Cohomology and Representation Theory

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The principal investigator will explore problems in the modular representation theory and cohomology of the symmetric group and related objects, including algebraic groups, Frobenius kernels, Schur algebras and superalgebras, and Iwahori-Hecke algebras. The project includes using connections between the representation theories of these different objects to obtain new results. The ordinary representation theory of the symmetric group is closely related to that of the general linear group through classical work of Schur and others. Recently this relationship has been extended to the modular representation theory, and even more recently to the corresponding cohomology. Although this interplay can lead to new results in either direction, it is especially useful in studying the symmetric group, where many of the basic problems remain completely open. The investigator will continue to refine and improve techniques to exploit this important relationship in order to obtain results in both areas. Recently Parshall and Scott proved that the celebrated Lusztig conjecture, in the case of the general linear group, is equivalent to a problem entirely in the realm of symmetric group representation theory. Meanwhile breakthroughs by Brundan and Kleshchev, Ariki, Grojnowski and others have led to a ``Lie-theoretic" approach to the symmetric group theory which is very different from the original approach of Gordon James and others. Together these developments suggest the next few years will be very exciting indeed in this field.This project is in an area of mathematics known as representation theory, in particular the representation theory of finite groups. Problems involving groups and their representations arises naturally in many diverse fields, including mathematical physics, chemistry, error correcting codes, cryptography and others. In particular, ideas used by mathematical physicists have played an important role in recent progress made in many of the areas described above.