Math 582CB/583CB
Root Systems and Coxeter Groups
Sara Billey and William McGovern
Winter/Spring 2005, Monday/Wednesday/Friday 10:30
Root systems, Weyl groups, affine Weyl groups, Coxeter groups, invariant
theory, and Kazhdan-Lusztig polynomials appear at the intersection between
combinatorics and representation theory. These topics are beautifully
described in the proposed textbook 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.
This is a two quarter course intended for first year graduate
students or beyond. Monty McGovern would teach the first quarter of the
course based on Chapter 1-3 and 7 of Humphrey's book during the winter quarter
of 2005 and Sara Billey would teach the second quarter in the spring on the
remaining chapters plus some additional material.
Prerequisites: Math 402-3-4 or a similar algebra class. No
combinatorics is required. Therefore, an advanced undergraduate could also
benefit from this course.
Outline:
- Winter Quarter:
- Root systems and finite reflection groups
- Classification of finite reflection groups using Dynkin diagrams/
Coxeter graphs.
- Polynomial invariants of finite reflection groups
- Kazhdan-Lusztig polynomials
- Spring Quarter:
- Affine Weyl groups
- Affine and parabolic Kazhdan-Lusztig polynomials
- Geometric representation of Coxeter groups
- Patterns in Coxeter groups
- Word problems
- Chip firing games on Coxeter groups.
Text: Reflection Groups and Coxeter Groups by James Humphreys