RESEARCH ON THE MODULAR REPRESENTATIONS OF FINITE GROUPS

(1)有限群上置换模的自同态环的研究Tsushima结合Iwahori-Hecke代数，建立了对称群的模表示的一些结果。特别是，他使用Young图上称为分支的操作构造了spect模块的一些简单组成部分。准确地说，将原来只对棒状分支型成立的Carter-Payne定理推广到柱状分支型。每个Specht模块都有简单的组成部分，其对应的Young图是原Young图的分支。此外，对q-Schur代数的中山猜想给出了一个完整的证明，这是因为James和Mathas对该猜想的原始证明中有一个缺口。(2)有限群的不可分解模的研究watanabe证明了当所考虑的群有一个素数阶自动机的阿贝尔Sylow p子群时，对于主p块的Alperin猜想是成立的，从而导出了Broue猜想在完全等距上的有效性。(3) Lie型群的表示理论kaneda在单连通的简单代数群的上同群上建立了Andersen-Haboush定理的量子类似，同时得到了Kempfs消失定理的量子类似。

(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.