For the solution of Broue's conjecture in representation theory of finite groups

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The head investigator Koshitani has gotten the following 3 results in this research.First, we get a result that if our block is the principal 3-block with elementary abelian defect group of order 9 of an arbitrary finite group, then Broue's conjecture holds anytime. This may be one of the most important results on the conjecture. This was published in the "Journal of Algebra" (Elsevier.).Because of the above result, it should be natural to consider Broue's conjectre as the next step for non-principal 3-block of finite groups with the same elementary abelian defect group of order 9. We did it for other concrete finite groups, say O'Nan's simple group and the Higman-Sims simple group which are sporadic, namely we proved Broue's conjecture is still true for non-principal 3'blocks of these finite simple groups with the same defect group. The result was published in the "Journal of Pure and Applied Algebra" (Elsevier).There are conjectures of Donovan and Puig, both of which are closely related to Broue's abelian defect group conjecture. As the third result in this research, we proved that these two conjectures hold for all principal 3-blocks of finite groups with the same Sylow 3-subgroups which are elementary abelian of order 9. This result was published in a journal "Communications inAlgebra (Dekker)".The head investigator Koshitani announced these 3 results in University of Durham, UK 2002,Torun University, Poland 2003,Banff Science Institute, Canada 2003 and also University of Oxford, UK 2004,as invited talks.We have had another important results due to the other investigators Kitazume and Matsuda, for instance.