Representations of finite groups and Lie algebras - from combinatorics to categorification

这一建议的主要主题是发现、解释和探索有限群的表示理论和李理论之间的联系。这些联系出现在许多不同的水平上：Grothendieck群水平上的数值不变量之间的组合重合，分级Grothendieck群水平上这些数值不变量的量化，以及范畴水平上的Morita派生或稳定等价。它在有限群、有限维半单李代数、代数群和超群的结构和表示理论以及量子群和无限维李代数的结构和表示理论上也有很多应用。提案中详细介绍了八个具体项目，其中许多项目通过分类的想法联系在一起。该提案中的几个项目涉及有限W-代数，它在过去几年里已经成为令人惊讶的不同数学和数学物理分支中的重要对象。也有项目在不同的语境下研究块和它们之间的等价性，包括布鲁的猜想。最后有一个项目讨论了它在Aschbacher极大子群规划中的应用。一般说来，有限群论和李论主要研究复杂结构的对称性。表象理论的思想是通过研究这些对称在已被理解的较简单结构中的阴影来理解这些复杂的对称。这个提议中的问题的动机之一是用基本范畴之间的等价性来解释各种完全不同的对象的表示理论之间的数值重合，例如有限群代数块和李代数。这种观点自然而然地导致了分类的思想，在这种思想中，组合不变量被范畴不变量所取代。通过延吉安和有限W-代数等对象，也有许多应用于数学和数学物理的其他领域。

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.