Coxeter Groups and Complex Reflection Groups

Lecture Notes

Presentations

Additional Reading Material

Syllabus

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:

• Coxeter Groups
• Bruhat Order
• Pattern avoidance in Coxeter groups
• Chip firing games on Coxeter groups.
• Kazhdan-Lusztig polynomials
• Complex reflection groups.
• Open problems.
• Applications of parabolic subgroups to the Solomon descent algebra, the Coxeter Complex, and affine Grassmannians (as time permits).

• "Combinatorics of Coxeter Groups" by Anders Bjorner and Francesco Brenti. Graduate Texts in Mathematics, 231. Springer, New York, 2005. This book is available electronically from the UW library via a site license.
• "Reflection Groups and Coxeter Groups" by James Humphreys. Cambridge studies in advanced mathematics, v. 29, Cambridge University Press, 1990.

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:

• Combinatorics Seminar at UW
• SAGE: Open Source Mathematical Software A collection of mathematical tools to do symbolic computation, graph manipulation, exact linear algebra, etc. Plus, its being developed right here at UW!
• Coxeter package for SAGE.
• John Stembridge's Maple packages for symmetric functions, posets,root systems, and finite Coxeter groups.
• GAP- Groups, Algorithms and Programming This is well written, reasonably fast, and free software for symbolic computation.
• Electronic Journal of Combinatorics
• Sloane's On-Line Encyclopedia of Integer Sequences
Graphviz Package Graph drawing software. Here is my dot file for Bruhat order. (Note, I haven't tried it with the latest version of Graphviz yet.)