Introduction to Representation Theory
Math 464 - Spring 2019
Announcements
The semester is over.
About the course
The aim of this course is to introduce students to the basic principles of representation theory which is concerned with the ways of writing a group as a group of matrices.
We will explore the properties of groups by using the structure of group algebras which are vector spaces by definition. This approach allows us to use linear algebra.
A significant part of this study will be done by characters, functions on the group that associates to each group element the trace of the corresponding matrix.
Suggested Problems
I will constantly assign exercise questions from the textbook. I strongly suggest that you shall work on these questions on a regular basis.
Chap 1: 3, 5, 6, 8, 9. Chap 2: 2, 3, 4, 5, 7, 9. Chap 3: 1, 2, 3, 5, 6, 7. Chap 4: 1, 2, 3. Chap 5: 1, 2, 3, 4. Chap 6: 1, 3, 4, 6. Chap 7: 1, 3, 4, 5. Chap 8: 1, 3, 4, 5, 7. Chap 9: 2, 3, 4, 5, 6, 7. Chap 10: 2, 3, 4, 5, 6. Chap 11: 2, 3, 5, 6. Chap 12: 1, 2, 3, 4, 5, 6. Chap 13: 1, 2, 3, 4, 6, 7, 9, 10. Chap 14: 1, 2, 4, 5, 6, 7, 8. Chap 15: 1, 2, 4. Chap 16: 2, 3, 4, 5, 6. Chap 17: 1, 2, 3, 4, 6, 7, 8. Chap 18: 1, 2, 3, 4, 5. Chap 19: 1, 2, 3, 4, 5.
Schedule of Lectures and Office Hours
Lectures: Tuesday 09:40-10:30 and Thursday 10:40-12:30 in M105.
Attendance will not be taken during the lectures. However, I strongly suggest that you shall attend the lectures regularly.
Office hours: Tuesday between 10:40-12:30 in M141.
If these office hours are not suitable for you, then you shall send me an email to fix an appointment.
Exams and Grading
Your final letter grade will be determined by two midterms and the final exam. The exam dates are as below.
Midterm 1 and Solutions - March 20 (%30)
Midterm 2 and Solutions - April 24 (%30)
Final - May 25 (%40)
Textbooks and Course Outline
The textbook is G. James and M. Liebeck - Representations and Characters of Groups, Cambridge University Press. It is available in the reserve part of the library with code QA176.J36 1993.
You can find the tentative course outline below. For each week, we will attempt to cover the content of the indicated pages of our textbook.
(Feb 12, Oct 14) Groups and homomorphisms.
(Feb 19, Feb 21) Vector spaces and linear transformations.
(Feb 26, Feb 28) Group representations. FG-modules.
(Mar 5, Mar 7) FG-submodules and reducibility. Group algebras.
(Mar 12, Mar 14) FG-homomorphisms. Maschke's Theorem.
(Mar 19, Mar 21) Schur's Lemma.
(Mar 26, Mar 28) Irreducible modules and the group algebra.
(Apr 2 - Apr 4) Conjugacy classes. Characters.
(Apr 9 - Apr 11) Inner product of characters.
(Apr 16 - Apr 18) The number of irreducible characters.
(Apr 23, Apr 25) Character tables and orthogonality relations.
(Apr 30, May 2) Normal subgroups and lifted characters.
(May 7, May 9) Some elementary character tables. Tensor products.
(May 14, May 16) Restriction to a subgroup. What is next?
Make-up and NA policy:
Only one make-up examination will be offered. The excuse for not attending an examination must be proved with documents. The make-up examination will take place shortly after the final exam. If you have missed both midterms, your letter grade would be NA. If you receive NA, then you can not take the final exam and the makeup.