Classical Problems in Group Theory

本文研究离散群的经典问题及其应用。在这项研究中，研究人员获得了以下结果。这些结果将按顺序排列和公布。在有限群的交叉伯恩赛德环上，（a）发现了它与群代数的量子对偶的关系;（B）证明了基本定理（嵌入到某个群的积中）;（c）得到了一个幂等公式，并将其应用于经典问题.我们已经安排他们作为预印本（交叉G-集和交叉伯恩赛德环），并在一些会议（西雅图，山形，草津）上发表演讲。2.在我们的经典问题和拓扑量子场论（TQFT）之间的关系，我们检查了Dijkgraaf-Witten不变量，在某些情况下，几乎代数整数。例如，一个3-环面的不变量肯定是一个有理数。此外，我们有一个弱的结果，循环规范群的情况下，然而，在这种情况下，原来的猜想进行了修订。这些声明将被发现在议事录的座谈会代数举行山形。在Schur函数上得到了许多重要的结果，特别是与仿射李代数有着深刻的联系。这些结果在明尼阿波利斯举行的组合学会议上发表。研究人员在与我们的项目相关的其他一些领域也有很多成果：环理论，真实的代数几何，monoidal范畴理论（熊本），动力系统与直觉逻辑之间的关系（札幌）.我们用这笔资金购买了一台工作站和一台个人电脑，用来运行一些公式处理程序（GAP，Mathematica）。

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).