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

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. (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.