# MA473 Reflection Groups

**Not Running 2019/20**

**Status for Mathematics students:** List C

**Commitment:** 30 lectures

**Assessment:** 3 hour exam.

**Prerequisites:** The only formal prerequisite is MA249 Algebra II. Some of the material is closely related to the material in MA453 Lie Algebras or MA3E1 Groups and Representations but neither of them is a formal prerequisite.

**Leads To: **

**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.