Applications of Cohomology groups and Shintani descent to Broue's conjecture

1（1）对于D型和F型低秩有限Chevalley群G，当p=5时，我们构造了G的主块与Sylow正规化子之间的完全等距.我们还确定了等距线的数量。（2）对于G型有限Chevaley群，当p=3时，我们建立了几个定义域的主块之间的Morita等价。(3)We证明了Broue猜想对具有9阶初等交换亏群的块成立。(4)We得到了对称群的Cartan矩阵的初等因子的一个简便公式。(5)We给出了群融合的Mislin定理的模理论证明。（6）将群上同调的基本概念推广到任意块2。（1）推广了Dade猜想，并检验了它对许多散在单群和有限Chevaley群成立。（2）对于具有八阶二面体亏群的块，我们在根链的正规化子的导出范畴之间构造函子，由此导出单模个数之间的关系。

1 (1)For low rank finite Chevalley groups G of type D and F, and for p=5, we constructed a perfect isometry between the principal blocks of G and a Sylow normalizer. We also determine the number of isometries. Moreover, for blocks of cyclic defect groups, we proved that for any bijection between irreducible characters there is a derived equivalence inducing the bijecyion, provided that the local structure of the blocks are the same,(2)For finite Chevaley groups of type G, and for p=3, we establish certain Morita equivalence between principal blocks for several defining fields.(3)We proved that Broue's conjecture holds for block with elementary abelian defect groups of order 9.(4)We have obtained an easy formula for elementary divisors of the Cartan matrices of symmetric groups.(5)We gave a module theoretic proof for the Mislin's theorem on group fusions.(6)The fundamental notions of group cohomologies are extended for arbitrary blocks.2 (1)We extend Dade's conjecture, and check that it holds for many sporadic simple groups and some finte Chevaley groups.(2)For a block with dihedral defect groups of order eight, we construct functors among the derived categories of normalizers of radical chains which induce the relations between the numbers of simple modules.