Integrated research of the general hypergeometric systems and nonlinear integrable systems

一般超几何系统与非线性可积系统的综合研究

基本信息

批准号：
11440058
负责人：
KIMURA Hironobu
金额：
$ 8万
依托单位：
Kumamoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440058/
关键词：
Painleve equation space of initial conditions confluence general hypergeometric function de Rham cohomology intersection theory generalized Airy function accessory parameter Gauss Manin系 Grassmann多様体 Airy関数 generating function 超幾何関

项目摘要

The objective of this research is 1) the study of the general hypergeometric functions and Okubo systems, 2) the study of nonlinear integrable systems including Painleve equations. The conjugacy classes of the centralizers of regular elements of GL(N,C) are determined by partitions of N. The general hypergeometric functions are functions on the Grassmannian manifold Gr(n,N) obtained by the Radon transformation of characters of universal covering groups of centralizers. We explicitly determined the algebraic de Rham cohomology groups associated with the integral representation of the general hypergeometric functions. This problem has been isolved in the case n=2 and in the case n>2 with the partitions (1,…,1), (N). For the case of partitions (q, 1,…,1), we proved the purity of the cohomology group, determined the dimension of the top cohomology and gave an explicit basis for it. This result will be important in constructing the Gauss-Manin system characterizing the general hypergeometri … More c functions. In the case where the partition is (N), we constructed the intersection theory of de Rham cohomology and expressed the intersection numbers in terms of skew Schur polynomials. In this computation, we recognized that an analogue of flat basis plays an important roles which appears in the theory of singularity. For the differential equation of Schlesinger type on P^1 without accessory parameters, we showed that the solutions have integral representations using the corresponding result for Okubo system. This integral representation is a particular case of that of GKZ hypergeometric functions. Thus it may be an interesting problem to understand the accessory parameter free equations in the framework of GKZ hypergeometric functions and to generalize this problem to the equations with irregular singularities.For the Painleve equations, we showed that there is a symplectic structure for the space of initial conditions for each Painleve equation and also showed that the geometry of the space of initial conditions determines the Painleve equation. We found an interesting phenomenon that a generating function for a series of rational solutions of Painleve II coincides with the asymptotic expansion at infinity of the function obtained from Airy function. Less

本研究的目标是 1) 一般超几何函数和 Okubo 系统的研究，2) 包括 Painleve 方程在内的非线性可积系统的研究 GL(N,C) 正则元素的集中器的共轭类由下式确定。一般的超几何函数是通过对中心化器的通用覆盖群的特征进行 Radon 变换而获得的格拉斯曼流形 Gr(n,N) 上的函数。确定了与一般超几何函数的积分表示相关的代数 de Rham 上同调群。该问题已在 n=2 的情况下和 n>2 的情况下与分区 (1,…,1), (N) 隔离。对于划分 (q, 1,…,1) 的情况，我们证明了上同调群的纯度，确定了顶上同调的维数，并为其给出了明确的基础，这个结果对于构造上同调群非常重要。表征一般超几何 c 函数的 Gauss-Manin 系统在划分为 (N) 的情况下，我们构建了 de Rham 上同调的交集理论，并用偏斜 Schur 多项式表示交集数。认识到平基的类似物在奇点理论中起着重要作用，对于没有附加参数的 P^1 上的 Schlesinger 型微分方程，我们使用相应的结果证明了解具有积分表示。 Okubo 系统。这种积分表示是 GKZ 超几何函数的特例，因此理解 GKZ 超几何函数框架中的辅助参数自由方程并将该问题推广到具有不规则奇点的方程可能是一个有趣的问题。对于 Painleve 方程，我们证明了每个 Painleve 方程的初始条件空间都存在辛结构，并且还表明初始条件空间的几何形状决定了 Painleve 方程。有趣的现象是 Painleve II 的一系列有理解的生成函数与从 Airy 函数获得的函数的无穷远渐近展开一致。