## Course overview

Coxeter groups are a class of abstract groups which enjoy remarkable algebraic and geometric properties. In the first part of the course, we will define Coxeter groups and study their properties, following the first three chapters of the textbook “Combinatorics of Coxeter Groups“.

The second and shorter part of the course will address one of the following applications of the theory of Coxeter groups. It is up to the participants to decide which direction we will follow:

- Finite reflection groups
- Iwahori-Hecke-algebras and knot theory
- Root systems and Lie theory
- Computations for Coxeter groups

The course is held online via zoom and open to anybody interested in the subject. People interested in following the course can inquiry the meeting details at felix.schremmer@ma.tum.de.

This course is jointly organized with Prof. Eva Viehmann.

## Weekly assignments

- April 13: introduction (slides)
- April 20: Sections 1.1, 1.2 (except for examples 1.2.10 and 1.2.11), 1.3 and 1.4. Exercises 2, 8 and 10 of Chapter 1. (slides)
- April 27: Sections 1.5 and 2.1. Exercises 1.18 and 2.3. (slides) Play a bit with the Bruhat cells of GL
_{2}and GL_{3}(Example 2.1.3):- Write down a few matrices and figure out in which Bruhat cell they lie in.
- Can you describe the Bruhat cells? What are their dimensions?

- May 4: Sections 2.2, 2.3 and the first two pages of 2.4, i.e. up to Proposition 2.4.4 (including its proof). Exercises 2.10 and 2.11.
- May 11:
**We meet at 9:00 s.t. because of the SVV.**Sections 2.4 and 2.5. Exercises 2.1 and 2.21.- Consider the Bruhat order on the symmetric group S
_{4}(Figure 2.4, page 31): How do the various actions of w_{0}according to Proposition 2.3.4 look like? - Similarly, how does Figure 2.7 behave under the action described in Proposition 2.5.4?

- Consider the Bruhat order on the symmetric group S
- May 18: We meet again at our regular time, 10 st. Please read sections 2.6 and 3.1 and have a look at Exercise 3.5 (the sections 2.7 and 2.8 will be skipped for this course).
- Exercise: In the Coxeter group S
_{5}, consider the elements x_{1}=13542 (a three cycle) and x_{2}=42531 (a four cycle).- Show that x
_{1}<x_{2}using the Tableau criterion. - It is given that x
_{2}=s_{3}s_{4}s_{2}s_{1}s_{2}s_{3}s_{4}is a reduced expression. Find a subexpression that constitutes a reduced expression for x_{1}.

- Show that x

- Exercise: In the Coxeter group S
- May 25: No course due to Whitsun Vacation.
- June 1st: Coxeter groups of type B
_{n}. Detailed assignment and exercises added: Solution sketches - June 8:
**Please complete the lecture survey.**Section 3.2 up to Figure 3.4, as well as all of section 3.3. Exercises 3.3 and 3.4. Note that in both exercises, the meets and joins are with respect to the right weak order. Moreover, the symbol ≤ in Exercise 3.3 refers to the right weak order. - June 15: Reading assignment and exercises
- June 22: Reading assignment and exercises
- June 29: Sections 4.5 and 4.6. Write down the root poset for D
_{4}and solve exercise 4.16. - July 6: Sections 4.7 and 4.8. Exercise 4.19. Consider a Coxeter group of type C̃
_{2}(so you have three nodes in a row with edge weights 4). Write down the set of small roots and all the nodes of the canonical automaton of distance at most 2 to the starting node (with all the edges between them). - July 13: Read section 4.9 as an outlook, and prepare the topics and questions you would like to discuss in the final review session (send them via email or bring them up in the session). Here is a little learning aid. Slides for the final review