The main theme of this proposal is to discover, explain and exploit connections between representation theory of finite groups and Lie theory. These connections arise at many different levels: combinatorial coincidences between numerical invariants at the level of Grothendieck groups, quantizations of these numerical invariants at the level of graded Grothendieck groups, and Morita, derived or stable equivalences at the categorical level. There are also many applications, especially to the structure and representation theory of finite groups, finite dimensional semisimple Lie algebras, algebraic groups and supergroups, and to quantum groups and infinite dimensional Lie algebras. There are eight specific projects detailed in the proposal, many of which are linked together by the idea of categorification. Several projects in the proposal are concerned with finite W-algebras, which have emerged in the last few years as important objects in surprisingly diverse branches of mathematics and mathematical physics. There are also projects studying blocks and equivalences between them in various contexts, including Broue's conjecture. Finally there is one project discussing applications to Aschbacher's maximal subgroups program.
Broadly speaking, finite group theory and Lie theory are concerned with studying the symmetry of complicated structures. The idea of representation theory is to understand such complicated symmetries by studying the shadows of these symmetries in simpler structures that are already understood. One of the motivations for the problems in this proposal is to explain observed numerical coincidences between the representation theory of various quite different objects such as blocks of finite group algebras and Lie algebras in terms of equivalences between the underlying categories. This point of view leads naturally to the idea of categorification in which combinatorial invariants are replaced by categorical ones. There are also a number of applications to other areas of mathematics and mathematical physics, via objects such as Yangians and finite W-algebras.
Representation theory is a core topic in mathematics with many connections to other areas of mathematics, mathematical physics, computer science and even biology. In the last few years the subject has been influenced heavily by ideas from higher category theory, leading to a new point of view of categorification. This has produced a new framework for the study of combinatorial representation theory of classical objects like symmetric and general linear groups. Intellectual merit: The projects encompassed by this grant take these classical objects as their starting point, but incorporate the new ideas of categorification into their study. Major results include new versions of Schur-Weyl duality encompassing W-algebras, the discovery of hidden gradings on such classical objects as group algebras of the symmetric group, and the development of new diagrammatic methods to study the general linear supergroup. Broader impacts: This award has had important educational impact through the training of graduate students and the ongoing efforts of both PIs in the training of other young researchers in this area. It has promoted knowledge of the methods and results of this beautiful subject area both nationally and internationally, through an active involvement of both PIs as organizers of major conferences and as editors of leading research journals.
