Endomorphismenringe von Permutationsmoduln auf den Nebenklassen einer p-Sylowgruppe
- Endomorphism rings of permutation modules on the cosets of a sylow p-subgroup
Naehrig, Natalie; Hiß, Gerhard (Thesis advisor)
Aachen : Publikationsserver der RWTH Aachen University (2008)
Dissertation / PhD Thesis
Aachen, Techn. Hochsch., Diss., 2008
When discussing Alperin's weight conjecture, J. L. Alperin himself has proposed to analyse endomorphism rings of permutation modules of groups on the cosets on its Sylow-p-subgroup. This thesis contributes to this approach. At first, fundamental elements of the ordinary and modular representation theory of such endomorphism rings are analyzed and compared with the corresponding structures of the underlying group. This enables us to derive criterions for projective simplicity for ordinary characters of the endomorphismring when reducing modulo a prime. Similar to the case of group algebras, these criterions are based on the ordinary character table. In the second part of this thesis we concentrate on Alperin's weight conjecture and discuss an equivalent assertion when the endomorphism ring is assumed to be quasi-Frobenius. Under additional assumptions on the relevant vertices and their normalizers, we are able to prove Alperin's weight conjecture in this case. Finally, the computational results are being discussed. The substantial amount of examples has revealed a pattern which associates the equivalence classes of weights and the isomorphism classes of simple modules of the group algebra via the isomorphism classes of the simple constituents of the endomorphism ring when regarded as a regular module. Indeed, with one exception (M11 modulo 3) in all examples, which have been analyzed, the number of such isomorphism classes of simple constituents of the socle is on the one hand equal to the number of equivalence classes of weights and on the other hand equal to the number of isomorphism classes of simple modules of the group algebra. This possibly permits a new approach of a proof of Alperin's weight conjecture for a class of groups which M11 modulo 3 does not belong to. In the last part of this thesis, all results of all analyzed examples are collected. Those results include the ordinary character table - as long as the example did not exceed the available memory space -, the Cartan- and decomposition matrices of the endomorphism rings, the socle- and radical constituents, and the determination of the Green-correspondents of weight modules.