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:

- Root systems and finite reflection groups
- Classification of finite reflection groups using Dynkin diagrams/ Coxeter graphs.
- Polynomial invariants of finite reflection groups
- Other Coxeter groups
- Kazhdan-Lusztig polynomials
- Applications of parabolic subgroups to the Solomon descent algebra, the Coxeter Complex, and affine Grassmannians (as time permits).

- "Reflection Groups and Coxeter Groups" by James
Humphreys. Cambridge studies in advanced mathematics, v. 29,
Cambridge University Press, 1990.
- "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 licence.

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.

- 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.)

Sara Billey