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

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.

表示论是对对称性的研究。对称群经常出现在物理学（例如球体的旋转）、化学（例如晶体学）和其他科学领域中。与具有对称性的对象相关的数据通常可以编码在称为表示的对象中。数学家研究表示之间的关系，以及如何从简单的、不可分割的表示中构建更大的表示，就像分子是由不可分割的原子构建一样。这些简单表示的许多属性（例如它们的维度）都是未知的，并且是深入研究的主题。表示及其结构可以打包在称为类别的集合中。过去半个世纪一个极其富有成效的工具是将表示论中的类别与代数几何中的类别相识别，从而允许使用强大的几何工具。但几何也有其局限性，特别是在涉及显式计算时。在过去的工作中，PI 使用图表方法从表示理论和几何中找到了新的、明确的类别描述。在图表中，一个非常大的矩阵或几何结构可以被编码为图片并以图形方式进行操作。这些描述使曾经困难的类别变得易于计算机代数系统使用。 PI 的合作者 Williamson 进行的计算机计算在简单表示的计算维度方面取得了数十年来的首次突破。 PI 将继续开发图解方法来研究表示理论和几何，提供新类别、结构和工具的显式构造，这些超出了当前其他方法的范围。该项目为学生提供研究培训机会。更具体地说，该提案将支持四个相关项目。第一个是提供通用工具，用于使用称为分支基础的细胞基础以图解方式研究一般的半简单幺半群类别，类似于 PI 和合作者之前构建的几个基础。然后，这些工具将应用于奇异 Soergel 双模的类别、辛群的表示以及 McKay 群的表示。第二个项目是引入 K 理论 Soergel 双模并研究它们与量子几何 Satake 等价的关系。第三个是产生 Khovanov-Lauda-Rouquier 代数的推广，它有可能对其他 Nichols 代数进行分类。第四个是研究李代数在各种重要类别上的作用，这些类别是由 PI 和 Qi 先前构建的。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，认为值得支持。