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Character Correspondents in Finite GroupsTwo of the most important conjectures in the representation theory of finite groups have been recently proven: Richard Brauer’s Height Zero Conjecture (1955) and John McKay’s Degree Conjecture (1971). The proofs of both rely on the Classification of Finite Simple Groups. A key idea in their resolution has been to associate, to each irreducible character of a simple group, another character of a local subgroup that, remarkably, behaves in a corresponding way. This was proposed by M. Isaacs, G. Malle and this author in 2007.In the case of the McKay Conjecture, it took nearly 20 years to fully verify this correspondence, a milestone finally achieved by M. Cabanes and B. Späth. A similar strategy was applied to Brauer’s Height Zero Conjecture for the prime 2 by L. Ruhstorfer, after the so called Gluck--Wolf--Navarro--Tiep theorem was proved, and a reduction of B. Späth and this author. For odd primes, however, G. Malle, A. Schaeffer-Fry, P. H. Tiep, and the present author had to develop an entirely different approach to finally establish the result.This idea of character correspondents is now being extended to tackle other global/local conjectures, including Alperin’s Weight Conjecture, several refinements of the McKay Conjecture, and even Feit’s Conjecture on fields of values. The purpose of this talk is to explain these developments in greater detail.Uploaded 06-10-26 Learn more about your ad choices. Visit megaphone.fm/adchoices
