Constructions of Tilting Complexes and Relative Projective Covers in Representation Theory

表示论中倾斜复形和相对射影覆盖的构造

• 批准号: 12640002
• 负责人: OKUYAMA Tetsuro
• 金额: $ 2.24万
• 依托单位: HOKKAIDO UNIVERSITY OF EDUCATION
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640002/
• 关键词: Represenrations of Finite Groups; Block Algebras; Tilting Complexes; Relative Projective Covers; Broue's Conjecture

In the research of this project, we studied theory of the relative projective covers of modules over group algebras to apply it for construction of tilting complexes over group algebras. And we obtained some good results as we shall describe in the following."Relative projective cover" is a generalization of usual projective cover. For the group SL(2,q), investigating structures of projective indecomposable modules in detail, we completed to check abelian defect conjecture by Broue for this group over the field of defining characteritic. We already decided unknown parameters in decomposition numbers of the group SU(3,q^2) over the field of odd characteritic deviding q^<+1>. Kunugi-Waki used this result to check the conjecture by Broue for this group. A similar situation as SU(3,q^2) occurs in the group Sp(4,q). Improving their method we could checked the conjecture for Sp(4,q). In the investigation, relative projective covers with respect to its parabolic subgroups were useful. We continued the investigation to the group G_2(q) which is closely related to SU(3,q^2) and obtained some interesting complexes, but could not check the conjecture.Using relative projective covers with respect to a family of modules, we obtained some results which can be used to investigate cohomology algebras of finite groups of small rank. And we could give a partial answer to the problem by Carlson on the existence of quasi regular sequences in cohomology algebras of finite groups.

在本课题的研究中，我们研究了群代数上模的相对投射覆盖理论，并将其应用于群代数上倾斜复形的构造。我们得到了一些很好的结果，我们将在下面描述。“相对投射覆盖”是一般投射覆盖的推广。对于SL（2，q）群，通过对投射不可分解模的结构的详细研究，在定义特征模的域上，完成了Broue对SL（2，q）群的Abel亏猜想的验证.我们已经确定了群SU（3，q^2）在奇特征整环q^<+1>上的分解数中的未知参数。Kunugi-Waki用这个结果来检查Broue对这个群的猜想。与SU（3，q^2）类似的情况也发生在群Sp（4，q）中。改进他们的方法，我们可以检验关于Sp（4，q）的猜想.在研究中，相对投射覆盖关于它的抛物子群是有用的。我们继续研究了与SU（3，q^2）密切相关的群G_2（q），得到了一些有趣的复形，但未能验证猜想.利用模族的相对投射覆盖，得到了一些可用于研究小秩有限群的上同调代数的结果.从而部分回答了Carlson关于有限群上同调代数中拟正则序列的存在性问题。

项目成果

M. Kitayama: "Induced vector fields and metric transformation"Applied Sciences, Geometry Balkan Press. 3(1). 27-35 (2001)

M. Kitayama：“诱导向量场和度量变换”应用科学，几何巴尔干出版社。

I. Hasegawa, K. Yamauchi: "Infinitesimal projective transformation on the tangent bundles with the horizontal lift connection"Journal of Hokkaido University of Education. 52(1). 1-5 (2001)

I.长谷川，K.山内：“具有水平升力连接的切丛上的无穷小射影变换”北海道教育大学学报。

M.Kitayama: "Induced vector fields and metric transformation"Applied Sciences, Geometry Balkan Press. 3(1). 27-35 (2001)

M.Kitayama：“诱导向量场和度量变换”应用科学，几何巴尔干出版社。

T. Okuyama, H. Sasaki.: "Relative projectivity of modules and cohomology algebras of finite groups"Algebras and Representation Theory. 4. 405-444 (2001)

T. Okuyama，H. Sasaki.：“有限群的模和上同调代数的相对射影性”代数和表示论。

I.Hasegawa, K.Yamauchi: "Infinitesimal projective, transformation on the tangent bundles with the horizontal lift connection"Journal of Hokkaido University of Education. 52(1). 1-5 (2001)

I.Hasekawa、K.Yamauchi：“无穷小射影、具有水平升力连接的切丛变换”北海道教育大学学报。