# Math 665: Coxeter Groups

### Fall 2019

 Course meets: Monday, Wednesday, Friday 12-1; 2866 East Hall Office hours: Monday and Wednesday 9:30-11:30 AM, Thursday 2:00-4:00 PM, 2844 East Hall. I am also glad to make appointments to meet at other times. At the moment, I am making all my office hours open to both my classes (593 and 665); if this causes a problem, I may restructure. Webpage: http://www.math.lsa.umich.edu/~speyer/665 Intended Level: Graduate students past the alpha algebra (593/594) courses. Students should be completely comfortable with abstract linear algebra and with the basic language of group theory. Expected Work: Problem sets will be assigned weekly, due on Fridays. Students will also be expected to take turns scribing lecture notes once we get past the first month. Escher's "Circle Limit IV", whose symmetry group is the Coxeter group (3,4,4) Euclid constructs the icosahedron, with symmetry group H3. Book 13, Proposition 18 of the Elements, image courtesy of the Clay Mathematics Institute.

 Course Outline: In the first month of the course, we will cover the structure theory and geometry of Coxeter groups in general. I taught this material two years ago. I expect these notes, based on class notes from that time, will be a good reference this time. We will then study weak order on Coxeter groups. This is a lattice, with many applications in cluster algebras, representation theory of quivers, and enumerative combiantorics. Finally, we will study Bruhat order, also known as strong order. This is a different partial order on a Coxeter group, with key importance in the geometry of flag varieties and the representation theory of Lie groups. The Tamari lattice, also known as associahedron, on parenthesizations of a five-fold product Back view of the associahedron.

## Problem Sets

 Homework Policy: You are welcome to consult each other provided (1) you list all people and sources who aided you, or whom you aided and (2) you write-up the solutions independently, in your own language. If you seek help from mathematicians/math students outside the course, you should be seeking general advice, not specific solutions, and must disclose this help. I am, of course, glad to provide help! I don't intend for you to need to consult books and papers outside your notes. If you do consult such, you should be looking for better/other understanding of the definitions and concepts, not solutions to the problems. You MAY NOT post homework problems to internet fora seeking solutions. Although I participate in some such fora, I feel that they have a major tendency to be too explicit in their help; you can read further thoughts of mine here. You may post questions asking for clarifications and alternate perspectives on concepts and results we have covered.
 Problem Set 1, due Friday September 13. Problem Set 2, due Friday September 20. Problem Set 3, due Friday September 27. Problem Set 4, due Friday October 4. Problem Set 5, due Friday October 11 . Problem Set 6, due Friday October 25 . Problem Set 7, due Friday November 1 . Problem Set 8, due Friday November 8 . Problem Set 9, due Friday November 15 . Problem Set 10, due Friday November 22 . No problem set over Thanksgiving. Problem Set 11, due Monday December 9. A representation of Ã2. The 24-cell, a four dimensional polytope with symmetry group F4. Image produced by Robert Webb's Stella software, and found via Wikimedia commons.

## Course Notes

With your aid, I will be compiling a set of notes for the course. For the first month, I believe my notes from last time will be adequate. However, I want to improve them. Therefore, each problem set will solicit requests for improvement.

After that, we will move to new material, which I will require you to record notes for. In past courses, students have found this very helpful, and I find it very helpful in my teaching. Those notes can be found here.

If you do not know LaTeX, you should learn! I can suggest sources; I also find TeX.stackexchange incredibly useful for specific questions.

I anticipate that this course will be image heavy. If you do not feel up to creating sophisticated figures, feel free to handdraw and scan images. Also, if I have brought an image to class in a digital format, feel free to ask me for a copy of it.

If you have forgetten when you are scheduled to scribe, you can check here.

## Handouts

These are images and documents I have distributed/anticipate distributing in lecture.

The A3 hyperplane system (also known as D3), shown in stereographic projection.
The B3 hyperplane system (also known as C3), shown in stereographic projection.
The H3 hyperplane system, shown in stereographic projection.
A3, with relecting hyperplanes and regions labeled. Names of reflecting hyperplanes are boxed.
B3, with relecting hyperplanes and regions labeled. Names of reflecting hyperplanes are boxed.
The Ã1 hyperplane arrangement, for the Cartan matrix with A12 = A21=-2
The Ã1 hyperplane arrangement, for the Cartan matrix with A12 = A21=-3
The rank two root systems drawn within their weight lattices: A2, B2, G2
The ideal triangle triangulation of the hyperbolic plane: Poincare model, Klein model
A "triangulation" of the hyperbolic plane using triangles whose sides meet "beyond infinity" Klein model
Tiling the hyperbolic plane with right angled pentagons Poincare model