This is a plan for a reading course in the Spring on Coxeter groups and reflection groups. ("Plan" might be too strong a word). There are rather a large number of books which cover this topic, for example:
- Kane "Reflection Groups and Invariant Theory". (Springer/CMS 2001).
- Borovik and Borovik "Mirrors and Reflections". (Springer 2010).
- Abramenko and Brown "Buildings". (Springer 2008).
- Humphreys "Reflection Groups and Coxeter Groups. (CUP 1990).
- Bjorner and Brenti "Combinatorics of Coxeter Groups". (Springer 2006)/
- Grove and Benson "Finite Reflection Groups, 2nd ed". (Springer 1985).
- Hiller "Geometry of Coxeter Groups". (Pitman 1982).
Note that the above list is not complete. The topic is covered tersely in Aschbacher "Finite Group Theory" and in Brouwer, Cohen and Neumaier "Distance-Regular Graphs"; there is a good discussion of root systems in Humphreys "Introduction to Lie Algebras and Representation Theory".
Part I: Theory
- Coxeter groups are reflection groups.
- Reflection groups are Coxeter groups.
- The classification of finite Coxeter groups.
- Invariant theory. [maybe]
Part II: Applications and Extensions
The major application is the classification of semisimple Lie algebras over the complex numbers, which we will not treat. Applications we may treat include:
- Root lattices.
- Classification of graphs with least eigenvalue at least -2.
- Coxeter graphs (as in BCN).
- Complex reflection groups.
- Reflections over finite fields.