Harvard University, FAS
Spring 2000

Syllabus

syllabus


OVERVIEW

This is a course on the (characteristic zero) representation theory of finite groups. Representation theory has been a dominant theme in mathematics since it was discovered about 100 years ago. It has also had a very important influence on physics and to a lesser extend on chemistry. Probably the best introduction to representation theory in general is the (characteristic zero) representation theory of finite groups. This is a beautiful, but relatively elementary subject which exhibits the most fundamental features of most other representation theories. It is also a subject that lends itself to the computation of non-trivial, but tractable examples.

TOPICS

Exactly what topics I cover will depend on the background, abilities and interests of the class. My initial plan is as follows. Topics 1)-4) will certainly be covered. Topics 5)-8) are open to variation.

1) Review of group theory and linear algebra.

2) Definitions and constructions including induction and tensor operations. Complete reducibility.

3) Characters. Including Schur's lemma and its applications, orthogonality relations, the regular representation, the ring of virtual represnetations, class functions and canonical decomposition.

4) Examples. Including general techniques for calculating character tables and the character table of GL_2(F_p).

5) Induction. Frobenius reciprocity, Mackey's theorem, Artin's theorem. [Sections 7.3, 7.4, 8.1, 8.2, 8.5, 9.2, 9.3 of S.]

6) The group algebra and applications of integrality. Including Burnside's theorem, Brauer's theorem and Frobenius' theorem. [Sections 6.4, 6.5, 8.3, 11.2, 12.3 and chapter 10 of S.]

7) Rationality questions. [Sections 12.1, 12.4, 13.1 and 13.2 of S.]

8) The representation theory of the symmetric group. [Chapter 4 of FH.]

[S=Serre's book and FH=Fulton's and Harris' book mentioned below.]

BOOKS

No one book is ideal for this course, and you are not required to have any book. The best book for the course is probably Serre's book Linear Representations of Finite Groups which covers everything except topic 8). This book is beautifully, if concisely written. We will only cover the first two thirds of the book.

A good book for topic 8), which also has some coverage of the rest of the course, is Fulton's and Harris' book Representation Theory. However we will only cover 5 of the 26 chapters in this book. The rest of book can be highly recommended to anyone who wants a taste of other directions in representation theory.

A more leisurely introduction to the subject is provided by James' and Liebeck's book Representations and characters of Groups. This book covers all the basics, but does not cover some of the more advanced topics I hope to talk about in the course.

Finally there are many other text books in this subject and you are encouraged to browse in the library. Please let me know if you find a particularly good book.

GRADES AND EXAMS

The course grade will comprise:

  • Homework 40%
  • Midterm 20%
  • Final 40%.
I plan to give homework assignments weekly. The midterm will be take home. The final will be a standard three hour exam.


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