Categorical and Diagrammatic Representation Theory

表征理论是对对称性的研究。对称群经常出现在物理学（如球体的旋转）、化学（如晶体学）和其他科学领域。与具有对称性的对象相关的数据通常可以编码在一个称为表示的对象中。数学家研究表征之间的关系，以及如何从简单的、不可分割的表征中构建更大的表征，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性质，如它们的维度，是未知的，也是激烈研究的主题。表示及其结构可以打包在一个称为类别的集合中。在过去的半个世纪里，一个非常富有成效的工具是用代数几何中的范畴来识别表征理论中的范畴，从而允许使用强大的几何工具。但几何也有它的局限性，尤其是在涉及显式计算的问题时。在过去的工作中，PI使用图解方法从表示理论和几何中发现了新的和明确的类别描述。在图解学中，一个非常大的矩阵或一个几何结构可以被编码为图像并以图形方式操作。这些描述使得曾经难以理解的分类可以用计算机代数系统来理解。由PI的合作者Williamson进行的计算机计算在计算简单表示的维度方面取得了几十年来的首次突破。PI将继续发展图解方法来研究表示理论和几何，提供超越当前其他方法范围的新类别、结构和工具的明确构造。本项目为学生提供研究训练机会。更具体地说，这项建议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的细胞基，类似于PI和合作者先前构建的几个基，以图表方式研究一般半简单单类类别。然后将这些工具应用于奇异Soergel双模的范畴、辛群的表示和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。第三是对Khovanov-Lauda-Rouquier代数的推广，它有可能对其他Nichols代数进行分类。四是研究李代数在各种重要范畴上的作用，这些范畴以前是由PI和Qi构造的。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。

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.