IntroSurvey of Representation Theory

Abstract

There could be thousands of Introductions/Surveys of representation theory, given that it is an enormous feld. This is just one of them, quite personal and informal. It has an increasing level of diffculty; the frst part is intended for fnal year undergrads. We explain some basics of representation theory, notably Schur–Weyl duality and representations of the symmetric group. We then do the quantum version, introduce Kazhdan–Lusztig theory, quantum groups and their categorical versions. We then proceed to a survey of some recent advances in modular representation theory. We fnish with 20 open problems and a song of despair.