MATH6032 - Topics in Algebra II - 2022/23

General Information

Lecturer

• Felix Schremmer
  • Office: LSB-232A
  • Tel: 3943-7978
  • Email:

Time and Venue

• Lecture: Wednesday 9:30-12:15, LSB-219

Course Description

By definition, a reflection group is a subgroup of the orthogonal group which is generated by a finite set of reflections. The concept of a Coxeter group is a purely group-theoretic analogue of those reflection groups. While the definition of a Coxeter group is fairly accessible, as it is a finitely presented group of a very specific shape, the theory is rather deep and still subject to research.

Typical examples of Coxeter groups are dihedral groups and symmetric groups. While the most important application of Coxeter groups is definitely Lie theory (coming from Weyl groups and affine Weyl groups), the theory is also relevant for classical geometry and knot theory.

The course will be divided into two parts. The first part will cover the fundamental structure theory of Coxeter groups, such as Bruhat order and parabolic subgroups. In the second part, we will turn our attention to more advanced topics related to Coxeter groups, depending on the participants' interests and backgrounds.

Textbooks

• Andreas Björner, Francesco Brenti: Combinatorics of Coxeter Groups
• James Humphreys: Reflection groups and Coxeter groups
• Nicholas Bourbaki: Groupes et algèbres de Lie

Lecture Notes

• Lecture Notes (last edit Mar 10)

Useful Links

• GroupProps Wiki
• SageMath Computer Algebra System