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.

Lectures: Thursday from 3:30 to 4:45 and Friday from 10:30 to 11:45
Classroom: Lecture Hall 6
Instructor: Priyavrat Deshpande.
Contact: Office: 403
phone: 962
email: pdeshpande AT cmi DOT ac DOT in
Office Hours:
Thursdays from 2:00 to 3:00 p.m.
Texts: 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, Springer 1985
  • J. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics.
Linear algebra, finite groups, LaTeX proficiency, and/or permission from the faculty advisor.
Grading: 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.

Course description

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.

Topics include:
  1. Isometries of Euclidean spaces and spheres.
  2. Hyperplane arrangements, mirrors and reflections.
  3. Root systems.
  4. Coxeter complexes.
  5. Classification of finite reflection groups.
  6. Combinatorics of Coxeter groups.
  7. Classification of affine reflection groups.
  8. 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 follow.
  1. Feel free to work together, but you should submit your own work.
  2. 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 due.
  3. Please turn in a neat stapled stack of papers. Refrain from using blank / printing paper. Use ruled paper.
  4. 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.
  5. 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.
  6. Do use commonly accepted notation (e.g., for functions, sets, etc.) and never invent new notation when there is already some available.
  7. 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.
  8. 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.
  9. 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 conclusion.
  10. Please be extra careful about the order in which you use your quantifiers.
  11. 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 proved.


  1. Homework 1. Due: Jan 18.
  2. Homework 2. Due: Jan 31.
  3. Homework 3. Due: Mar 07.
  4. Homework 4. Due: Mar 14.
  5. Homework 5. Due: Apr 14.

Course Project

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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