Monoidal Categories and Categorification in Classical Representation Theory

群是对称研究中出现的基本数学对象。一些最重要和最普遍的例子包括对称群，它们产生于有限集合的对称性，和一般线性群，它们产生于有限维向量空间的对称性；这两个例子都是本项目的重点。经典表示理论研究群（和相关代数），关注它们如何作用于其他数学对象，如拓扑空间。非正式地说，表示是从特定有利位置获取的底层组（或代数）的快照。在过去十年左右的时间里，分类的概念在这一领域变得极其重要，并导致了现在被称为高级表征理论的发展。这涉及到群在称为范畴的数学结构上的行为，不仅利用这些结构之间的关系（函子），而且利用这些关系之间的关系（自然变换）。这是一个新兴的领域，在数学内外都有许多应用，包括有限群论、环论、组合学、李论、范畴论、表示论、结论，甚至计算机科学和理论物理。本课题将从对称群、一般线性群以及相关的李代数和李超代数等方面研究表示理论中出现的一些重要而丰富的范畴。该项目反映了最近高等表征理论的趋势。特别强调一元范畴和绝对的Kac-Moody作用。同时，该项目也深深植根于表征理论中的重要经典问题，如有限群的Broue的Abelian缺陷群猜想，以及有限单群的极大子群的研究。该项目的几个部分涉及表征理论中自然发生的类别块之间的衍生等价研究，并利用衍生类别上的编织群作用。该项目的一部分是更基础的性质，在一个新的方向上发展最高权重类别的一般理论框架。

Groups are fundamental mathematical objects that arise in the study of symmetry. Some of the most important and universal examples include symmetric groups, which arise from symmetries of finite sets, and general linear groups, which arise from symmetries of finite-dimensional vector spaces; both of these examples are a particular focus for this project. Classical representation theory studies groups (and related algebras) by focusing on how they act on other mathematical objects, such as topological spaces. Informally, a representation is a snap-shot of the underlying group (or algebra) taken from a particular vantage point. In the last decade or so, the idea of categorification has become extremely important in this general area and has led to the development of what is now known as higher representation theory. This involves actions of groups on mathematical structures called categories, utilizing not only the relations between these structures (functors) but also relations between these relations (natural transformations). It is a burgeoning area with many applications inside and outside of mathematics, including finite group theory, ring theory, combinatorics, Lie theory, category theory, representation theory, knot theory, and even computer science and theoretical physics.This project will study some important and rich categories appearing in representation theory arising from symmetric groups, general linear groups, and related Lie algebras and Lie superalgebras. The project reflects the recent trend towards higher representation theory. There is special emphasis on monoidal categories and categorical Kac-Moody actions. At the same time, the project is firmly rooted in important classical problems in representation theory, such as Broue's Abelian Defect Group Conjecture for finite groups, and the study of maximal subgroups of finite simple groups. Several parts of the project are concerned with the study of derived equivalences between blocks of naturally occurring categories in representation theory, and exploit braid group actions on derived categories. One part of the project is of a more foundational nature, developing the general theoretical framework of highest weight categories in a new direction.