Constructions of Tilting Complexes and Relative Projective Covers in Representation Theory

在这个项目的研究中，我们研究了群代数模的相对投射覆盖理论，并将其应用于构造群代数上的倾斜复形。我们得到了一些好的结果，如下所述。“相对射影覆盖”是通常的射影覆盖的推广。对于群SL(2，q)，详细研究了投射不可分解模的结构，在定义特征域上完成了对Broue关于这一群的交换缺陷猜想的检验。我们已经确定了奇特征域上群SU(3，Q^2)的分解数中的未知参数。库努吉-瓦基利用这一结果验证了布鲁对这一组的猜想。Sp(4，q)群中出现了与SU(3，q^2)类似的情况。改进了他们的方法，我们可以检验Sp(4，q)的猜想。在研究中，相对于抛物子群的相对射影覆盖是有用的。我们继续对与SU(3，q^2)密切相关的群G_2(Q)进行研究，得到了一些有趣的复形，但不能证明猜想的正确性。利用关于一族模的相对射影覆盖，得到了一些可用于研究小秩群的上同调代数的结果。Carlson对有限群上同调代数中拟正则序列的存在性问题给出了部分回答。

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.