Representations of finite groups
This is an introduction to representation theory, the theory of actions of groups and other algebraic structures on vector spaces, through the study of representations of finite groups.
Provisional syllabus: Each of the topics listed below will occupy roughly one-two weeks of course time.
1. Review of linear algebra and finite groups
2. Basic notions of representation theory: definitions, irreducible representations, Schur's Lemma, Maschke's theorem
3. Characters and orthogonality
4. The group algebra, Wedderburn theory
5. Algebraic integers and applications
6. Induced representations, Frobenius reciprocity
7. Representations of symmetric groups
8. Representations of GL(2,k) where k is a finite field.
9. Applications: Burnside's theorem
Prerequisites: Modern Algebra I will be assumed from the beginning; topics from Modern Algebra II will be introduced gradually.
Textbook: Gordon James, Martin Liebeck, Representations and Characters of Groups
Second edition, Cambridge University Press
Other useful references include
Jean-Pierre Serre, Linear Representations of Finite Groups
William Fulton, Joseph Harris, Representation Theory: A First Course
Midterm: March 11
Final: to be announced
1st week (due January 28)
2nd week (due February 4)
3rd week (due February 11)
4th week (due February 18)
5th week (due February 25)
6th week (due March 4)
(Midterm: no homework)
7th week (due March 25)
8th week (due April 1)
9th week (due April 8)
10th week (due April 15)
11th week (due April 22)
due Thursday, May 7 at 12 noon