Modular Representation Theory and Categorification with Applications

模块化表示理论及其分类及其应用

• 批准号: 2101791
• 负责人: Alexander Kleshchev
• 金额: $ 27.05万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101791&HistoricalAwards=false
• 关键词: Modular; Representation; Theory; Categorification; Applications

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.

群是对称研究中产生的数学对象。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，作为有限集合的对称性，和一般线性群，作为有限维向量空间的对称性。表示理论通过群体在其他数学对象（如向量空间）上的行为来研究群体。非正式地说，一个群体的表现是从不同方向拍摄的群体快照。在过去的几年里，分类的思想变得非常重要，并导致了高级表征理论的发展。这涉及群在称为范畴的高等数学结构上的行为，不仅利用这些结构之间的关系（函子），而且利用关系之间的关系（自然变换）。特别是，Khovanov-Lauda-Rouquier （KLR）代数在表征理论的大部分基础上编码了更高的对称性，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。该项目的研究在计算机科学和理论物理方面具有潜在的更广泛的影响。更直接地说，该奖项将通过培训研究生和指导该领域的年轻研究人员产生重要的教育影响。更详细地说，本项目涉及李代数、有限群和相关对象（如Hecke代数、量子群、Schur代数和KLR代数）的表示理论中的几个不同项目。我们的观点借鉴了高级表征理论的最新进展，即分类，各种图表定义的单类和2类起着突出作用。另一方面，许多应用是在表示理论中的经典问题，如有限群的块论和Schur代数，以及有限群的结构理论。我们将研究Schur代数块的局部描述，直至导出等价，KLR代数的倒代数，厚Heisenberg分类，超Kac-Moody 2-范畴及其在对称群双盖块上的应用，KLR代数的同调性质，分解数，以及从拟单群到子群的不可约限制。该项目将应用于其他数学领域，包括有限群论（及其应用）、李论、组合学、表示论、结论和范畴论。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。