Classical Problems in Group Theory

群论中的经典问题

• 批准号: 08454001
• 负责人: YOSHIDA Tomoyuki
• 金额: $ 3.33万
• 依托单位: HOKKAIDO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1996
• 资助国家: 日本
• 起止时间: 1996 至 无数据
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08454001/
• 关键词: finite group; Burnside ring; crossed G-set; Mackey functor; Frobenius theorem; TQFT; monoidal category; modular representaion; Burnside ring; Mackey functor; Frobenius theorem; modular representation

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).

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，研究者获得了以下结果。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二重的关系；(b)证明了基本定理（群代数积的嵌入）；(c)得到了一个幂等公式，并将其应用于经典问题。我们将它们整理成预印本（交叉g集和交叉Burnside环），并在一些会议（西雅图，山形，草津）上发表演讲。在我们的经典问题和拓扑量子场论（TQFT）之间的关系上，我们检查了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数。此外，对于循环规范群，我们得到了一个弱结果；然而，在这种情况下，原来的猜想不得不被修改。这些论述可在山形召开的代数研讨会论文集中找到。我们得到了许多关于舒尔函数的重要结果，特别是与仿射李代数的密切联系。这些结果是在明尼阿波利斯举行的组合学会议上发表的。研究者在其他一些与我们的项目相关的领域有很多成果：环理论，实代数几何，一元范畴理论（熊本），动力系统和直觉逻辑之间的关系（札幌）。使用设备经费，我们购买了一台工作站和一台个人电脑，用于运行一些公式处理程序（GAP,Mathematica）。

项目成果

Hirofumi YAMADA: "Higher Specht polynomials (with S.Ariki, T.Terasoma)" Hiroshima Math.J.27. 177-188 (1997)

Hirofumi YAMADA：“高光谱多项式（与 S.Ariki、T.Terasoma）”Hiroshima Math.J.27。

M. SAITO: "Symmetric algebras of normal A-hypergeometric systems" Hokkaido Math. Journal. 25・3. 591-619 (1996)

M. SAITO：“普通 A 超几何系统的对称代数”北海道数学杂志 25・3（1996）。

Hirofumi YAMADA: "Reduced Schurfunctions and the Littlewood-Richardson coefficient (with S.Ariki, T.Nakajima)" J.London Math.Soc.(in printing).

Hirofumi YAMADA：“约化 Schur 函数和 Littlewood-Richardson 系数（与 S.Ariki、T.Nakajima 合作）”J.London Math.Soc.（印刷中）。

Y. YOSHIDA: "Classical Problems in Group Theory" SUGAKU Expositions. 9・2. 169-186 (1996)

Y. YOSHIDA：“群论中的经典问题”SUGAKU Expositions 9・2（1996）。

H. YAMADA: "Higher Specht polynomials" Hiroshima Math. J.27・4. 177-188 (1997)

H. YAMADA：“高光谱多项式”广岛数学 J.27・4（1997）。