# Representation theory II

This page contains the information about the course **V4A4 Representation Theory II**, taught by Prof. Dr. Catharina Stroppel.

This course will give an introduction into some aspects of geometric representation theory. We will start by studying flag varieties and Grassmannians. Important hereby are in particular the Schubert cells which are the cells in some affine cellular decomposition of these varieties. Moreover we will spend some time on describing these varieties as varieties with a torus action. Then the connection to cohomology theories will be made.

We will introduce the socalled Bott-Samelson varieties, which are certain resolutions of singularities arising in this context. We will study them from a combinatorial, representation theoretic and geometric point of view. These varieties (in connection with Soergel bimodules) turned out to be extremely useful in proving and disproving long-outstanding conjectures in representation theory. The second part will most likely introduce equivariant cohomology and explain the important localisation theorem.

The motto throughout the course will be that we want to use geometry to do representation theory.

**Prerequisites** Basic knowledge on algebraic varieties and projective spaces, some familiarity with representation theory. The material from courses like Algebra I and Algebra II are assumed to be known. Basic knowledge in Lie theory (like in Humphreys book) helps for the applications.

## Lectures

- Wednesday, 10–12
- SR 1.008, Endenicher Allee 60
- Friday, 10–12
- SR 1.008, Endenicher Allee 60

The first lecture is **October 9**.

## Tutorials

The first tutorial takes place in the **second week of the semester** (i.e. October 14–18)

- Tuesday, 8–10, SR 0.011
- Tuesday, 12–14, SR 0.011

The tutors are

- Calvin Pfeifer
- Jendrik Stelzner

## Exercise sheets

Solutions to the homework must be handed in *before the lecture on Friday*. The first time this happens is on October 18, you will get this homework after the first lecture. In the first tutorial session (i.e. in the second week of the semester) there will be problems to be solved during the tutorial.

- tutorial exercises
- sheet 1
- sheet 2
- sheet 3
- sheet 4
- sheet 5
- sheet 6
- sheet 7
- sheet 8 (fixed variables in problem 38)
- sheet 9
- sheet 10 (fixed semidirect product in problem 46)
- sheet 11
- sheet 12 (fixed problem 55: you can hand in your solutions to this problem on January 21 in class)
- sheet 13

Here is a detailed solution of problem 36.

Here is some extra material on extensions of sections.

## Exam

The exam will be an **oral exam**, taking place on Monday February 3 and Tuesday February 4.

## References

Andersen: Introduction to Equivariant cohomology in algebraic geometry, arXiv:1112.1421

Chriss–Ginzburg: Representation Theory and Complex geometry

Fulton: Young tableaux

Lakshmibai–Brown: Flag varieties: An interplay of geometry, combinatorics and representation theory

Björner–Brenti: Combinatorics of Coxeter groups