Foundations in Representation Theory
(V3A3 / F4A1)
First LectureFriday, Oct 11. Tutorials start the week after.Instructor Prof. Dr. Catharina Stroppel
Assistant Dr. Johannes Flake
Lectures
- Mondays, 12–2 pm, Kleiner Hörsaal, Wegeler Str. 10
- Fridays, 10–12 am, Hörsaal 6, Friedrich-Hirzebruch-Allee 5
- Mondays, 10 am–12 pm
- Thursdays, 8–10 am
- Thursdays, 12–2 pm
- Fridays, 8–10 am
References
- Fulton–Harris: Representation Theory: A First Course
- Serre: Linear Representations of Finite Groups
- Linear Algebra I & II
- Algebra I (semisimplicity, tensor products, basics in homological algebra; suitable references are Lang: Algebra and Weibel: An Introduction to Homological Algebra)
This course will be an introduction to representation theory. We will consider actions of groups (finite, algebraic and also of their Lie algebras) and algebras (quiver representations and semisimplicity questions) from a structural and categorical point of view. The course will provide general theory, technical toolkits and examples which will be important and helpful for the follow-up courses in representation theory. Starting with the representation theory of finite groups, we will explain modern approaches (combinatorial and categorical) to the study of (categories of) representations. The symmetric groups will provide important examples, with the theory of Vershik and Okounkov as our main focus. We will then introduce affine algebraic groups and study some of their representation theory (as comodules over the ring of regular functions). This brings together basic notions from algebraic geometry with combinatorics and with algebra. The main example will be rational representations of general linear groups over the field of complex numbers. The famous Schur-Weyl duality will serve as a motivation. We will study modules over Hopf algebras and more general tensor categories, where we also encounter open problems and approaches from current research.
To follow the course no knowledge in representation theory is assumed, but standard algebraic notions will not be recalled. Basic categorical knowledge is helpful, but more important is the willingness to learn how to work with categories in practise and in a way suitable for representation theory.