**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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