Math 665: Coxeter Groups

Professor: David E Speyer

Fall 2017

Course meets: Monday, Wednesday, Friday 1-2; 3866 East Hall

Office Hours: 2844 East Hall, Wednesday 9-12, Thursday 1-4.

Webpage: http://www.math.lsa.umich.edu/~speyer/665

Optional Textbook: Reflection Groups and Coxeter Groups, by James Humphreys. I'm not sure how much use I'll make of this; I intend it as a back up for when I say something incomprehensible.

Intended Level: Graduate students past the alpha algebra (593/594) courses. Students should be completely comfortable with abstract algebra and with the basic language of group theory.

Expected Work: Problem sets will be assigned weekly, due on Wednesdays. Students will also be expected to take turns scribing lecture notes.

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.


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 15.

Problem Set 2, due Monday September 25 (later date due to Rosh Hoshanah).

Problem Set 3, due Friday September 29.

Problem Set 4, due Friday October 6.

Problem Set 5, due Friday October 13.

No problem set for October 20, enjoy Fall break.

Problem Set 6, due Friday October 27.

Problem Set 7, due Friday November 3.

Problem Set 8, due Friday November 10.

Problem Set 9, due Friday November 17.

Problem Set 10, due Friday December 8.
A non-diagonalizable represenation of Ã2

Course Notes

With your aid, I will be compiling a set of notes for the course in the Course Notes file.

All students will be required to take turns scribing notes for this file. When it is your turn to scribe, download the template file and write in a summary of what happened in class that day. Then e-mail it to me. The deadline for editing the update is 24 hours after the lecture. I will, in turn, proofread and edit your entries in the next 24 hours and post them back to this webpage, so that the class always has a good record of what we have covered. You are welcome to download and read the source of the notes but please do your writing in the template file; my experience is that it is easier for me to resolve merge conflicts when I copy your text into the master file than if you edit the master file directly.

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 write the update, you can check here.

Handouts

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

The A3 hyperplane system, shown in stereographic projection.
The B3 hyperplane system, shown in stereographic projection.
The classification of finite Coxeter diagrams.
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
The Weyl Denominator Identity for the classical crystallographic types
A lemma about the determinant of the Jacobian matrix.
The Coxeter plane in A3 and B3.