RESEARCH ON THE MODULAR REPRESENTATIONS OF FINITE GROUPS

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(1) Study of endomorphism rings of permutatuion modules over finite groupsIn connection with the Iwahori-Hecke algebras, Tsushima has established some results on the modular representations of symmetric groups. In particular he constructs some simple constituents of the Specht modules using the operation on the Young diagram called branch. To be precise, Carter-Payne's theorem which is known to be true only for bar branch type is extended to pillar branch type. Also it is shown that each Specht module has simple constituents whose corresponding Young diagrams are branches of the original Young diagram. Moreover a complete proof has been given to the Nakayama conjecture for the q-Schur algebras, which is done because the original proof to the conjecture given by James and Mathas contains a gap.(2) Study of indecomposable modules of finite groupsWatanabe has shown that Alperin conjecture is true for the principal p-block if the group under consideration has an abelian Sylow p-subgroup with automizer of prime order, which induces the validity of Broue's conjecture on perfect isometry.(3) Representation theory of groups of Lie typeKaneda has established the quantum analogue of Andersen-Haboush's theorem on the cohomology group of the simply connected simple algebraic groups, which yields at the same time the quantum analogue of Kempfs vanishing theorem.