Coxeter Groups and Complex Reflection Groups

Spring, 2021

Prof. Sara Billey

Monday, Wednesday, Friday 2:30-3:20

Class Link

Lecture Notes


Additional Reading Material


Summary: Coxeter groups, Root systems, Weyl groups, affine Weyl groups, invariant theory, and Kazhdan-Lusztig polynomials appear at the intersection between combinatorics, geometry and representation theory. These topics are beautifully described in the proposed text books and are very relevant to current research. As time permits, we hope to cover an introduction to the classification of finite complex reflection groups as well. The goal of this course is to introduce students to the basic material in this area and connect with some of the current open problems. It is impossible to be fully comprehensive in 10 weeks, so we will only scratch the surface of this deep elegant theory. The motivation for this material comes from studying symmetry of polytopes and Lie algebras/Lie groups The necessary prerequisites are just Abstract Algebra and basic Combinatorics (graphs, partitions, compositions, posets, etc). The main topics we will cover include:

Reference Texts: Presentations: This course is surprisingly close to the frontier of research in this area. Each student will be expected to present one lecture on a research paper in this area. Students can choose an article on their own subject to approval or select one from those handed out. To get the ball rolling, some classic papers and recent journal articles will be posted at the link above entitled "additional reading material" which are related to the current topic. Occasionally, problems on the homework will related to this reading.

Exercises: The single most important thing a student can do to learn mathematics is to work out problems. I will pose exercises throughout the lectures. These will be collected on Canvas every Wednesday. The second most important thing to do for this class is to read the book. This is an excellent book, which an aspiring author may wish to emulate. Doing extra exercises from the textbook is encouraged to solidify your understanding. You may turn these in also.

Grading: The grade will be appropriate for a topics course for graduate students. It will be based on the homework and the presentations. We will discuss the inclusion of a final exam in class. If we do one, it will be Tue, Jun 8 at 2:30 to 4:20PM.

Computing: Use of computers to verify solutions, produce examples, and prove theorems is highly valuable in this subject. Please turn in documented code if your proof relies on it. If you don't already know a computer language, then try Sage, Python, or Mathematica.

Interesting Web Sites:

Sara Billey