Final Report Summary - COUNTING CONJECTURES (Counting conjectures and characters of almost simple groups)
The Representation Theory of Finite Groups is a thriving subject, with many fascinating and deep open problems, and some recent successes. In 1963 Richard Brauer formulated a list of deep conjectures about ordinary and modular representations of finite groups. These have lead to many new concepts and methods, but basically all of his main conjectures are still unsolved to the present day. The aim of the project was to make breakthrough contributions to these long-standing conjectures.
The project reached almost all of its stated Key Intermediate Goals, and achieved further unforeseen progress. It resulted in the proof of two of the major conjectures on characters of finite groups: in joint work of the PI and his collaborators, we were able to prove the original McKay conjecture on characters of odd degree from 1972. In joint work with a long term research visitor of the group, Professor Radha Kessar, the PI was able to settle one direction of Brauer's height zero conjecture from 1955 on defect groups of blocks. Furthermore, the joint work of the research group with collaborators led to the previously out of reach reduction of all fundamental counting conjectures to statements about simple groups, which can now be studied on the basis of the classification theorem for finite simple groups. A considerable part of the necessary results have already been obtained, thus giving hope for an eventual solution of the remaining conjectures. The project also led to the understanding of the action of outer automorphisms on the cuspidal irreducible characters of quasi-simple groups.
Moreover, the work of the research group let to an almost complete classification of simple endotrivial modules, a goal that had not been invisioned but which could be solved as a byproduct of the other results obtained in the project.
The project reached almost all of its stated Key Intermediate Goals, and achieved further unforeseen progress. It resulted in the proof of two of the major conjectures on characters of finite groups: in joint work of the PI and his collaborators, we were able to prove the original McKay conjecture on characters of odd degree from 1972. In joint work with a long term research visitor of the group, Professor Radha Kessar, the PI was able to settle one direction of Brauer's height zero conjecture from 1955 on defect groups of blocks. Furthermore, the joint work of the research group with collaborators led to the previously out of reach reduction of all fundamental counting conjectures to statements about simple groups, which can now be studied on the basis of the classification theorem for finite simple groups. A considerable part of the necessary results have already been obtained, thus giving hope for an eventual solution of the remaining conjectures. The project also led to the understanding of the action of outer automorphisms on the cuspidal irreducible characters of quasi-simple groups.
Moreover, the work of the research group let to an almost complete classification of simple endotrivial modules, a goal that had not been invisioned but which could be solved as a byproduct of the other results obtained in the project.