Introduction to Reflection groups
January - April 2013
The project reports are located here.
For homeworks click here.
A LaTeX template for your project report: tex
file and the pdf output. Make all the necessary changes.
||Thursday from 3:30 to 4:45 and Friday from 10:30 to 11:45
||Lecture Hall 6
|email: pdeshpande AT cmi DOT ac DOT in
| Thursdays from 2:00 to 3:00
||Useful reading material (in alphabetical order):
- A. Bjorner, F. Brenti, Combinatorics Of Coxeter
Groups, Springer 2005
- A. Borovick, A. Borovick, Mirrors and Reflections,
Springer UTM 2009
- N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4
-6, Springer-Verlag 2002
- M. Davis, The Geometry and Topology of Coxeter Groups,
LMS monograph series
- L. Grove, C. Benson, Finite Reflection Groups,
- J. Humphreys, Reflection groups and Coxeter groups,
Cambridge Studies in Advanced Mathematics.
|Linear algebra, finite
groups, LaTeX proficiency, and/or permission from the
||The homework will be assigned roughly every 2 weeks. Each
student will have to do a project. This includes a report
(prepared in LaTeX) and a 40 minutes board-talk.
- Homework 50%
- Project report 25%
- Presentation 25%
About Reflection groups
A reflection group is a discrete group which is generated by a set
of reflections of a finite-dimensional (Euclidean) space. Examples
of finite reflection groups include the symmetry groups of regular
polytopes, and the Weyl groups of simple Lie algebras. Examples of
infinite groups include the triangle groups corresponding to regular
tessellations of the Euclidean plane and the hyperbolic plane, and
Weyl groups of infinite-dimensional Kac–Moody algebras.
Discrete isometry groups of more general Riemannian manifolds
generated by reflections are also interesting. The most important
class arises from Riemannian symmetric spaces of rank 1: the
n-sphere (finite reflection groups), the Euclidean space
(corresponding to affine reflection groups), and the hyperbolic
space (hyperbolic reflection groups).
Coxeter groups grew out of the study of reflection groups. A
reflection group is a subgroup
of a linear group generated by reflections while a Coxeter group
is an abstract group generated by involutions (i.e. reflections),
and whose relations have a certain form (corresponding to
hyperplanes meeting at a non-obtuse angle). All finitely generated
reflection groups are Coxeter groups and vice versa. We will
explore how the geometry and topology of reflection groups helps
us understand their group theoretic properties.
The main aim of the course will be to understand the classification
of finite (or spherical) reflection groups using Coxeter-Dynkin
diagrams. The course will begin with a review of Euclidean geometry
and the theory of polytopes. The emphasis will be on the language of
hyperplane arrangements (this should also help create a necessary
background for the study of buildings). In the latter half, after
the introduction of Coxeter presentation, some combinatorial aspects
will be covered. For example, reduced words, exchange property, weak
and Bruhat orders, special subgroups etc. Finally towards the end,
if time permits, we will cover classification of Euclidean and
Hyperbolic Coxeter groups.
- Isometries of Euclidean spaces and spheres.
- Hyperplane arrangements, mirrors and reflections.
- Root systems.
- Coxeter complexes.
- Classification of finite reflection groups.
- Combinatorics of Coxeter groups.
- Classification of affine reflection groups.
- Reflections in hyperbolic plane.
Homework problems will be assigned during the class. It is your duty
to submit the solutions on time. Copying and/or plagiarism will not
be tolerated. Here are a few writing guidelines you might want to
- Feel free to work together, but you should submit your own
- Your questions/comments/suggestions are most welcome. I will
also be fairly generous with the hints. However, do not expect
any kind of help, including extensions, on the day a homework is
- Please turn in a neat stapled stack of papers. Refrain from
using blank / printing paper. Use ruled paper.
- Your final finished version should be as polished as you can
make it. This probably means that you cannot submit sketchy
solutions or sloppily written first versions. Please expect to
do a fair amount of rewriting. Do not hand in work with parts
crossed out; either use a pencil and erase or rewrite.
- Please write complete sentences that form paragraphs and so
forth. It might be a good idea to use short simple sentences;
avoid long complicated sentences.
- Do use commonly accepted notation (e.g., for functions, sets,
etc.) and never invent new notation when there is already some
- Make sure you provide a statement clearly indicating
precisely what it is that you are about to prove. You can, if
you want, label your statement as a Theorem or
Claim or whatever. Write the word Proof,
and then give your proof.
- Throughout your writing, constantly tell the reader (i.e. me,
:-)) exactly what it is that you are about to demonstrate. Be
sure to indicate the end of your proof.
- Your answers should combine “logic”, your hypotheses, and
possibly other mathematical facts (e.g., theorems that we have
proved in class) into an argument that establishes the asserted
- Please be extra careful about the order in which you use your
- In your arguments you can make free use of anything that we
have proven in class, and of course all logic rules and basic
axioms and definitions. Anything else that you use should be
- Homework 1. Due: Jan 18.
- Homework 2. Due: Jan 31.
- Homework 3. Due: Mar 07.
- Homework 4. Due: Mar 14.
- Homework 5. Due: Apr 14.
The project reports can be found here.
A project should consist of a short introduction to a topic (or a
survey) or proof of a result and its implications. For example, a
project on Artin groups may include their definition, examples,
basic properties and statements of (some of the) important results
etc. On the other hand a project on the three reflections theorem
should include a detailed proof and its implications.
Students are welcome to choose topics on their own (even outside
the above list). A topic should be finalized before the end of
January. The final report (6-8 pages) is due in the last week of
March. Presentations will be scheduled during the month of April.
It is students' responsibility to keep the instructor updated
regarding the project work.
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