Categorification in Representation Theory

点击翻译按钮获取中文摘要

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.