Introductory description

A reflection is a linear transformation that fixes a hyperplane and multiplies a complementary vector by -1. The dihedral group can be generated by a pair of reflections. The main goal of the module is to classify finite groups (of linear transformations) generated by reflections. The question appeared in 1920s in the works of Cartan and Weyl as the Weyl group is a finite crystallographic reflection group.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Finite reflection groups
Root systems
Coxeter groups and their geometric realisation
Fundamental chamber
Classification of finite Coxeter groups
Crystallographic groups

Learning outcomes

By the end of the module, students should be able to:

• Understand reflections in Euclidean spaces and groups generated by them
• Use Coxeter groups to describe reflection groups
• Identify fundamental chambers, corresponding to the positive roots
• State and prove classification of finite Coxeter groups and finite reflections groups
• Recognise crystallographic groups

Indicative reading list

J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, 1992.

Subject specific skills

This module lays a solid foundation to the study of Cartan-Killing combinatorics. It develops in depth the notions that allow one to investigate Lie algebras, Coxeter groups, buildings and crystallography. Students taking the module will learn some of the techniques required for working on a large-scale research project. These techniques are partly theoretical, and partly computational.

Transferable skills

Clear and precise thinking, and the ability to follow complex reasoning, to construct logical arguments, and to expose
illogical ones. The ability to retrieve the essential details from a complex situation and thereby facilitate problem
resolution.