# Representation Theory, Math 9140b, Winter 2010

This course will study the representation theory of finite groups and Lie groups, with a focus on the symmetric groups and linear groups, and the relationships between these two. Basic background in group theory, modules and linear algebra will be assumed.
• Instructor: Dan Christensen
• E-mail: jdc@uwo.ca
• Office: Middlesex 103b.
• Office Phone: 519-661-2111 x86530.
• Office Hours: after class or by appointment.
• Class times and location: Mondays and Wednesdays, 10:00 to 11:30, MC108. Starts Jan 6.
• Prerequisites: Group theory, modules and linear algebra.

## Outline:

Basic representation theory: irreducibility, complete reducibility, Schur's lemma, character theory, induced representations, etc.

Representations of the symmetric groups: partitions, Young tableaux, Young symmetrizers, Specht modules, and lots more.

Representations of the special and general linear groups: polynomial and rational representations, complete reducibility, characters, tensor representations, and lots more.

Text: There is no text for the course. Some references you may like to use:

• W. Fulton, Young tableaux, LMSST 35, Cambridge Univ. Press, 1997.
• G.D. James and A. Kerber, The representation theory of the symmetric group, Addison-Wesley or Cambridge Univ. Press, 1981. 532pp.
• Shlomo Sternberg, Group theory in physics. A pleasant book with a conversational style and lots of applications to motivate the material.
• G.D. James, The representation theory of the symmetric groups, Lecture Notes in Math., Vol. 682, 1978. 161pp.
• W. Fulton and J. Harris, Representation theory, GTM 129, Springer, 1991.
• And there are lots more!

Homework: Homework will be due roughly every two weeks. I encourage discussion of the class material with other students, and I encourage you to consult textbooks to learn more. But don't copy solutions from students or texts, and when a student or a text helps you solve a problem, you should give credit.

Presentations: In the second half of the semester, each student will give a presentation on a topic related to the course.

Exam: There will be a final exam at the end of the course.

Evaluation: Evaluation will be based upon homework (40%), presentations (30%) and the final exam (30%).