2010
DOI: 10.1093/imrn/rnq089
|View full text |Cite
|
Sign up to set email alerts
|

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

Abstract: We consider the q-Painlevé III equation arising from the birational representation of the affine Weyl group of type (A 2 + A 1 ) (1) . We study the reduction of the q-Painlevé III equation to the q-Painlevé II equation from the viewpoint of affine Weyl group symmetry. In particular, the mechanism of apparent inconsistency between the hypergeometric solutions to both equations is clarified by using factorization of difference operators and the τ functions.

Help me understand this report
View preprint versions

Search citation statements

Order By: Relevance

Paper Sections

Select...
4
1

Citation Types

4
66
2

Year Published

2011
2011
2024
2024

Publication Types

Select...
9

Relationship

0
9

Authors

Journals

citations
Cited by 39 publications
(72 citation statements)
references
References 32 publications
4
66
2
Order By: Relevance
“…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
“…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%
“…. 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%
“…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%
“…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%