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Classification problem of hypergeometric differential systems

超几何微分系统的分类问题

基本信息

批准号：
12640149
负责人：
SAITO Mutsumi
金额：
$ 2.24万
依托单位：
HOKKAIDO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

项目摘要

With support of many examples with a computer, and by communication with world-wide experts in several fields, we obtained the following results.Mutsumi Saito generalized the classification theorem of A-hypergeometric systems to the cases when A is inhomogeneous and/or when we work in the analytic category. He also gave a dimension formula for the log-free series solutions when A is homogeneous, and a rank formula and the proof of the equivalence of Cohen-Macaulayness with the condition that the ranks are the same at all parameters, when A is homogeneous, and the convex hull of A is a simplex.Hiroshi Yamashita obtained some results useful to know when an isotropy representation is irreducible. Furthermore he systematically constructed nonzero quotient representations of isotropy representations attached to discrete series.Keiji Matsumoto clarified a pairing between twisted cohomology groups associated with generalized Airy functions. Writting a base of twisted cohomology groups by Young diagrams, he showed that for the base, the pairing can be explicitly written by skew-Schur polynomials.Youichi Shibukawa solved the classification problem for R operators.For the simplest affine Lie algebra A_1^<(1)> , using two of its realizations, Hiro-Fumi Yamada discovered the weight vectors are written by a modular version of Schur functions and Schur's Q-functions respectively.Akihito Wachi has studied the structure of generalized Verma modules, in particular, their irreducibility, emphasizing their relations with invariant functions.

在计算机上的许多例子的支持下，通过与几个领域的全球专家的交流，我们得到了以下结果。Mutsumi Saito将A-超几何系统的分类定理推广到A是非齐次的和/或在解析范畴内工作的情况。他还给出了当a为齐次时无对数级数解的维数公式，给出了当a为齐次且a的凸包为单纯形时，在各参数阶数相等条件下的秩公式和Cohen-Macaulayness等价性的证明。Hiroshi Yamashita得到了一些有用的结果，用于了解各向同性表示何时不可约。在此基础上，系统构造了离散级数上各向同性表示的非零商表示。Keiji Matsumoto澄清了与广义Airy函数相关的扭曲上同群之间的配对。在杨氏图上，他写出了一个扭曲上同群的基，并证明了对基的对偶可以用斜舒尔多项式显式表示。Shibukawa解决了R算子的分类问题。对于最简单的仿射李代数A_1^<(1)>， hiroo - fumi Yamada利用它的两种实现，发现权向量分别由舒尔函数和舒尔q函数的模表示。Akihito Wachi研究了广义Verma模的结构，特别是它们的不可约性，强调它们与不变函数的关系。