Math 235 Reflection Groups

Semester: Spring 2016

E_6 root system
Lecture 01 (21412)
Inst : Prof. Asher Auel
asher * auel AT yale * edu
Time : Tue Thu 1:00 - 2:15 pm
Loct : LOM 205
Office : LOM 210
Phone : (203) 432-4187
hours :
Tue  2:15 - 3:45 pm
Wed 2:15 - 3:45 pm
Text : Finite Reflection Groups, 2nd Edition
L. C. Grove and C. T. Benson
Springer-Verlag, Graduate Texts in Mathematics 99
ISBN-10: 1-44-193072-8; ISBN-13: 978-1-4419-3072-9 (Softcover)
ISBN-10: 0-38-796082-1; ISBN-13: 978-0-3879-6082-1 (Hardcover)
Download ebook from SpringerLink (with Yale access)
Course syllabus and homework schedule.
Final exam review session: Thursday, May 5, 3:00-5:00 pm, LOM 205
Awesome resources:

Description of course: A reflection is a linear transformation describing the mirror image about a line in the plane, about a plane in 3-dimensional space, or generally, about a hyperplane in n-dimensional space. A reflection group is a discrete group generated by reflections. These include symmetry groups of regular polytopes (e.g., Platonic solids) and of regular tilings (e.g., wallpaper patterns). A kaleidoscope is an example of an optical device for visualizing certain reflection groups in the plane. The theory of reflection groups links linear algebra, abstract algebra (particularly group theory), Lie algebras, and representation theory in a beautiful way.

The main topics covered will be orthogonal transformations, reflections in real Euclidean space, Coxeter groups, crystallographic groups, root systems, and the classification of finite Coxeter groups. Along the way, the course will cover the basics of (finite) group theory as well as certain advanced topics in linear algebra. This will be a heavily proof-based course with homework requiring a significant investment of time and thought. This course is only appropriate to students who have taken a first course in linear algebra. A previous course in abstract algebra is not necessary. While the course is primarily targeted at students interested in studying higher mathematics, the subject matter would be of interest (and possible use) in subjects such as chemistry, computer science, materials science, and theoretical physics.

Expected background: The official prerequisite is linear algebra, either Math 222 or 225. The unofficial prerequisites are a mature mathematical mind, some experience with writing proofs, and the desire to work hard.

Homework 20%
Midterm exam (31 Mar)   30%
Final exam (08 May) 50%
Grades: Your final grade will be based on weekly homework, a midterm exam, and a final exam. Notice that more overall emphasis is placed on exams than on weekly homework assignments in computing your final grade. On the other hand, completing your weekly homework will be crucial to your success on the exams.
Group work, honestly: Working with other people on mathematics is highly encouraged and fun. You may work with anyone (e.g., other students in your section, in the course, not in the course, tutors, bums on the street, ...) on your homework problems. If done right, you'll learn the material better and more efficiently working in groups. The golden rule is:
Work with anyone on solving your homework problems,
but write up your final draft by yourself.
Writing up the final draft is as important a process as figuring out the problems on scratch paper with your friends, see the guidelines below. Mathematical writing is very idiosyncratic -- we will be able to tell if papers have been copied -- just don't do it! You will not learn by copying solutions from others or from the internet! Also, if you work with people on a particular assignment, you must list your collaborators on the top of the first page. This makes the process fun, transparent, and honest.


(or otherwise the small print)

Homework: Weekly homework will be due at the start of class on Friday. Each assignment will be posted on the syllabus page the week before it's due.

Late or improperly submitted homework will not be accepted. If you know in advance that you will be unable to submit your homework at the correct time and place, you must make special arrangements ahead of time. Under extraordinary circumstances, late homework may be accepted with a dean's excuse.

Your homework must be stapled, with your name clearly written on the top. Consider the pieces of paper you turn in as a final copy: written neatly and straight across the page, on clean paper, with nice margins and lots of space, and well organized.

Your lowest homework score from the semester will be dropped.

Exams: The midterm exam will take place in class on Thursday 31 March 2016. The final exam will take place 02:00 pm - 05:30 pm on Sunday 08 May 2016 in a location to be decided by the registrar.

Make-up exams will only be allowed with a dean's excuse.

The use of electronic devices of any kind during exams is strictly forbidden and would be pointless anyway.

Homework guidelines: Generally, a homework problem in any math course will consist of two parts: the creative part and the write-up.

  • The creative part: This is when you "solve" the problem. You stare at it, poke at it, and work on it until you understand what's being asked, and then try different ideas until you find something that works. This part is fun to do with your friends; you can do it on the back of a napkin. If you're having trouble, even in understanding what the problem's asking, use the resources available to you: my office hours, teaching assistants' office hours, weekly tutoring sessions, etc. Ask for help as early as you can! This part should all be done on "scratch paper."

  • The write-up: Now that everything about the problem is clear in your mind, you go off by yourself and write up a coherent, succinct, and nicely written solution on clean sheets of paper. Consider this your final draft, just as in any other course. This part you should definitely NOT do with your friends.