MATH-335 / 5 credits

Teacher: Lachowska Anna

Language: English

## Summary

Study groups generated by reflections

## 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.

## 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

## 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

• Semester: Spring
• Exam form: Written (summer session)
• Subject examined: Coxeter groups
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks

## Reference week

 Mo Tu We Th Fr 8-9 9-10 10-11 11-12 12-13 13-14 CM1100 14-15 15-16 CM1100 16-17 17-18 18-19 19-20 20-21 21-22

Monday, 13h - 15h: Lecture CM1100

Monday, 15h - 17h: Exercise, TP CM1100

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