## Abstract

By considering the normalizers of reflection subgroups of types A^{(1)}_{1} and A^{(1)}_{3} in W~ (D^{(1)}_{5} ), two subgroups: W~ (A_{3} × A_{1})^{(1)} × W(A^{(1)}_{1} ) and W~ (A_{1} × A_{1})^{(1)} × W(A^{(1)}_{3} ) can be constructed from a (A_{3} × A_{1} × A_{1})^{(1)} type subroot system. These two symmetries arose in the studies of discrete Painlevé equations (Kajiwara K, Noumi M, Yamada Y. 2002 q-Painlevé systems arising from q-KP hierarchy. Lett. Math. Phys. 62, 259-268; Takenawa T. 2003 Weyl group symmetry of type D^{(1)}_{5} in the q-Painlevé V equation. Funkcial. Ekvac. 46, 173-186; Okubo N, Suzuki T. 2018 Generalized q-Painlevé VI systems of type (A_{2}n+_{1} + A_{1} + A_{1})^{(1)} arising from cluster algebra. (http://arxiv.org/abs/quant-ph/1810.03252)), where certain non-translational elements of infinite order were shown to give rise to discrete Painlevé equations. We clarify the nature of these elements in terms of Brink-Howlett theory of normalizers of Coxeter groups (Howlett RB. 1980 Normalizers of parabolic subgroups of reflection groups. J. London Math. Soc. (2) 21, 62-80; Brink B, Howlett RB. 1999 Normalizers of parabolic subgroups in Coxeter groups. Invent. Math. 136, 323-351). This is the first of a series of studies which investigates the properties of discrete integrable equations via the theory of normalizers.

Original language | English |
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Article number | 20190299 |

Number of pages | 18 |

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

Volume | 475 |

Issue number | 2229 |

Early online date | 4 Sept 2019 |

DOIs | |

Publication status | Published - 27 Sept 2019 |

## Keywords

- Coxeter groups
- Discrete Painleve equation
- Normalizer

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