On the structure of a character ring of a finite group

本课题的目的是研究有限群的特征环的结构。下面我们总结一下我们的结果。首先研究了有限群的特征环的单位群的结构。由于已知字符环上的有限阶单位，我们考虑了字符环上的无限阶单位。实际上，我们可以在可解群的字符环中构造无限阶的单位。我们打算就这个结果写一篇论文。其次，研究了有限群普通表示中的归纳法定理。利用麦基分解定理，对格林的一个定理作了另一个证明，并写了一篇论文。（K.Yamauchi， J.A.Green定理的另一个证明，Journal of Algebra, 235(2000), 829-832）第三，我们考虑了Weidman和Saksonov关于有限群的Brauer特征环的一个定理的推广。我们得到了有用的新结果，并写了一篇论文。(山内克：《论布劳尔字环对另一个字环的同构》，j。最后，我们考虑了艾萨克斯关于有限群表示的扩展定理的推广。我们得到了一些新的结果，并写了一篇论文。（K.Yamauchi, On the extension of Group Representations over Arbitrary Fields II, the Bulletin of Education，千叶大学学报48(2000),11-19）其他同事也对该研究项目做出了贡献，并在各自的领域取得了优异的成果。

The objective of this research project is to study the structure of a chracter ring of a finite group. In the following we summarize our results.First we studied the structure of the units group of a character ring of a finite group. Since the units of finite order in a chracter ring are well known, we considered the units of infinite order in a character ring. Actually we were able to construct units of infinite order in a character ring of a solvable group. We are going to write a paper on this result.Secondly we studied induction theorems in ordinary representations of finite groups. By using Mackey decomposition theorem we have obtained another proof of a theorem of J.A.Green and written a paper. (K.Yamauchi, Another proof of a theorem of J.A.Green, Journal of Algebra 235 (2000), 829-832)Thirdly we considered a generalization of a theorem of Weidman and Saksonov with respect to a Brauer character ring of a finite group. We have obtained useful and new results and written a paper. (K.Yamauchi, On isomorphisms of a Brauer character ring onto another, J.Alg. (to appear))Finally we considered a generalization of a theorem of I.M.Isaacs about the extensions of representations of finite groups. We have obtained some new results and written a paper. (K.Yamauchi, On the Extensions of Group Representations over Arbitrary Fields II, The Bulletin of the Faculty of Education, Chiba Univerrsity 48 (2000), 11-19)Other co-workers also have made contribution to this research project and obtained excellent results on their field as well.