Monoidal Categories and Categorification in Classical Representation Theory

群是对称性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，它产生于有限集合的对称性，以及一般线性群，它产生于有限维向量空间的对称性；这两个例子都是本项目特别关注的焦点。经典表示论研究群(和相关的代数)如何作用于其他数学对象，如拓扑空间。非正式地，表示是从特定的有利位置获取的基础组(或代数)的快照。在过去的十年左右，分类的概念在这一一般领域变得极其重要，并导致了现在所知的更高表征理论的发展。这涉及到群体在称为范畴的数学结构上的动作，不仅利用这些结构(函子)之间的关系，而且利用这些关系之间的关系(自然变换)。它是一个新兴的领域，在数学内外有着广泛的应用，包括有限群论，环论，组合学，李论，范畴论，表示论，纽结理论，甚至计算机科学和理论物理。本课题将研究对称群，一般线性群，以及相关的李代数和李超代数在表示论中出现的一些重要和丰富的范畴。该项目反映了最近更高代表性理论的趋势。其中特别强调了么一范畴和范畴Kac-Moody作用。同时，该项目扎根于表示理论中的重要经典问题，如Broue关于有限群的Abelian亏群猜想，以及有限单群的极大子群的研究。该项目的几个部分涉及到在表示理论中研究自然发生的范畴的块之间的派生等价，并利用派生范畴上的辫子群作用。该项目的一个部分是更具基础性的，以新的方向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.