Categorification in Representation Theory

在过去的20年里，通过范畴化的过程，在表示论、低维拓扑和相关领域取得了重大进展。这是指考虑具有额外信息层的更高类别对象的过程，其去类别化的阴影（当忘记额外信息时）描述了人们感兴趣解决的原始问题。在实践中，这通常意味着不是考虑通过线性变换作用于向量空间的群或代数，而是研究所谓的2-范畴（一个小工具，有对象，1-态射和2-态射-1-态射分类群或代数中的元素）通过函子作用于范畴。在特定的例子中，（例如，李代数或Hecke代数的2-范畴分类），在长期未解决的问题上取得了重大突破，如Hecke代数分解数的计算，所有Coxeter群的Kazhdan-Lusztig定理的证明，和詹姆斯猜想的反例受此启发，人们一直在努力开发一种抽象的2-表征理论，以捕捉成功的例子，并为未来的例子提供一个框架。在这个项目中，学生将研究抽象2-表示理论的两个问题，并将其应用于经典表示理论中相关的例子。

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.