Categorical and Diagrammatic Representation Theory

分类和图解表示理论

• 批准号: 2201387
• 负责人: Benjamin Elias
• 金额: $ 27.7万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2201387&HistoricalAwards=false
• 关键词: Categorical; Diagrammatic; Representation; Theory; Categorical; Diagrammatic; Representation; Theory

Representation theory is the study of symmetries. Symmetry groups arise frequently in physics (e.g. rotations of a sphere), chemistry (e.g. crystallography), and other scientific fields. Data related to the objects possessing symmetry can often be encoded in an object called a representation. Mathematicians study the relationships between representations, and how bigger representations can be built from simple, indivisible ones, much as a molecule is built from indivisible atoms. Many properties of these simple representations, such as their dimensions, are unknown and the topic of intense research. The representations and their structure can be packaged in a collection called a category. An extremely fruitful tool of the last half century has been to identify categories in representation theory with categories from algebraic geometry, allowing the use of powerful geometric tools. But geometry also has its limits, especially when it comes to matters of explicit computation. In past work, the PI has found new and explicit descriptions of categories from representation theory and geometry, using diagrammatic methods. In diagrammatics, a very large matrix or a structure from geometry could be encoded as a picture and manipulated graphically. These descriptions make once-difficult categories accessible computer algebra systems. Computer calculations performed by the PI's collaborator Williamson have led to the first breakthroughs in computing dimensions of simple representations in decades. The PI will continue to develop diagrammatic methods to study representation theory and geometry, providing explicit constructions of new categories, structures, and tools which are beyond the current scope of other approaches. This project provides research training opportunities for students.More concretely, this proposal will support four related projects. The first is to provide general tools for studying generically semisimple monoidal categories diagrammatically using a cellular basis called the branching basis, akin to several bases previously constructed by the PI and collaborators. These tools will then be applied to the categories of singular Soergel bimodules, representations of symplectic groups, and representations of McKay groups. The second project is to introduce K-theoretic Soergel bimodules and to study their relationship to the quantum geometric Satake equivalence. The third is to produce a generalization of Khovanov-Lauda-Rouquier algebras, which has the potential to categorify other Nichols algebras. The fourth is to study the actions of lie algebras on various important categories, which were previously constructed by the PI and Qi.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.

表示理论是对称性的研究。对称组经常出现在物理学（例如球体的旋转），化学（例如晶体学）和其他科学领域。与具有对称的对象有关的数据通常可以在称为表示的对象中编码。数学家研究了表示形式之间的关系，以及如何从简单的，不可分割的形式中建立更大的表示，就像分子是由不可分割的原子构建的一样。这些简单表示的许多属性，例如它们的维度，是未知的，也是激烈研究的主题。表示及其结构可以包装在称为类别的集合中。上半个世纪的一种极其富有成果的工具是识别代表理论中的类别，并具有代数几何形状的类别，从而允许使用强大的几何工具。但是几何也有其限制，尤其是在明确计算的问题上。在过去的工作中，PI使用示意图方法发现了从表示理论和几何形状的类别的新的和明确的描述。在示意图中，可以将非常大的矩阵或几何结构编码为图片并以图形方式进行操纵。这些描述使曾经难以置信的类别可访问的计算机代数系统。 PI合作者威廉姆森（Williamson）执行的计算机计算导致了数十年来简单表示的计算维度的首次突破。 PI将继续开发示意方法来研究表示理论和几何形状，从而提供超出其他方法范围的新类别，结构和工具的明确结构。该项目为学生提供了研究培训机会。更具体地说，该提案将支持四个相关项目。首先是提供通用工具，用于使用称为分支基础的蜂窝基础来示意地研究一般的半神经单体类别，类似于PI和合作者先前构建的几个基础。然后，这些工具将应用于单数Soergel双模型的类别，符号组的表示以及McKay组的表示。第二个项目是引入K理论的soergel双模型，并研究它们与量子几何萨克等效性的关系。第三个是对Khovanov-Lauda-Rouquier代数的概括，该代数有可能对其他Nichols代数进行分类。第四个是研究Lie代数对各种重要类别的行为，这些类别以前是由PI和QI构建的。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估审查标准来通过评估来支持的。