### Abstract

In this paper, we show how to relate n-dimensional cubes on which ABS equations hold to the symmetry groups of discrete Painlevé equations. We here focus on the reduction from the four-dimensional cube to the q-discrete third Painlevé equation, which is a dynamical system on a rational surface of type A
_{5}
^{(1)}
with the extended affine Weyl group W̃ ((A
_{2}
+ A
_{1}
)
^{(1)}
). We provide general theorems to show that this reduction also extends to other discrete Painlevé equations at least of type A.

Original language | English |
---|---|

Article number | 505201 |

Pages (from-to) | 505201-505216 |

Number of pages | 16 |

Journal | Journal of Physics. A. Mathematical and Theoretical |

Volume | 47 |

Issue number | 50 |

DOIs | |

Publication status | Published - 25 Nov 2014 |

Externally published | Yes |

### Keywords

- geometric reductionPACS numbers: 02.30.Ik, 02.20.Qs
- discrete Painlevé equations
- ABS equations

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## Cite this

*n*-cube to discrete Painlevé systems.

*Journal of Physics. A. Mathematical and Theoretical*,

*47*(50), 505201-505216. [505201]. https://doi.org/10.1088/1751-8113/47/50/505201