Investigations on the conjectures of McKay and Alperin-McKay

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The proposed research project is located in the representation theory of finite groups. Our aim is the investigation of two prominent and long standing open conjectures, the McKay conjecture and its refinement by Alperin, the Alperin-McKay conjecture, respectively. These conjectures were formulated in the mid 70th of the last century. In its original formulation, the McKay conjecture postulates that two seemingly unrelated sets of objects, constructed from the representation theory of a group, have the same number of elements. One of these sets is defined by “local data”, i.e. by data constructed from proper subgroups, the other set only by “global data” of the group itself. That global data should be determined by local data, is the philosophy behind the McKay conjecture and other famous conjectures of representation theory. Two recent developments have inspired the proposed project. Firstly, in 2007, Isaacs, Malle and Navarro published a powerful reduction theorem for the McKay conjecture. This leaves one to verify some rather complicated conditions for the finite simple groups. Secondly, the results of Späth’s 2007 PhD thesis provide a means to verify these complicated conditions in the local situation, at least in special cases.