## Bilinear Forms and Representation Theory (Math 4657), Fall 2015
Instructor: Andrew Obus
email: obus [at] virginia.edu
office: Kerchof Hall 208 |

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.

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).

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 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.Collab
allows anonymous comments to be submitted to instructors. Please
take advantage of this if there is something you want to say!