MAGIC108: Real and Complex Reflection Groups

The course will introduce real and complex finite reflection groups as well as some of the corresponding invariant theory. The main purpose will be to describe various parts of the classification of these groups.

In the real world, a reflection in Euclidean space is an orthogonal transformation that fixes every point of a codimension 1 subspace. Such subspaces are called hyperplanes. So just as we see in 2-dimensions, reflections in Euclidean space have order 2. The finite groups generated by such reflections were classified by Coxeter in the 1930s. Such groups appear in various branches of algebra and geometry. For example, they appear as Weyl groups in algebraic groups. 

The notion of a complex reflection came along later. These are transformations of a complex space that fix every vector of a hyperplane. They no longer have to have order 2. The finite groups generated by complex reflections were determined by Shephard and Todd in the 1950s. Remarkably they appear in normalizers of certain maximal tori in the finite groups of Lie type. 

Required: Undergraduate Linear Algebra, Group Theory and Ring theory. 
Advantageous: Lie algebras, Representation Theory. 

1.  Finite groups acting on inner product spaces.
2.  Reflections and reflection groups.
3.  Orthogonal decompositions of a reflection group.
4.  Examples: $\Sym(n)$, $2\wr \Sym(n)$, $\Dih(2n)$, $B_n$, $G(p,m,n)$ something in %characteristic $p$.
5.   Coxeter groups; real reflection groups.
6.   Root systems.
7.  The classification of root systems.
8.  Classification of Coxeter groups.
9.  Examples of indecomposable root systems.
10.  Presentations of coxeter groups.
11.   Complex reflection groups.
12.   Invariants.
13.