MAT 401: We will study the elementary aspects of the representation theory of finite groups. Groups acting on a structure is one of the oldest and most fruitful mathematical concepts. The presence of symmetry, a notion that can be often made precise in terms of group actions, usually greatly simplifies the complexit y involved in describing an object or a situation. It is remarkable that symmetry is naturally present in many contexts. One of the first steps in trying to exploit this situation is to ``linearize" the problem, that is to replace the action on the structure by an action on a vector space naturally associated, by local or global methods, to the situation. Here's an example: a group acts on a manifold and a point is fixed under this action; then the group also acts on the tangent space to the manifold at the fixed point. Mathematically, one speaks of a linear representation of the group. In this seminar we are going to study the basics of finite dimensional linear representations of finite groups: that is group homomorphisms from a finite group to the group of, usually complex, invertible square matrices. The structure that emerges is both rich and beautiful and is the model for the study of the representations of general groups and algebras, a very active field of research. I will lecture on the theoretical aspects, but also on many concrete examples. Students are expected to lecture on some of these examples as well as on selected more theoretical results. I will assist the students in preparing their presentations.
Prerequisites: MAT 200 (C or higher), MAT 310, AND 312 (C or higher). Or permission of the instructor.
Textbook: Linear representations of finite groups. J.P. Serre. This and other books have been placed on reserve in the library. I will discuss this in class.
Grade: Based on the in-class presentations and on in-class participation and, possibly, on assignments.
Schedule of exams. There will be no exams.
Schedule of in-class presentations. To be announced.
Office Hours: By appointment: on TU and TH 10:15-11:15 and on TU 2:15-3:15. MAT Tower 5-108 (on TU am I could be in P-143).
Etiquette: Be punctual. No cell-phones. No food.
Special needs. If you have a physical, psychological, medical, or learning disability that may impact on your ability to carry out assigned course work, you are strongly urged to contact the staff in the Disabled Student Services (DSS) office: Room 133 in the Humanities Building; 632-6748v/TDD. The DSS office will review your concerns and determine, with you, what accommodations are necessary and appropriate. All information and documentation of disability is confidential. Arrangements should be made early in the semester (before the first exam) so that your needs can be accommodated.