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arXiv · 1402.6084

Geometric Reductions of ABS equations on an $n$-cube to discrete Painlev\'e systems

Nalini Joshi·Nobutaka Nakazono·Yang Shi
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Abstract

In this paper, we show how to relate $n$-dimensional cubes on which ABS equations hold to the symmetry groups of discrete Painlev\'e equations. We here focus on the reduction from the 4-dimensional cube to the $q$-discrete third Painlev\'e equation, which is a dynamical system on a rational surface of type $A_5^{(1)}$ with the extended affine Weyl group $\widetilde{\mathcal W}\bigl((A_2+A_1)^{(1)}\bigr)$. We provide general theorems to show that this reduction also extends to other discrete Painlev\'e equations at least of type A.

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Nalini Joshi, Nobutaka Nakazono, Yang Shi. 2014-02-25. Geometric Reductions of ABS equations on an $n$-cube to discrete Painlev\'e systems. https://doi.org/10.1088/1751-8113/47/50/505201

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