A Study on Broue's Conjecture in Representation Theory of Finite Groups

在本课题的研究中，我们研究了Broue的一个猜想：有限群的交换亏群的块代数与其Brauer对应块代数等价。取得了如下成果：1.给出了构造两个对称代数的倾斜复形的方法，使它们稳定等价于Morita型。作为我们方法的应用，我们证明了对于具有9阶初等交换Sylow 3-子群的有限群，如Matieu群M11，M21（=SL（3，4）），M22，M23和Hignan-Sims群HS的主3-块，上述结果也成立.我们还可以检验群SL（2，q）在定义特征上的主块猜想.有许多块的例子是不知道是稳定等价于它的布劳尔对应。研究了Janko最小群的主2-块。利用Rickard给出的SL（2，4）的极好的倾斜复形，我们可以证明这个猜想对这个块是成立的.我们还可以确定群U（3，q）的分解数中的未知参数。我们相信我们的结果可以用来研究这个群的猜想.与Broue猜想相关的是块代数的张量分解。利用可分代数理论和块源代数的Puigis理论，得到了关于这一问题的一些结果。

In the research of this project, we studied a conjecture by Broue which says that a block algebra with abelian defect group of a finite group is derived equivalent to its Brauer correspondent. And we obtained good results as follows :1. We could give some methods to construct tilting complexes for two symmetric algebras which are stably equivalent of Morita type. As applications of our methods, we could show that the conjectures are true for principal 3-blocks of some finite groups with elementary abelian Sylow 3-subgroup of order 9, for example, Matieu groups M11, M21(=SL(3,4)), M22,M23 and Hignan-Sims group HS. We could also checked the conjecture for the principal blocks of the groups SL(2,q) over defining characteristic.2. There are many examples of blocks which are not known to be stably equivalent to its Brauer correspondent. We studied the principal 2-block of the smallest group of Janko. Using the splendid tilting complex for SL(2,4) given by Rickard , we could show that the conjecture is true for this block. We could also decide unknown parameters in the decomposition numbers of the group U(3,q) . We believe that our result can be used to study the conjecture for this group.3. Related to Broue's conjecture, there have been studied on tensor decompositions of blocks as algebras. We obtained some result on this problem by making use of theory of separable algebras and Puigis theory of source algebras of blocks.