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

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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.