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Reflection groups
January-April 2016

Course description: We will study finite reflection groups, i.e. finite subgroups of the group of isometries of $\mathbb{R}^n$. This includes root systems, parabolic subgroups, length function, parabolic subgroups and classification of finite reflection groups using Coxeter graphs. We will also study general Coxeter groups, Weyl groups and Bruhat ordering. If time permits, we will discuss affine reflection groups.

Prerequisite: Linear algebra and group theory.

Lectures:	Tuesdays and Thursdays 9:10 - 10:25 a.m.
Classroom:	LH 2
Instructor:	Kavita Sutar-Deshpande
Email:	ksutar AT cmi DOT ac DOT in
Reference books:	'Reflection groups and Coxeter groups' - James Humphreys. 'Finite reflection groups' - L.C. Benson, C.T. Grove. 'Mirrors and reflections: The geometry of finite reflection groups' - Alexandre Borovik, Anna Borovik.
Teaching Assistant:	Shraddha Srivastava (email: shraddha AT cmi DOT ac DOT in)
Grading:	40% - Homework 30% - Midterm 30% - Final exam All grades will be uploaded on Moodle.

Homework

Homework will be posted on this website every Sunday night and will be due in one week (on the following Monday).
You may pick up the graded homework on Tuesdays.

Late submissions will carry penalties.
You may work alone or in groups to do your homework. Feel free to approach the instructor or TA if you have questions/doubts. In any case, you must write your own solutions.

Attendance

Attendance to classes is not required but absences will be noted. Explanation for longer absences is required.

Warning against copying/plagiarism

Please be warned that I will not tolerate any kind of plagiarism in your work. You are free to search for ideas on the internet and in books but the details and work have to be your own.

According to the Merriam-Webster OnLine Dictionary, to plagiarize means:

to steal and pass off (the ideas or words of another) as one's own
to use (another's production) without crediting the source;
to commit literary theft;
to present as new and original an idea or product derived from an existing source.

Synonyms: copying, appropriation, infringement, piracy, counterfeiting, theft, borrowing.

Please visit www.plagiarism.org for more details.

Schedule

Week	Topic
1 (04/01 - 08/01)	Isometries of $R^n$, reflections and reflection groups. Homework 1
2 (11/01 - 15/01)	Root systems No homework Holiday on 15 January (Pongal).
3 (18/01 - 22/01)	Positive and simple systems Homework 2
4 (25/01 - 29/01)	Holiday on 26 January. Deadline for course registration - 30 January
5 (01/02 - 05/02)	Generation by simple reflections No homework
6 (08/02 - 12/02)	The length function, deletion and exchange conditions, Coxeter presentation for a finite reflection group. Homework 3
7 (15/02 - 19/02)
8 (22/02 - 26/02)	Mid-semester exams week.
9
10
11
12
13
14
15	Holiday on 14 April (Dr. Ambedkar Jayanthi). Holiday on 18 April (Good Friday).
16
17 ()	Final exams week.

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