Bilinear Forms and Representation Theory (Math 4657), Fall 2015
Instructor: Andrew Obus
email: obus [at] virginia.edu
office: Kerchof Hall 208
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!
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.
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
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
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.
assignment will consist of two parts. The first part will consist
of a few straightforward problems, which are not to be handed in.
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
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
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
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