Coxeter Groups

Course guarantor:

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 Borel-de 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:

Time-table for winter semester 2024/2025:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon
Tue
Wed
Thu	roomTR:211 Hrivnák J. 14:00–15:50 (lecture parallel1) Trojanova 13
Fri

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

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