# MA673 Reflection Groups

**Lecturer: **Robert Kropholler

**Term(s): **Term 1

**Commitment:** 30 lectures

**Assessment:** 100% oral examination

**Formal registration prerequisites: **None

**Assumed knowledge: **The groups theory and some geometric ideas from the second year Maths core:

MA251 Algebra I: Advanced Linear Algebra:

- Euclidean spaces
- Abelian groups

MA249 Algebra II: Groups and Rings:

- Groups, generators and relations.

**Useful background: **Interest in Group Theory:

MA3K4 Introduction to Group Theory:

- Semidirect products

**Synergies: **The following modules go well together with Reflection Groups:

**Content**: 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. In fact, if you have done MA453 Lie Algebras then you are already familiar with classification of semisimple Lie algebras, which is essentially the classification of crystallographic reflection groups.

Besides classifications, we will concentrate on examples and polynomial invariants.

**Reference**: R. Goodman, *The Mathematics of Mirrors and Kaleidoscopes*, American Mathematical Monthly.

www.math.rutgers.edu/~goodman/pub/monthly.pdf

**Book**:

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