SPRING 2014

Math 464 – Introduction to Representation Theory

Updated on February 19th, due to the size of the class

Instructor: Semra Öztürk Kaptanoðlu, M 138

Office Hours: See my schedule in my web page

Credits: (3-0)3 ,  Prerequisite: Math 367

This course brings together linear algebra and groups together, it provides some of the background for students who are planning to study algebra or algebra related topics such as number theory, group theory, or physics, even chemistry.

Briefly, a (linear) representation of a finite group G is a homomorphisms from G to GL(n, F). Hence for each group element there is matrix as the image of the homomorphism. Each representation gives a character of G, that is a function on G under which each element tahes the value of the trace of the corresponding matrix. Characters are class functions, that is they are constant on each conjugacy class of the group.

Cataloque Contents: Group representations, FG-modules, Maschke's Theorem, irreducible modules, group algebras. Characters, inner products of characters, the number of irreducible characters, character table, induced modules and characters, algebraic integers and real representations.

Textbook: Representations and Characters of Groups, by G. James and M. Liebeck, Cambridge University Press It includes  solutions to the excercises.

Grading: Grading will be based on three exams each one is %30 of the grade,  two  midterm and a final exams, and homeworks of %10 if there is a teaching assistant otherwise grade will be given over 90.    Attendance: If you do not attend %70 of the lectures you will not be allowed to take the exams.  In case of an overlap with another course this does not apply.                                                                                                                                                                                                 Make-up: No make – up exams.  In case of a very serious reason that I approve,  an oral presentation may be assigned instead of a written exam.                                                               Course Outline

Week 1

Introduction, Groups and homomorphisms, Vector spaces  and linear transformations

Week 2

Group representations

Week 3

FG-Modules and submodules, irreducible modules

Week 4

Group algebras, FG-homomorphisms

Week 5

Maschke’s Theorem

Week 6

Schur’s Lemma

Week 7

Conjugacy Classes

Week 8

Characters

Week 9

Inner Products of Characters

Week 10

The Number of Irreducible Characters

Week 11

Character Tables and Orthogonality Relations

Week 12

Normal Subgroups and Lifted Characters

Week 13

Real representations

Week 14

Some applications in  group theory