Categorification in Representation Theory

Abstract

In this survey article, we give an introduction to the relatively young subject of $2$-representation theory, which studies categorifications of important players in classical representation theory. In particular, we provide a streamlined exposition of the main results leading to the classification of simple $2$-representations for Soergel bimodules associated to finite Weyl groups in characteristic $0$, obtained in [M. Mackaay, V. Mazorchuk, V. Miemietz, D. Tubbenhauer and X. Zhang, Simple transitive $2$-representations of Soergel bimodules for finite Coxeter types. Proc. London Math. Soc. 126 (2023)].