Global/Local Conjectures in Representation Theory of Finite Groups

Group Theory is essentially the theory of symmetry for mathematical and physical systems, with major impact in diverse areas of mathematics. The Representation Theory of Finite Groups is a central area of Group Theory, with many fascinating and deep open problems, and significant recent successes. In 1963 R. Brauer [B] formulated a list of deep conjectures about ordinary and modular representations of finite groups. These conjectures have led to many new concepts and methods, but basically all of his main conjectures are still unsolved to the present day. A new wealth of difficult problems, relating global and local properties of finite groups, was opened up in the seventies and eighties (of the 20th century) by subsequent conjectures of J. McKay [Mc], J. Alperin [A2], M. Broué [Br], and others, all remain open up to date. The classification of finite simple groups raised the hope that one should be able to reduce some of the aforementioned conjectures to statements about simple groups, and subsequenly establish these statements by exploring deep knowledge about simple groups provided by the Deligne-Lusztig theory and other recent fundamental results in group theory. This hope has recently materialized when the first three of the above mentioned conjectures have been reduced to questions about finite simple groups (see M. Isaacs, Malle, and G. Navarro [IMN], Navarro and P.H. Tiep [NT2], Navarro and Späth [NSp], and B. Späth [Sp2, Sp3]). Since then, further substantial progress on some of the remaining steps towards proving these conjectures has been achieved.