Actions of semisimple groups and Weyl groups and research on representations

研究结果如下：(1)J.Sekiguchi和T.Fukui研究了实射影平面上八直线系统的构型与E8型根系之间的关系。我们首先定义了具有一定条件的E8型根系的10个根的集合到具有一定条件的配置的连通分支集的全映射。此外，我们还证明了该映射是W(E8)-等变的。在8次对称群的作用下，证明了此类连通分支的轨道数为2160。T.Fukui在第12届国际计算机代数应用会议上发表了关于这一结果的报告。(2)首席研究员(Sekiguchi)将三维仿射空间上由三个向量场生成的具有一定条件的多项式系数的Lie进行了分类。他在莫斯科和俄罗斯克拉斯诺亚尔斯克举行的RFBR-JSPS联合研讨会(复杂代数簇的几何和分析)上发表了讲话。他将结果推广到Arnold的一些特殊奇点的情形，并在京都大学RIMS的RFBR-JSPS联合研讨会上发表了演讲。(3)J.Sekiguchi研究了E6型Weyl群在有限素域上射影平面上六线系的组态上的作用。他在仙台举行的国际会议(代数组合学)和仙台数论和代数组合学的小型研讨会上发表了演讲。

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.