Categorification in Representation Theory

表示论中的分类

• 批准号: 2440089
• 金额: --
• 依托单位: University of East Anglia
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2440089
• 关键词: Categorification; Representation; Theory

In the last 20 years, major progress in representation theory, low-dimensional topology and related areas has been made through the process of categorification. This refers to the process of considering higher categorical objects with extra layers of information, whose decategorified shadows (when forgetting the extra information) describe the original problem one is interested in solving. In practice, this usually means that instead of considering a group or an algebra acting on a vector space by linear transformations, one studies a so-called 2-category (a gadget with objects, 1-morphisms and 2-morphisms - the 1-morphisms categorifying the elements in the group or algebra) acting on categories by functors.In specific examples (e.g. 2-categories categorifying Lie algebra or Hecke algebras), major breakthroughs in long open problems have been achieved in this way, such as the computation of decomposition numbers for Hecke algebras, a proof of the Kazhdan-Lusztig conjectures for all Coxeter groups, and counterexamples to James' conjecture. Inspired by this, there has been an ongoing effort to develop an abstract 2-representation theory that captures the successful examples and provides a framework for future ones. In this project, the student will work on both questions from abstract 2-representation theory, and on applying those to examples relevant in classical representation theory.

在过去的20年中，通过分类过程取得了代表理论，低维拓扑和相关领域的重大进展。这是指考虑具有额外信息层的更高分类对象的过程，其脱离的阴影（忘记额外信息时）描述了人们对解决的原始问题。在实践中，这通常意味着，与其考虑通过线性转换在矢量空间上作用的群体或代数，不如研究一个所谓的2类分类（具有对象，1个态度和2个年度主义的小工具，1多态性 - 1多态性 - 对小组或派系的元素进行分类的元素。代数或Hecke代数），以这种方式实现了长期开放问题的重大突破，例如Hecke代数分解数量的计算，这是所有Coxeter群体的Kazhdan-Lusztig猜想的证明，以及对James的猜想的反附件。受到这一点的启发，一直在不断努力开发一种抽象的2个代表理论，该理论捕获了成功的示例并为将来的框架提供了一个框架。在这个项目中，学生将研究来自抽象2代表理论的两个问题，并将这些问题应用于经典表示理论相关的示例。