Coxeter groups
Summary
Study groups generated by reflections
Content
- Orthogonal transformations in a real Euclidean space
- Groups generated by reflections. Coxeter groups, root systems. Crystallographic groups. Fundamental regions for Coxeter groups.
- Coxeter graphs. Root systems, simple and positive roots.
Classification of finite root systems. Classification of finite crystallographic Coxeter groups. Order and structure of irreducible Coxeter groups. Generators and relations of Coxeter groups.
- Applications and connections with other fields.
Keywords
Orthogonal transformations, reflection, regular polytop, root system, simple root, positive root, Coxeter group, Coxeter graph, crystallographic group, Weyl group, fundamental region, simply laced root system, the longest element of a Coxeter group, Coxeter element, Coxeter plane, Coxeter number, root lattice, highest root, finite Dynkin diagrams.
Learning Prerequisites
Required courses
Linear algebra I-II, Group theory
Recommended courses
Linear algebra I-II, Geometry I-Ii, Group theory, Lie algebras, Linear representations of finite groups
Learning Outcomes
By the end of the course, the student must be able to:
- Apply concepts and ideas of the course
- Reason rigorously using the notions of the course
- Choose an appropriate method to solve problems
- Identify the concepts relevant to each problem
- Apply known methods to solve problems similar to the examples shown in the course and in the problem sets
- Solve new problems using the ideas of the course
- Implement appropriate methods to identify and study the groups generated by reflections
Teaching methods
Lectures and exercise sessions
Assessment methods
Take-home test 15%.
Final written exam 85%.
Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés.
Supervision
| Office hours | No |
| Assistants | Yes |
| Forum | Yes |
Resources
Bibliography
1. J. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, 1990.
2. C.T. Benson, L.C. Grove, Finite Reflection Groups. Second Edition, Springer, 2010.
3. A. Bjorner, F. Brenti, Combinatorics of Coxeter Groups. Springer, 2005.
Ressources en bibliothèque
- (electronic version)
- Reflection Groups and Coxeter Groups / Humphreys
- Finite Reflection Groups / Benson & Grove
- Combinatorics of coxeter groups / Björner & Brenti
Moodle Link
Dans les plans d'études
- Semestre: Printemps
- Forme de l'examen: Ecrit (session d'été)
- Matière examinée: Coxeter groups
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines