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Actions of semisimple groups and Weyl groups and research on representations

半单群和Weyl群的作用及表示研究

基本信息

批准号：
17540013
负责人：
SEKIGUCHI Jiro
金额：
$ 1.92万
依托单位：
Tokyo University of Agriculture and Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540013/
关键词：
root systems configurations lines Weyl group vector fields finite prime field projective plane 射影平面帯球関数接続公式

项目摘要

We obtained the following results on this research.(1)J.Sekiguchi and T.Fukui studied the relationship between configurations of systems of eight lines on a real projective plane and the root system of type E8. We first defined a map of the totality of sets of 10 roots of the root system of type E8 with some conditions to the set of connected components of the configurations with certain conditions. Moreover we showed that this map is W(E8)-equivariant. Then we proved that there are 2160 number of orbits of such connected components by the action of the symmetric group of 8th degree. T.Fukui gave a talk on this result at the occasion of 12th International conference on Applications of Computer Algebra. Moreover we wrote a paper on this result and submitted it to Serdical Journal of Computing.(2) The head investigator(Sekiguchi) classified Lie albegras generated by three vectors fields on three dimensional affines space with polynomial coefficients with some conditions. He gave talks at the occasions of RFBR-JSPS Joint symposium(Geometry and Analysis on Complex Algebraic Varieties) held in Moscow and Krasnoyarsk (Russia). He extended the results to the case of some of exceptional singularities due to Arnold and gave a talk at the RFBR-JSPS Joint symposium at RIMS, Kyoto University.(3)J.Sekiguchi studied the action of Weyl group of type E6 on the set of configurations of systems of six lines on a projective plane over a finite prime field. He gave talks at the occasions of International conference (Algebraic Combinatorics) held at Sendai and a small workshop on Sendai number theory and algebraic combinatorics.

在本研究中，我们得到了以下结果：Sekiguchi和T.Fukui研究了实投影平面上八线体系的构型与E8型根系的关系。我们首先定义了具有一定条件的E8型根系的10根集合的总体到具有一定条件的构型的连通分量集合的映射。此外，我们还证明了该映射是W(E8)-等变的。然后利用8次对称群的作用证明了这样的连通分量有2160个轨道。T.Fukui在第十二届计算机代数应用国际会议上发表了关于这一结果的演讲。我们还就这一结果写了一篇论文，并将其提交给了Journal of Computing。(2)首席研究员（Sekiguchi）在一定条件下对三维仿射空间上由三个矢量场生成的多项式系数李氏白图进行了分类。他曾在莫斯科和克拉斯诺亚尔斯克（俄罗斯）举行的RFBR-JSPS联合研讨会（复数代数变种的几何和分析）上发表演讲。他将结果扩展到一些由于Arnold引起的特殊奇点的情况，并在京都大学RIMS的RFBR-JSPS联合研讨会上发表了演讲。Sekiguchi研究了E6型Weyl群对有限素数域上射影平面上六直线系统组形集的作用。曾在仙台举行的代数组合学国际会议和仙台数论与代数组合学小型研讨会上发表演讲。