
Q Science > QA Mathematics (inc Computing science) > QA171 Representation theory Q Science > QC Physics > QC20 Mathematical Physics 
Divisions: 
Faculties > Sciences > School of Mathematics Statistics and Actuarial Science Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Applied Mathematics Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Pure Mathematics 
Depositing User:  Andrew N W Hone 
Date Deposited:  20 Jun 2014 22:47 UTC 
Last Modified:  17 Jan 2017 13:05 UTC 
Resource URI:  https://kar.kent.ac.uk/id/eprint/41481 (The current URI for this page, for reference purposes) 
Hone, Andrew N.W.:  https://orcid.org/0000000197807369 
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