Induced modules over group algebras and induced characters of finite groups

群代数上的归纳模和有限群的归纳特征

• 批准号: 13640008
• 负责人: YAMAUCHI Kenichi
• 金额: $ 2.11万
• 依托单位: Chiba University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640008/
• 关键词: finite group; group algebra; irreducible character; representations of finite groups; induced modules; character ring; modular representations of finite groups; induced characters

The objective of this research project is to study the induced modules over group algebras and the induced characters of finite groups. In the following we summarize our results.First we proved that the Jacobson radical of the character ring of a finite group is zero, by making use of a theorem of B.Banaschewski. (B.Banaschewski, On the character rings of finite groups, Canad.J.Math. 15 (1963), 605-612.) We wrote a paper on this result.(K.Yamauchi, The bulletin of the faculty of education, Chiba Univ. Vol.51 (2003),315-317.)Secondly we concretely constructed the units of infinite order in the character ring of a cyclic group of order p, by making use of units in Z[ω] where Z is the ring of rational integers and ω is a primitive p-th root of unity and p(【greater than or equal】 5) is a prime number, and we also proved that R(G) has units of infinite order, if the order of G/G' is divisible by a prime number p(【greater than or equal】5) where R(G) is the character ring of a finite group G and G' is the commutator subgroup of a finite group G. We wrote a paper on these results which is submitted for publication.Finally we considered to what extent the existence of an isomorphism of R(G) onto R(H) for two finite groups G and H, can influence the modular representations of G and H. We wrote a paper on this theme which will be submitted for publication elsewhere.Other co-workers also have made contribution to this research project and obtained excellent results on their field as well.

本课题主要研究群代数上的诱导模和有限群的诱导特征。首先利用B. Banaschewski的一个定理证明了有限群的特征标环的Jacobson根为零。（B.Banaschewski，On the character rings of finite groups，Canad.J.Math. 15（1963），605-612.）我们就这个结果写了一篇论文。（山内光，教育学部公报，千叶大学，2003年第51卷，第315 -317页）其次，利用Z[ω]中的单位，具体构造了p阶循环群的特征标环中的无穷阶单位，其中Z是有理整数环，ω是本原p次单位根，p（[大于或等于] 5）是素数，我们还证明了R（G）有无穷阶单位，如果G/G'的阶能被素数p（5）整除，这里R（G）是有限群G的特征标环，G'是有限群G的换位子群.最后我们考虑了两个有限群G和H的R（G）到R（H）的同构的存在性在多大程度上影响G和H的模表示。我们已经就此课题撰写了一篇论文，并将在其他地方发表。其他同事也为这个研究项目做出了贡献，并在他们的领域取得了优异的成绩。

项目成果

越谷 重夫: "Conjectures of Donovan and Puig for principal 3-blocks with abelian defect groups"Communications in Algebra. 31 No.5. 2229-2243 (2003)

Shigeo Koshigaya：“Donovan 和 Puig 对于具有阿贝尔缺陷群的主 3 块的猜想”通讯 31 No.5 (2003)。

北詰 正顕(共著): "3-state Potts model, Moonshine vertex operator algebra and 3A-elements of the Monster group"International Mathematical Research Notices. 23. 1269-1303 (2003)

Masaaki Kitazume（合著者）：“3 态 Potts 模型、Moonshine 顶点算子代数和 Monster 群的 3A 元素”国际数学研究通知 23. 1269-1303 (2003)。

Kenichi Yamauchi: "On the Jacobson radical of the character ring of a finite group"The Bulletin of the Faculty of Education, Chiba University. 51. 315-317 (2003)

山内健一：“论有限群字符环的雅各布森根式”千叶大学教育学部通报。

Ken-ichi Maruyama: "Stability properties of maps between Hopf spaces"The Quarterly Journal of Mathematics, Oxford. Second Series.. 53. 47-57 (2002)

Ken-ichi Maruyama：“Hopf 空间之间映射的稳定性特性”《数学季刊》，牛津。

Masaaki Kitazume (with M.Miyamoto): "3-transposition automorphism groups of VOA, in "Finite Groups Theory and Combinatorics in honor of Michio Suzuki""Advanced Studies in Pure Mathematics. 32. 315-324 (2001)

Masaaki Kitazume（与 M.Miyamoto）：“美国之音的 3 转置自同构群，在“有限群理论和组合学，纪念铃木道夫”中，“纯数学高级研究”。