27) REPRESENTATION THEORY I.
Lecturer: Matyas Domokos
No. of Credits: 3 and no. of ECTS credits: 6
Prerequisite Topics in Algebra
Course Level: Intermediate PhD
Brief introduction to the course:
The course gives an introduction to the theory of group representations, in a manner that provides
useful background for students continuing in diverse mathematical disciplines such as algebra,
topology, Lie theory, differential geometry, harmonic analysis, mathematical physics,
combinatorics.
The goals of the course:
Develop the basic concepts and facts of the complex representation theory of finite groups,
compact toplogical groups, and Lie groups.
The learning outcomes of the course:
By the end of the course, students are enabled to do independent study and research in fields
touching on the topics of the course, and how to use these methods to solve specific problems. In
addition, they develop some special expertise in the topics covered, which they can use efficiently
in other mathematical fields, and in applications, as well. They also learn how the topic of the
course is interconnected to various other fields in mathematics, and in science, in general.
More detailed display of contents (week-by-week):
Week 1. Definition of linear representations, irreducible representations, general constructions.
Week 2. Properties of completely reducible representations.
Week 3. Finite dimensional complex representations of compact groups are unitary.
Week 4. Products of representations, Schur Lemma and corollaries
Week 5. Spaces of matrix elements. Of representations.
Week 6. Decomposition of the regular representation of a finite group.
Week 7. Characters, orthogonality, character tables, a physical application.
Week 8. The Peter-Weyl Theorem
Week 9. Representation of the special orthogonal group of rank three.
Week 10. The Laplace spherical functions.
Week 11. Lie groups and their Lie algebras
Week 12. Repreentations of the complex special linear Lie algebra SL(2,C)
Reference: E. B. Vinberg: Linear Representations of Groups