Coxeter Groups
Code  Completion  Credits  Range  Language 

02COX  Z  2  2+0  Czech 
 Garant předmětu:
 Jiří Hrivnák
 Lecturer:
 Jiří Hrivnák
 Tutor:
 Jiří Hrivnák
 Supervisor:
 Department of Physics
 Synopsis:

The course is an introduction to the theory of Coxeter groups and their invariant theory. The case of the finite Coxeter groups  the reflection groups and their properties are studied. The notions of the Weyl chamber and length are defined. General theory of the Coxeter groups, the corresponding bilinear forms and the theory of their classification represent abstract generalization of the reflection groups. The study of affine Weyl groups and related objects forms basic example of infinite Coxeter groups. As an introduction to the invariant theory the MacDonald identity and the Weyl identity are presented.
 Requirements:

Knowledge on the level of the course of linear algebra and geometry, basics of the group theory.
 Syllabus of lectures:

1. Reflections and reflection groups
2. Root systems, crystallographic roots systems
3. Weyl chambers and fundamental systems
4. Length and reflecting hyperplanes
5. Parabolic subgroups, Isotropy subgroups
6. Coxeter groups and Coxeter systems
7. Bilinear forms of Coxeter systems
8. Classification of Coxeter systems and reflection groups
9. Weyl groups, root lattice, fundamental weights and the weight lattice
10. The Classification of crystallographic root systems
11. Affine Weyl groups, affine root systems, alcoves
12. The Borelde Siebenthal theorem
13. The MacDonald identity, the Weyl identity
 Syllabus of tutorials:
 Study Objective:

Knowledge:
The fundamentals of the theory of the Coxeter groups and their invariants
Abilities:
Ability of orientation in the related mathematical field and recent literature, ability to comprehend and evaluate abstract material, ability of analytical approach to problems
 Study materials:

Key references:
[1] R. Kane, Reflection Groups and Invariant Theory , CMS books in Mathematics, Springer, 2001
Recommended references:
[2] J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge Advanced Studies in Mathematics, no. 29, Cambridge University Press, Cambridge, 1990.
[3] C. T. Benson, L. C. Grove, Finite Reflection Groups , Second Edition, Springer, 2010;
 Note:
 Timetable for winter semester 2023/2024:
 Timetable is not available yet
 Timetable for summer semester 2023/2024:
 Timetable is not available yet
 The course is a part of the following study plans:

 Aplikovaná algebra a analýza (elective course)
 Matematická fyzika (elective course)