Root Systems, Weyl Groups and Coxeter Groups

Fall, 2017

Prof. Sara Billey

Monday, Wednesday, Friday 12:30-1:20

Padelford room C-36


Additional Reading Material


Summary: Root systems, Weyl groups, affine Weyl groups, Coxeter 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 book and are very relevant to current research. 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. 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. Approximately, several recent journal articles will be handed out which are related to the current material. Occasionally, problems on the homework will related to this reading. 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.

Exercises: The single most important thing a student can do to learn mathematics is to work out problems. The second most important thing to do for this class is to read the book. It is the model textbook which every author should emulate in my opinion. Therefore, the goal for the quarter is to read every page and do every exercise in the textbook by Humphreys. Occasionally, I will add extra exercises. These will be collected and read every Wednesday.

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.

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, Maple or GAP. All are installed on zeno. John Stembridge has a nice package for Maple called "weyl" which may be useful to you. There is also a lot of tools for Coxeter groups in SAGE.

Interesting Web Sites:

Sara Billey