Comprehensive Research of Hypergeometric and Painleve Systems

超几何和Painleve系统的综合研究

• 批准号: 14204009
• 负责人: NOUMI Masatoshi
• 金额: $ 28.45万
• 依托单位: Kobe University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-14204009/
• 关键词: Painleve equation; discrete Painleve equation; elliptic hypergeometric functions; monodromy manifold; affine Weyl group symmetry; special solutions of Riccati type; 一般超幾何; モジュライ空間; 超幾何函数解; qパンルヴェ方程式; q超幾何函数; ベックルント変換; 楕円パンルヴェ方程式; 楕円超幾何関数; 代数

1.Representation theoretic study of integrable systems and Painleve systems : (1) Construction of a new Lax pair associated with the affine Lie algebra <so>^^^^(8) for the Painleve equation P_<VI> which reveals a group-theoretic origin of the affine Wey1 group symmetry. (2) Determinant representation of special solutions of hypergeometric and algebraic type to the Painleve equation P_<VI>.2.Geometric study of moduli spaces : (1) Geometric investigation of the monodromy manifold for the Painelve equation P_<VI>, and the Riemann-Hilbert correspondence : (2) Foundation of higher dimensional discrete systems based on the Gremona transformations and the birational Weyl group actions on the configuration spaces of points in projective spaces. (3) Study of moduli spaces of vector bundles and Fourir-Mukai transformations. Their applications to instanton counting.3.Discrete integrable systems and discrete Painleve systems: (1) Introduction of higher order q-Painleve systems with affine Weyl group symmetry of type A. (2) Description of the elliptic Painleve equation of type E_8 in terms of γ-functions. Construction of special solutions of Riccati type, expressed in terms of elliptic hypergeoemetric functions. (3) Construction of q-hypergeometric solutions to the hierarchy of q-Painleve equations in Sakai's list. (4) Theoretic study of totally positive Weyl group actions and totally positive R-matrices, together with applications to combinatorics and solvable lattice modeles.

1.可积系统和 Painleve 系统的表示理论研究： (1) 为 Painleve 方程 P_<VI> 构造一个与仿射李代数 <so>^^^^(8) 相关的新 Lax 对，该方程揭示了一个群-仿射 Wey1 群对称性的理论起源 (2) Painleve 方程的超几何和代数类型特殊解的行列式表示。 P_<VI>.2.模空间的几何研究： (1) Painelve 方程 P_<VI> 的单向流形的几何研究，以及黎曼-希尔伯特对应关系： (2) 基于Gremona 变换和射影空间中点的构型空间上的双有理 Weyl 群作用。 (3) 向量丛的模空间和 Fourir-Mukai 变换在瞬子中的应用。 3.离散可积系统和离散Painleve系统：（1）引入具有A型仿射Weyl群对称性的高阶q-Painleve系统。（2）用γ函数描述E_8型椭圆Painleve方程以椭圆超几何函数表示的 Riccati 型特殊解的构造 (3) 的层次结构的 q 超几何解的构造。 Sakai 列表中的 q-Painlevel 方程 (4) 全正 Weyl 群作用和全正 R 矩阵的理论研究，以及在组合学和可解晶格模型中的应用。

项目成果

A.Kuniba: "Geometric crystal and tropical R for D^<(1)>_n"Int.Math.Res.Not.. 48. 2565-2620 (2003)

A.Kuniba：“D^<(1)>_n 的几何晶体和热带 R”Int.Math.Res.Not.. 48. 2565-2620 (2003)

Backlund transformations of the sixth Painleve equation in terms of Rieman-Hilbert correspondence

第六 Painleve 方程的 Rieman-Hilbert 对应关系的 Backlund 变换

• 发表时间: 2004
• 期刊: Int. Math. Res. Not. 1
• 作者: M.Inaba

Box-ball system with reflecting end

带反射端的盒球系统

• 发表时间: 2005
• 期刊: J.Nonlin.Math.Phys. 12
• 作者: Shirai;Satoko;A.Kuniba
• 通讯作者: A.Kuniba

M.Noumi et al.: "Backlund transformations and the manifolds of Painleve systems"Funkcial. Ekvac.. 45. 237-258 (2002)

M.Noumi 等人：“Backlund 变换和 Painleve 系统的流形”Funkcial。

Y.Kajihara: "Multiple elliptic hypergeometric series. An approach from the Cauchy determinant"Indag.Math.. 14. 395-421 (2003)

Y.Kajihara：“多重椭圆超几何级数。柯西行列式的一种方法”Indag.Math.. 14. 395-421 (2003)