Combinatorics of Coxeter Groups
Winter, 2009
Prof. Sara Billey
Monday, Wednesday, Friday 11:30-12:20 More
Hall room 219
Syllabus
Summary:This course will introduce Coxeter groups from a combinatorial
point of view. The course will build on the course taught this past
fall by Monty McGovern, however, it is possible to take this class
without having seen the first course. The necessary prerequisites are
just Abstract Algebra and basic Combinatorics (graphs, partitions,
compositions, posets, etc). The main topics we will cover include:
- The basics of Coxeter groups.
- Reduced words and the exchange property
- Bruhat order on Coxeter groups, quotients and parabolic subgroups.
- Weak order on Coxeter groups.
- Roots, games and automata.
- Enumeration aspects of Coxeter groups.
- The word problem (if time permits or strong class interest).
- Revisiting Kazhdan-Lusztig polynomials (if time permits or strong
class interest).
Reference Texts:
- "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 library.
- "Reflection Groups and Coxeter Groups" by James
Humphreys. Cambridge studies in advanced mathematics, v. 29,
Cambridge University Press, 1990.
Additional Reading/Presentations: This course is surprisingly close to the frontier
of research in this area. Approximately, five 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. One or two exercises will be
assigned during each lecture. These will be collected and read every
Monday.
Grading: The grade will be appropriate for an advanced topics course
for graduate students. It will be based on the homework and the
presentations.
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 Maple or GAP. Both 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:
Last modified: Thu Jan 29 17:27:00 PST 2009