Root Systems, Weyl Groups and Coxeter Groups
Prof. Sara Billey
Monday, Wednesday, Friday 12:30-1:20
Padelford room C-36
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/
- 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
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.
- "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
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:
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
On-Line Encyclopedia of Integer Sequences
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.)