## Abstract

In this paper, we consider the discrete power function associated with the sixth Painlevé equation. This function is a special solution of the so-called cross-ratio equation with a similarity constraint. We show in this paper that this system is embedded in a cubic lattice with W (3A^{(1)} _{1} ) symmetry. By constructing the action of W (3A^{(1)} _{1} ) as a subgroup of W (D^{(1)} _{4} ), i.e. the symmetry group of P_{VI}, we show how to relate W (D^{(1)} _{4} ) to the symmetry group of the lattice. Moreover, by using translations in W (3A^{(1)} _{1} ), we explain the odd–even structure appearing in previously known explicit formulae in terms of the t function.

Original language | English |
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Article number | 20170312 |

Number of pages | 19 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 473 |

Issue number | 2207 |

DOIs | |

Publication status | Published - 22 Nov 2017 |

Externally published | Yes |

## Keywords

- ? function
- ABS equation
- Affine Weyl group
- Discrete power function
- Painlevé equation
- Projective reduction