Representation Theory, Math 9140A, Fall 2026

This course will study the representation theory of finite groups as well as some applications.

It is aimed at MSc and PhD students in Mathematics. The prerequisites are undergraduate group theory and linear algebra.

• Instructor: Dan Christensen
• Semester: September 10 to December 8, except for reading week, October 10-18.
• Class times and location: Time and location on the OWL site.
• E-mail: jdc@uwo.ca
• Office: Middlesex 130
• Office Hours: After class.
• Prerequisites: Undergraduate group theory and linear algebra.

Outline:

Here are the topics that may be covered. Additional topics will be covered in presentations.

• Representations of finite groups: irreducibility and complete reducibility.
• Schur's lemma.
• Character theory and the orthogonality relations.
• Induced representations and Frobenius reciprocity.
• Fourier analysis on finite groups.
• Applications to group theory, including Burnside's theorem.
• Probability and random walks on graphs (if time permits).

Here is the official course outline.

Text: The text is Representation Theory of Finite Groups: An Introductory Approach, by Benjamin Steinberg, 2012, Springer. From on campus, you should be able to download it for free via this link.

Other references you may find useful, but which are not required:

• J.P. Serre, Linear Representations of Finite Groups, GTM 42, Springer, 1977.
• A. Baker, Representations of Finite Groups, pdf course notes.
• Shlomo Sternberg, Group theory in physics. A pleasant book with a conversational style and lots of applications to motivate the material.
• W. Fulton and J. Harris, Representation theory, GTM 129, Springer, 1991. Starts with a very fast treatment of the representation theory of finite groups.

Homework: Problem sets will be due roughly every 2 weeks. Doing problems and talking about the material are both essential for learning the material in this course, so you are encouraged to discuss the problems with classmates and with me. But you must write up the solutions on your own and must not look at other students' written solutions nor should you attempt to find solutions to problems online or in textbooks. You should not use an AI assistant to help with the completion of any course assessments. Your solutions should be clear and carefully written and you should give credit to those who helped you and to any references you used. Homework will be graded based on both correctness and clarity. Late problem sets will not be accepted unless arranged in advance for a good reason.

Presentations: In the second half of the semester, each student will give a 50 minute presentation on a topic related to the course. Here is a list of potential topics.

Exam: The final exam will be a three hour exam, held during the December examination period (December 11 to 22, 2026).

Evaluation: Evaluation will be based upon homework (30%), presentations (35%) and the final exam (35%).