Two variations on (A3 × A1 × A1)(1) type discrete Painlevé equations

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By considering the normalizers of reflection subgroups of types A(1)1 and A(1)3 in W~ (D(1)5 ), two subgroups: W~ (A3 × A1)(1) × W(A(1)1 ) and W~ (A1 × A1)(1) × W(A(1)3 ) can be constructed from a (A3 × A1 × A1)(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 (A2n+1 + A1 + A1)(1) arising from cluster algebra. (, 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 languageEnglish
Article number20190299
Number of pages18
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2229
Early online date4 Sept 2019
Publication statusPublished - 27 Sept 2019


  • Coxeter groups
  • Discrete Painleve equation
  • Normalizer


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