Bilinear Forms and Representation Theory (Math 4657), Fall 2015

Instructor: Andrew Obus
email: obus [at] virginia.edu
office: Kerchof Hall 208
phone: 434-424-4930
website: http://people.virginia.edu/~aso9t/math4657f15.html

TTh 12:30-1:45, Monroe Hall 110.  Please ask questions if anything in lecture is unclear. Lectures will run the entire 75 minutes. Please show up on time!

Textbook

There will be no textbook per se for the course.  The main reference will be Ben Webster's lecture notes (posted on Collab), and we will follow them relatively closely. However, you may find the following books useful as additional references.

• Linear Representations of Finite Groups by Jean-Pierre Serre (available at the bookstore).  This book is thin, terse, and contains a wealth of material.  Most of the material on representation theory that we will cover is contained in Chapter 1.  Dense reading, but does not require much background.
• Representation Theory of Finite Groups: An Introductory Approach, by Benjamin Steinberg (more leisurely and elementary than Serre).
• Representations and Characters of Groups, by Gordon James and Martin Liebeck (includes lots of background, takes a more module-theoretic approach).
• Representation Theory: A First Course by William Fulton and Joe Harris.  This is a more advanced book, probably better suited for reading after this course, and concurrently with (graduate) Algebra I.
• Algebra by Michael Artin.  Chapter 8 is a good reference for bilinear forms, and Chapter 10 is good for the beginnings of representation theory.

Content

Representation theory is what you get when you introduce group theory and linear algebra to each other.  In particular, it is the study of group actions on vector spaces (for instance, the group Z/2 acts on R^n by having the nontrivial element act by point reflection around the origin).  In some sense, this is studying groups the way that they were "meant" to be studied, that is, by seeing how they act on geometric objects.  Aside from being a beautiful topic in its own right, representation theory has lots of applications in mathematics as well as the sciences (indeed, the first chapter of Serre's textbook above was written for quantum chemists!).  Strikingly, representation theory can also help prove results in pure group theory, and was instrumental in the classification of finite simple groups, completed in the 1980s. We will hopefully be able to give a few applications at the end of the course.

In this class, we will focus on the representation theory of finite groups.  In order to prove the basic results in the representation theory of finite groups, it is helpful to have a solid background in bilinear and sesquilinear forms over the complex numbers (these are generalizations of the familiar dot and cross products over the real numbers).  We will spend most of the first half of the class on this.  Bilinear forms are closely related to quadratic forms, and we will probably take a detour for a lecture or two to discuss some relationships between quadratic forms and number theory (we may give the "one sentence" proof that all primes congruent to 1 mod 4 can be written as sums of two squares).

Office Hours

Tuesdays 11-12, Wednesdays 4:45-5:45. Kerchof Hall 208 (my office). If these times do not work for you, please make an appointment with me.

Homework

Each homework assignment will consist of two parts.  The first part will consist of a few straightforward problems, which are not to be handed in.  These are meant to check your understanding --- if you have questions about them, I am happy to help!  The second part will have about 3 or so more difficult problems (usually proofs) which I will collect and grade.  Please make homework as neat as possible, and please make your arguments as clear as you can (be honest when you don't know how to do something)!

Homework will be graded in a somewhat non-traditional way.  Each problem will get a grade of "Correct" or "Redo."  "Correct" means that there are only very minor problems with the argument, which I can point out quickly.  "Redo" means that there is a reasonably serious gap in your reasoning, or that the argument does not make sense (I will try to write some comments in this case).  Problems marked "Redo" can be handed in within a week of being returned for 80% credit.  If a problem handed in a second time gets a "Redo," then it can be handed in a third time for 60% credit, a fourth time for
40% credit, etc.  If you elect not to redo a problem, then it earns 0% credit.  My feeling is that this allows for better learning than standard homework procedure, and is workable for a small class.

Homework will be assigned more or less weekly, and will be posted on Collab.  Grades will be posted there as well.  There may be occasional extra credit problems assigned (these will be more difficult than the standard handed-in homework problems).

Homework will usually be due on Thursdays in class.

Exams

There will be a take-home midterm due Thursday October 15th in class.  There will be a take-home final due Monday December 7th at noon in my office.

Your grade will be calculated as follows: 10% Participation, 25% Midterm, 30% Homework, 35% Take-Home Final.  This is a small class, so it does not make sense to have a strict curve.  If you all show a good understanding of the material, you can all earn high grades!