Actions of semisimple groups and Weyl groups and research on representations

我们在这项研究中获得了以下结果。（1）J.Sekiguchi和T.福井研究了在真实的投影平面上的八条线系统的构形与E8型根系之间的关系。我们首先定义了E_8型根系在一定条件下的10个根的集合之和到一定条件下构型的连通分支集合的映射。此外，我们还证明了该映射是W（E8）-等变的.然后证明了在8次对称群的作用下，这类连通分支的轨道数为2160。T.福井在第12届计算机代数应用国际会议上就这一结果发表了讲话。此外，我们写了一篇关于这个结果的论文，并将其提交给Serdical Journal of Computing。(2)首席研究员（关口）分类李雅格拉斯产生的三个向量场的三维仿射空间多项式系数的一些条件。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（复杂代数变种的几何和分析）上发表演讲。他将结果扩展到Arnold的一些特殊奇点的情况，并在京都大学RIMS的RFBR-JSPS联合研讨会上发表了讲话。（3）J.Sekiguchi研究了E6型Weyl群对有限素域上射影平面上六线系的构形集的作用。他发表讲话的场合国际会议（代数组合学）在仙台举行的一个小型讲习班仙台数论和代数组合学。

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.