MATHEMATICS
W4044, Spring 2015
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
Homework
assignments
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
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