Fall 2025

Graduate Seminar: Representations of SL₂ (S4A2)

Seminar times. Wednesdays, 2pm, R1.007, Endenicher Allee 60 Instructors. Johannes Flake Jonathan Gruber Course content. This is a seminar on the representation theory of the algebraic group SL₂(k) over a field k (mostly algebraically closed, including in positive characteristic), with two main objectives. First, much of the general representation theory of algebraic groups crucially relies on a good understanding of representations of SL₂(k). We aim to lay the foundations for learning about this area of representation theory, giving an overview of the most important tools and techniques and illustrating more general results using the example of SL₂(k). Second, representations of SL₂(k) play an important role in the structure theory of tensor categories. We will provide an overview of the applications of SL₂(k) to the study of tensor categories, including Temperley-Lieb diagrams, Verlinde categories and their generalizations. The topics include:

• basics on algebraic groups
• tori, weights, Borel subgroups
• induction functors
• simple modules, good filtrations, tilting modules
• Temperley-Lieb diagrams, Jones-Wenzl projectors
• Verlinde categories, Ostrik's theorem, generalized Verlinde categories
• SL₂ over finite fields
• quantum SL₂, link invariants