Coursebooks 2018-2019

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Coxeter groups

MATH-335

Lecturer(s) :

Lachowska Anna

Language:

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, fundamental region, symmetric group, simply laced root system, Bruhat order, the longest element of a Coxeter group, Coxeter element, Coxeter plane, Coxeter number, McKay correspondence  

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:

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.

Ressources en bibliothèque

In the programs

Reference week

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      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