Categorification in Representation Theory

近20年来，表征理论、低维拓扑及相关领域的重大进展都是通过范畴化的过程取得的。这指的是考虑具有额外信息层的更高类别对象的过程，这些对象的非分类阴影（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（一个具有对象、1-态射和2-态射的小器具——对群或代数中的元素进行分类的1-态射）作用于由函子作用于范畴的问题。在具体的例子中（如2范畴分类李代数或Hecke代数），长开放问题的重大突破都是通过这种方式实现的，例如Hecke代数分解数的计算，所有Coxeter群的Kazhdan-Lusztig猜想的证明，以及James猜想的反例。受此启发，人们一直在努力发展一种抽象的2-表示理论，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-表征理论中的两个问题，并将这些问题应用于经典表征理论中的相关例子。

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.