MAGIC108: Real and Complex Reflection Groups

Course details

A specialist MAGIC course


Autumn 2021
Monday, October 4th to Friday, December 10th


Live lecture hours
Recorded lecture hours
Total advised study hours


11:05 - 11:55 (UK)
14:05 - 14:55 (UK)


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.


  • Professor Christopher Parker

    Professor Christopher Parker

    University of Birmingham
    Main contact
  • SS

    Professor Sergey Shpectorov

    University of Birmingham


Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.


The assessment for this course will be released on Monday 10th January 2022 at 00:00 and is due in before Sunday 23rd January 2022 at 23:59.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.


Only current consortium members and subscribers have access to these files.

Please log in to view course materials.


Please log in to view lecture recordings.