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，并集中于1980年代末S所提出的Alperin权猜想，这个猜想将局部结构-这里所谓的权-与整体结构-群代数的单模联系起来。作为正在进行的研究的结果，我们已经能够证明在群代数的单模和某个置换模的自同构环的单子模(两者都采用同构)之间存在双射。双射是通过关于自同态环的子集上的偏序的极大元来实现的。在这个建议中，我们的目标是关于相同偏序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抖动理论的理论方法似乎非常有希望。

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.