Research on defect groups of blocks of a group ring

群环块缺陷群研究

• 批准号: 10640004
• 负责人: FUKUSHIMA Hiroshi
• 金额: $ 1.09万
• 依托单位: Gunma University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640004/
• 关键词: character; finite group; representation; block; modular; defect; group ring; radical

Blocks of defect zero are easily understood objects in modular representation theory, since they are isomorphic to full matrix rings over the underlying field. Brauer's "Problem 19" asks for group-theoretic criteria to determine when a group has such blocks. In connection with this problem, we proved the following result : let G be a solvable p-nilpotent group for some prime p, such that OィイD2pィエD2(G) = 1. If certain groups is not involved in G, an element in OィイD2p'ィエD2(G) with a centraliser of p'-order is found. Hence G has a p-block of defect zero.Next we proved the following result : Let G be a finite solvable group and p, a prime. Then G has a character of p-defect 0 if and only if G has a chain of subgroups G ⊇ GィイD21ィエD2 ⊇ GィイD22ィエD2 ⊇ … ⊇ GィイD2nィエD2, which satisfies the some conditions. By using the theorem, we can investigate without charactertable whether given groups have blocks of defect zero.

在模表示理论中，缺陷零块是易于理解的对象，因为它们与底层域上的满矩阵环同构。Brauer的“问题19”要求用群体理论标准来确定一个群体何时有这样的街区。针对这一问题，我们证明了以下结果：设G是某素数p的一个可解的p幂零群，使得O O * * D2p * * * D2(G) = 1。如果某些基团不参与G，则在O′′中找到一个具有p′阶集中器的元素。因此G有一个缺陷为零的p块。接着证明了以下结果：设G是有限可解群，p是素数。则当且仅当G有一个子群G的链，且满足若干条件时，G具有p-缺陷0。利用该定理，我们可以无特征地研究给定群是否存在缺陷为零的块。

项目成果

Hiroshi Fukushima: "On the existence of p-blocks of defect 0 in solvable groups"J. Algebra. (to appear).

福岛浩：“关于可解群中缺陷 0 的 p 块的存在性”J.

S.Owa,布川護他2名: "Starlikeness of certain integrals"Mathematical Research Hot-Line. 2(1). 23-26 (1998)

S.Owa、Mamoru Fukawa 等 2 人：“某些积分的恒星相似性”数学研究热线 2(1) 23-26 (1998)。

布川護: "Gamma-starlike関数について"京都大学数理解析研究所講究録. 1112. 44-50 (1999)

Mamoru Fukawa：“关于类伽马星函数”京都大学数学科学研究所 Kokyuroku. 1112. 44-50 (1999)。

Takashi Itoh: "Completely positive decompositions from duals of CィイD1*ィエD1-algebras to von Neumann algebras"Math. Japonica. 51. 89-98 (2000)

Takashi Itoh：“从 C-D1*D1-代数的对偶到冯诺依曼代数的完全正分解”Math。 51. 89-98 (2000)

N.E.Cho. 布川他2名: "Augular estimate of certain analytic functions"Indian Journal of Pure and Applied Mathematics. 29(3). 321-331 (1998)

N.E.Cho. Fukawa 等人 2：“某些分析函数的预测”，《印度纯粹与应用数学杂志》29(3) (1998)。