Modular Representation Theory and Categorification with Applications

群是在对称性研究中出现的数学对象。这个项目的主要焦点是群的一些最基本和最普遍的例子：对称群，它是由有限集合的对称性产生的，以及一般线性群，它是由有限维向量空间的对称性产生的。表示论通过群对其他数学对象的作用来研究群，例如向量空间。非常非正式地，组的表示是从不同方向拍摄的组的快照。在过去的几年里，范畴化的思想变得非常重要，并导致了更高表征理论的发展。这涉及到群体在称为范畴的更高数学结构上的行动，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)。特别地，Khovanov-Lauda-Rouquier(KLR)代数编码了很大一部分表示理论背后的更高的对称性，包括对称和一般线性群等经典对象。这个项目的目标是进一步建立这些代数和其他代数的理论，并应用它来提高我们对群论经典对象的理解。该项目的研究可能在未来对计算机科学和理论物理产生更广泛的影响。更直接的是，该奖项将通过培训研究生和指导这一领域的年轻研究人员产生重要的教育影响。更详细地说，这个项目涉及李代数、有限群和相关对象的表示理论中的几个不同的项目，如Hecke代数、量子群、Schur代数和KLR代数。我们的观点借鉴了更高表征理论的最新进展，即范畴化，各种图解定义的一元范畴和2-范畴发挥了突出的作用。另一方面，它在表示论中的许多经典问题，如有限群的块论、Schur代数以及有限群的结构理论中都有许多应用。我们将研究到导出等价的Schur代数块的局部刻划，KLR代数的尖端代数，粗Heisenberg范畴，超Kac-Moody 2-范畴及其在对称群的双覆盖块上的应用，KLR代数的同调性质，分解数，以及拟单群到子群的不可约限制。该项目将应用到其他数学领域，包括有限群论(及其应用)、李理论、组合学、表示理论、纽结理论和范畴理论。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Groups are mathematical objects arising in the study of symmetry. The main foci of this project are some of the most fundamental and universal examples of groups: symmetric groups, which arise as symmetries of finite sets, and general linear groups, which arise as symmetries of finite-dimensional vector spaces. Representation theory studies groups via their actions on other mathematical objects, such as vector spaces. Very informally, representations of a group are snap-shots of the group taken from different directions. In the past few years, the idea of categorification has become very important and has led to the development of higher representation theory. This involves actions of groups on higher mathematical structures called categories, utilizing not only the relations between these structures (functors) but also relations between the relations (natural transformations). In particular, Khovanov-Lauda-Rouquier (KLR) algebras encode higher symmetries underlying a large part of representation theory, including classical objects like symmetric and general linear groups. The goal of this project is to further build the theory of these and other algebras and apply it to improve our understanding of the classical objects of group theory. The research in this project has potential future broader impacts in computer science and theoretical physics. More directly this award will have important educational impact through the training of graduate students and mentoring young researchers in this area. In more detail, this project is concerned with several diverse projects in representation theory of Lie algebras, finite groups, and related objects such as Hecke algebras, quantum groups, Schur algebras and KLR algebras. Our perspective draws on recent advances in higher representation theory, namely categorification, with various diagrammatically defined monoidal categories and 2-categories playing a prominent role. On the other hand, many applications are to classical problems in representation theory such as block theory of finite groups and Schur algebras as well as structure theory of finite groups. We will study local description of blocks of Schur algebras up to derived equivalence, cuspidal algebras for KLR algebras, thick Heisenberg categorification, super Kac-Moody 2-categories and applications to blocks of double covers of symmetric groups, homological properties of KLR algebras, decomposition numbers, and irreducible restrictions from quasi-simple groups to subgroups. The project will have applications to other areas of mathematics including finite group theory (and its applications), Lie theory, combinatorics, representation theory, knot theory and category theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.