Investigations on Alperin's Weight Conjecture by means of Auslander-Reiten Quivers

这一建议定位于有限群和有限维代数的模表示理论，并着重于1980年代末提出的Alperin的权猜想。这个猜想将局部结构——这里所谓的权重——与整体结构——群代数的简单模块——联系起来。由于正在进行的研究，我们已经能够证明在群代数的单模和某置换模的自同构环的单子模（均取同构）之间存在双射。在自同态环的一个子集上，利用偏序的极大元实现了双射。在这个建议中，我们的目标是在相同偏阶的权重和最小元素之间的双射。这个想法是基于对大量计算实验的分析。通过Auslander-Reiten颤抖理论的理论方法似乎很有希望。

This proposal is located in the modular representation theory of finite groups and finite dimensional algebras and focuses on Alperin’s weight conjecture, stated in the late 1980’s. This conjecture puts local structures - here the so called weights - into relation with global structures - the simple modules of a group algebra. As a result of ongoing research, we have been able to show the existence of a bijection between the simple modules of a group algebra and the simple submodules (both taken up to isomorphism) of the endomorphism ring of a certain permutation module. The bijection has been realised by means of maximal elements with respect to a partial order on a subset of the endomorphism ring. Within this proposal we are aiming at a bijection between weights and minimal elements with respect to the same partial order. The idea is based on the analysis of a substantial number of computational experiments. A theoretical approach via the theory of Auslander-Reiten quivers seems very promising.