Two variations on ( A 3 × A 1 × A 1 ) (1) type discrete Painlevé equations

By considering the normalizers of reflection subgroups of types A (1) 1 and A (1) 3 in W ~ ( D 5 ( 1 ) ) , two subgroups: W ~ ( A 3 × A 1 ) ( 1 ) ⋉ W ( A 1 ( 1 ) ) and W ~ ( A 1 × A 1 ) ( 1 ) ⋉ W ( A 3 ( 1 ) ) 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.