IntroSurvey of representation theory

TL;DR

This paper provides a personal and informal survey of representation theory, covering basics, quantum versions, Kazhdan-Lusztig theory, quantum groups, and recent advances, aimed at a broad audience including final year undergraduates.

Contribution

It offers a comprehensive, accessible overview of various aspects of representation theory, including recent developments and open problems, with a unique personal perspective.

Findings

Explains Schur-Weyl duality and symmetric group representations

Introduces quantum groups and Kazhdan-Lusztig theory

Surveys recent advances in modular representation theory

Abstract

There could be thousands of Introductions/Surveys of representation theory, given that it is an enormous field. This is just one of them, quite personal and informal. It has an increasing level of difficulty; the first part is intended for final 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 finish with twenty open problems and a song of despair.