Derived Equivalences, Braid Relations, and Stability Conditions

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Representation theory is the mathematical study of symmetry and of the various ways symmetry manifests itself in nature. A wonderful blend of algebra, geometry, and combinatorics, it enjoys fruitful interactions with physics and chemistry.The proposed research introduces a completely new approach to some fundamental unsolved problems in representation theory, based on modern methods of derived equivalences developed in the recent proof of Broue's conjecture for symmetric groups. This approach applies in particular to Lusztig's famous and influential conjecture on characters of irreducible modules for general linear groups in prime characterisctic, which has inspired major advances in mathematics even outside representation theory proper and continues to be a subject of intense study.The first part of the research concerns extensions and applications of the derived equivalence methods in several directions, including a proof of Broue's conjecture for general linear groups in nondefining characteristic and, most significantly, a uniform proof of the existence of braid group actions on derived categories in a variety of Lie-type representation theories. The braid group actions should provide a foundation around which to build a deeper understanding of the whole family of theories.Thanks to the work on Broue's conjecture mentioned above, the famous numerical conjectures of Lusztig and James can be reformulated in terms of some small and manageable wreath products. In order to exploit this surprising connection, the second part of the research investigates homological properties of these wreath products, using the braid relations appearing in the first part together with an exciting new idea coming from mathematical physics, the stability conditions of Bridgeland and Douglas.