Coursebooks

Coxeter groups

Lachowska Anna

English

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. Classification of finite root systems. Classification of finite crystallographic Coxeter groups. Order and structure of irreducible Coxeter groups. Generators and relations of Coxeter groups.

- Affine Coxeter groups. Classification.

- 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, affine Weyl group, the highest root, finite and affine 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

Written exam

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 No

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.

In the programs

• Mathematics, 2019-2020, Bachelor semester 6
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Coxeter groups
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks

Reference week

MoTuWeThFr
8-9
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
Under construction
Lecture
Exercise, TP
Project, other

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• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German