Classical Problems in Group Theory

本研究的目的是研究离散群的经典问题及其应用。在本研究中，研究人员获得了以下结果。这些结果将按顺序排列和公布。在有限群的交叉Burnside环上，(A)我们发现了它与群代数的量子对偶的关系；(B)我们证明了基本定理(嵌入到某些群代数的乘积中)；(C)我们得到了一个幂等公式并将它应用于经典问题。我们已经将它们安排成预印本(交叉G-集和交叉Burnside环)，并在一些会议(西雅图、山形、草津)上作了演讲。在我们的经典问题和拓扑量子场论(TQFT)之间的关系上，我们检验了Dijkgraaf-Witten不变量在某些情况下几乎是代数整数。例如，3-环面的不变量肯定是有理整数。此外，对于循环规范群的情形，我们得到了一个弱的结果；然而，在这种情况下，我们不得不修正原来的猜想。这些声明将在山形举行的代数研讨会的会议记录中找到。我们得到了许多关于Schur函数的重要结果，特别是与仿射李代数有很深的联系。这些结果是在米尼阿波利斯举行的组合学会议上发表的。研究人员在环论、实代数几何、么半群范畴理论(熊本)、动力系统与直觉逻辑之间的关系(札幌)等其他领域也取得了许多成果。用购买设备的资金，我们购买了一台工作站和一台个人计算机，用于运行一些公式处理程序(GAP、MATHEMICA)。

The purpose of this reserch was the study of classical problems for discrete groups and its applications. In this research, the investigators obtained the following results. These results will be arranged and published in order.1. On the crossed Burnside ring of a finite group, (a) we discovered its relation with the quantum double of the group algebra ; (b) we proved the fundamental theorem (an embedding into the products of some group alegabras) ; (c) we obtained an idempotent formula and applied it to the classical problems. We have arranged them as a preprint (Crossed G-sets and crossed Burnside rings) and gave lectures on them in some conferences (Seattle, Yamagata, Kusatsu).2. On a relationship between our classical problems and Topological Quantum Field Theory (TQFT), we checked that Dijkgraaf-Witten invariants are, in some cases, almost algebraic integers. For example, the invariant for a 3-torus is surely a rational integer. Furthermore, we have a weak result for cyclic gauge group case ; however, in this case, the original conjecture had to be revised. These statements will be found in the proceeding of Symposium on Algebra held in Yamagata.3. We obtained many important results on Schur functions, especially a deep connection with affine Lie algebras. These results was expressed in a conference on combinatorics held in Mineapolis.4. Investigators have a lot of results in some other area which related with our project : ring theory, real algebraic geometry, theory of monoidal categories (Kumamoto), a relationship between dynamical system and intuitional logic (Sapporo).5. Using the funds for equipment, we purchased a workstation and a personal computer, which were used to run some formula manipulation programs (GAP,Mathematica).