Projective Reduction of the Discrete Painleve System of Type (A2 + A1)(1)

Kajiwara, Kenji; Nakazono, Nobutaka; Tsuda, Teruhisa

doi:10.1093/imrn/rnq089

Cited by 39 publications

(72 citation statements)

References 32 publications

(39 reference statements)

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“…Even in the second order equations, there are various discrete Painlevé equations which are not directly investigated in this paper, e.g., equations arising from the translations with different directions or length in the root lattice [55,109,121]. Also, we do not deal with higher order or multi-variable generalizations, which are now actively studied in relation with soliton equations [19,20,57,58,87,119,124,126], geometry of space of initial values [71,114], geometry of flag varieties [89], or general theory of monodromy preserving deformations [65].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Geometric aspects of Painlevé equations

Kajiwara¹,

Noumi²,

Yamada³

2017

J. Phys. A: Math. Theor.

123

250

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In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlevé equations, with a particular emphasis on the discrete Painlevé equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on P 1 × P 1 and classified according to the degeneration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeometric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.

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Section: Introductionmentioning

confidence: 99%

“…. Such a phenomenon occurs when the mapping is finer than translations of the underlying root lattice ("projective reduction"), as shown below [55,121].…”

Section: Introducing the Four Variablesmentioning

confidence: 99%

Geometric aspects of Painlevé equations

Kajiwara¹,

Noumi²,

Yamada³

2017

J. Phys. A: Math. Theor.

123

250

View full text Add to dashboard Cite

show abstract

“…In this way, we find geometric reductions from partial difference equations posed on an n-dimensional quadrilateral lattice (known 19 (three kinds of discrete equations for the E 8 type, two kinds each for the types E 7 and E 6 , and excluding the three affine root systems of type A 0 on the last column) correspond to the symmetries of discrete Painlevé equations [28].…”

Section: Introductionmentioning

confidence: 99%

Reflection groups and discrete integrable systems

2015

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Abstract. We present a method of constructing discrete integrable systems with crystallographic reflection group (Weyl) symmetries, thus clarifying the relationship between different discrete integrable systems in terms of their symmetry groups. Discrete integrable systems are associated with space-filling polytopes arise from the geometric representation of the Weyl groups in the n-dimensional real Euclidean space R n . The "multi-dimensional consistency" property of the discrete integrable system is shown to be inherited from the combinatorial properties of the polytope; while the dynamics of the system is described by the affine translations of the polytopes on the weight lattices of the Weyl groups. The connections between some well-known discrete systems such as the multi-dimensional consistent systems of quad-equations [1] and discrete Painlevé equations [28] are obtained via the geometric constraints that relate the polytope of one symmetry group to that of another symmetry group, a procedure which we call geometric reduction.

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“…Most discrete analogs of Painlevé equations have already been obtained, but discrete analogs of P D in a q-analog of P V (q-P V ) [15], where a; b; g; t; q A C Â are parameters. We term this method of reduction a projective reduction [4]. It is well known that P D…”

Section: Introductionmentioning

confidence: 99%