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

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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.