A Study on Derived Equivalences of Blocks and Shintani Descent in Group Algebras

在本课题的研究中，我们研究了有限群表示理论中的主要问题之一--“阿贝尔亏格猜想”。我们讨论了Shintani后裔和Glauberman对应。1.我们构造了有限小秩Chvalley群的单侧倾斜复形，例如U(3，q^2)、Sp(4，q)、G_2(Q)。我们可以构造与这些复合体重合的华丽倾斜复合体作为单边复合体，并研究由它们诱导的特征环之间的完美等距关系。2.文1中所描述的情况可以被认为是派生范畴版本的Glauberman Corents的问题。我们研究了可解群情形，推广了Harris和Linckelmann的结果。我们还研究了一个问题，给出了不同类型的2阶5-Sylow正规化子的箭图表示，以应用于构造一些有限Chvalley群的倾斜复形，例如^2F_4(Q)，其中5-Sylow正规化子是4S_4型的。

In the research of this project, we studied "the abelian defect conjecture" which is one of main problems in representation theory of finite groups. We were concerned with Shintani descent and Glauberman correspondent.1.We have constructed one sided tilting complexes for finite Chevalley groups of small rank, for example, U(3,q^2)、Sp(4,q)、G_2(q). We could construct Splendid tilting complexes which coincide with these complexes as one side complexes and studied perfect isometries between character rings induced by them.2.The situations described in 1 can be considered problems of derived category version of "Glauberman correspondent". We have studied solvable group case and obtain some generalization of results by Harris and Linckelmann. And we clarify the strategy raised by Rouquier in this setting.3.We also worked on a problem giving quiver presentations for various types of 5-Sylow normalizers of rank 2 to apply for constructing tilting complexes of some finite Chevalley groups, for example, ^2F_4(q) of which 5-Sylow normalizer is of 4S_4-type.