A &lt;i&gt;q&lt;/i&gt;-Analogue of the Higher Order Painlev&amp;eacute; Type Equations with the Affine Weyl Group Symmetry of Type &lt;i&gt;D&lt;/i&gt;

Masuda, Tetsu

doi:10.1619/fesi.58.405

Cited by 11 publications

(11 citation statements)

References 16 publications

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“…, 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: Introductionmentioning

confidence: 99%

“…The normalizer procedure can be used to obtain generalizations of Painlevé equations as subsystems systems with bigger symmetry groups. Such a system was in fact given by Masuda in hisD (1) n generalization of Sakai's W(D (1)5 ) q-PVI equation[5].…”

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confidence: 97%

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

Shi

2019

Proc. R. Soc. A.

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“…Higher order generalizations of the q-Painlevé equations has been proposed from some points of view; birational representations of affine Weyl groups ( [13,14,15,31]), a cluster mutation ( [6]), a q-analogue of an isomonodromy deformation ( [24]), similarity reductions of discrete integrable systems ( [26,27,28]) and a Pade method ( [19,20]). However there doesn't exist any theory which governs all of them unlike in the case of 2nd order ( [23]).…”

Section: Introductionmentioning

confidence: 99%

Generalized $q$-Painlevé VI systems of type $(A_{2n+1}+A_1+A_1)^{(1)}$ arising from cluster algebra

Okubo¹,

Suzuki²

2018

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In this article we formulate a group of birational transformations which is isomorphic to an extended affine Weyl group of type (A 2n+1 + A 1 + A 1 ) (1) with the aid of mutations and permutations to a mutation-periodic quiver on a torus. This group provides four types of generalizations of Jimbo-Sakai's q-Painlevé VI equation as translations of the extended affine Weyl group. Then the known three systems are obtained again; the q-Garnier system, a similarity reduction of the lattice q-UC hierarchy and a similarity reduction of the q-Drinfeld-Sokolov hierarchy.

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“…Among the above four families, q-analogues of the Garnier system, the Fuji-Suzuki-Tsuda system, and the Sasano system have already been constructed and studied by several authors [14,17,18,11]. The aim of this paper is to obtain a q-analogue of the matrix sixth Painlevé system (abbreviated to matrix P VI ).…”

Section: Introductionmentioning

confidence: 99%

A $q$-analogue of the matrix sixth Painlevé system

Kawakami

2020

Preprint

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We derive a q-analogue of the matrix sixth Painlevé system via a connection-preserving deformation of a certain Fuchsian linear q-difference system. In specifying the linear q-difference system, we utilize the correspondence between linear differential systems and linear q-difference systems from the viewpoint of the spectral type. The system of non-linear q-difference equations thus obtained can also be regarded as a non-abelian analogue of Jimbo-Sakai's q-P VI .

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A <i>q</i>-Analogue of the Higher Order Painlevé Type Equations with the Affine Weyl Group Symmetry of Type <i>D</i>

Cited by 11 publications

References 16 publications

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

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

Generalized $q$-Painlevé VI systems of type $(A_{2n+1}+A_1+A_1)^{(1)}$ arising from cluster algebra

A $q$-analogue of the matrix sixth Painlevé system

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A <i>q</i>-Analogue of the Higher Order Painlev&eacute; Type Equations with the Affine Weyl Group Symmetry of Type <i>D</i>

Cited by 11 publications

References 16 publications

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

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

Generalized $q$-Painlevé VI systems of type $(A_{2n+1}+A_1+A_1)^{(1)}$ arising from cluster algebra

A $q$-analogue of the matrix sixth Painlevé system

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A <i>q</i>-Analogue of the Higher Order Painlevé Type Equations with the Affine Weyl Group Symmetry of Type <i>D</i>

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

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