Generalizedq-Painlevé VI Systems of Type (A2n+1+A1+A1)(1) Arising From Cluster Algebra

Okubo, Naoto; Suzuki, Takao

doi:10.1093/imrn/rnaa283

Cited by 11 publications

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“…In §4, the normalizer theory developed in §3 is placed in the context of discrete Painlevé equations. In particular, we discuss the quasi-translational nature of the elements which give rise to discrete Painlevé equations given by Takenawa [8], and Okubo & Suzuki [6] in terms of the normalizer theory. Concluding remarks and some future directions are given in §5.…”

Section: ) With An Underlying (mentioning

confidence: 99%

“…(b) Okubo-Suzuki system n = 1 case, (A 3 × A 1 × A 1 ) (1) In 2018 [6], Okubo and Suzuki proposed a new (A 2n+1 × A 1 × A 1 ) (1) type generalisation of Sakai's W(D (1) 5 ) q-PVI equation from the framework of Cluster algebra. This we refer as the OS-system.…”

Section: )mentioning

confidence: 99%

“…In our notation, the four directions T i (1 ≤ i ≤ 4) given in [6] are, First let us look at the T 1 direction. We have…”

Section: )mentioning

confidence: 99%

“…, and generalizations of Painlevé equations such as Kajiwara, Noumi and Yamada's W(A n × A m ) (1) system (KNY) [3], Sasano [4] and Masuda's [5] D n type systems, and more recently Okubo and Suzuki's (A 2n+1 × A 1 × A 1 ) (1) system [6].…”

Section: Introductionmentioning

confidence: 99%

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

Shi

2019

Proc. R. Soc. A.

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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.

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Section: ) With An Underlying (mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

“…In our notation, the four directions T i (1 ≤ i ≤ 4) given in [6] are, First let us look at the T 1 direction. We have…”

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Shi

2019

Proc. R. Soc. A.

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“…For the q-Garnier system, a symmetry structure was clarified recently. In [9] we formulated a birational representation of an extended affine Weyl group W(A (1) 2n+1 × A (1) 1 × A (1) 1 ) with the aid of a cluster mutation and derived the q-Garnier system as translations. We also generalized this representation to that of W(A (1) mn−1 + A (1) m−1 + A (1) m−1 ) in [10].…”

Section: Introductionmentioning

confidence: 99%