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 |