Representation Theory

Information on
Exercise classes

Lecture Notes

First lecture:
Wednesday, 23/02/2011.

On Thu, 31 March 2011 the class will take place in HG G 19.1.

First exercise classes:
Friday, 25/02/2011.

Requirements for the certificate ("Testat"): Reasonable attempt to solve 60% of all exercises.

Course Description

The ideas of representation theory are among the most important unifying concepts in current mathematical research, and they are relevant to extremely diverse fields, from geometry and quantum mechanics to number theory, besides being an extremely important subject by itself. The course will present an introduction to various aspects of the representation theory of groups.

The goal of the course is to present the basic ideas and concepts of the representation theory of groups, together with examples of some of its important applications. The student will then be able to apply these basic facts, and will be ready to study other aspects of representation theory not covered during the class, using the intuition and knowledge presented during the lectures (for instance, the general representation theory of reductive groups, or more sophisticated applications in physics).

The planned course outline is as follows:
* Introduction and motivation with some simple examples
* Representation of finite groups in characteristic zero
* Examples and sample applications (e.g., Burnside's theorem on groups of order divisible by at most two primes)
* Representations of general compact groups and applications
* Representations of compact Lie groups and Lie algebras and applications (e.g. sample application to quantum mechanics)
* Introduction to the representations of locally-compact, non-compact, groups through the example of SL(2,R)

Prerequisites for the class are basic algebra, especially basic group theory, and, for the representation theory of compact or locally compact groups, the basic theory of Hilbert spaces and of integration. (In particular, no prior knowledge of quantum mechanics is required.)

References

• J-P. Serre: "Linear representations of finite groups", Springer GTM, vol 42
• W. Fulton and J. Harris: "Representation theory. A first course", Springer GTM/Readings in Mathematics
• A.W. Knapp: "Representation Theory of Semisimple Groups: An Overview Based on Examples", Princeton
• H. Weyl: "The theory of groups and quantum mechanics", Dover