Markus Schmidmeier
Mathematical Sciences
Florida Atlantic University

# Introduction to Representation Theory

Fall 2014

Hi, here is some information about my course Introduction to Representation Theory (CRN: 96928, MAS 9396, 3 credits). We meet Mondays, Wednesdays and Fridays, 11:00 - 11:50 a.m. in BU 403.

The goal of these lectures is to introduce the beginner to finite dimensional representations of groups, Lie groups and Lie algebras. Representation theory is the study of the ways a given group can act on vector spaces. Groups and group actions have been studied since the early 20th century by mathematicians with interests in a variety of topics --- hence representation theory is interconnected with many areas within mathematics and outside. The approach in this course will be to concentrate on examples, and to develop the general theory sparingly, and then mainly as a useful and unifying language to describe phenomena already encountered in the examples.

Prerequisite

Introductory courses in abstract algebra and in linear algebra.

Textbook and Topics

Textbook: William Fulton and Joe Harris, Representation Theory. A First Course, Springer, Graduate Texts in Mathematics 129, ISBN 0-387-97495-4, (1991).

For group representations, I find that the following textbook provides a short but excellent introduction: Jean-Pierre Serre, Linear Representations of Finite Groups, Springer, Graduate Texts in Mathematics 42, ISBN 0-387-90190-6 (1977).

A friendly introduction to groups and their representations is the book by Gordon James and Martin Liebeck, Representations and Characters of Groups, Cambridge, 2nd edition, ISBN 0-521-00392-X.

An introduction to Lie algebras, including the representation theory of the sl(2,C), is the book by Karin Erdmann and Mark J. Wildon, Introduction to Lie algebras, Springer SUMS, ISBN-13: 978-1-84628-040-5.

Topics:

 Appendix B Multilinear Algebra(1 week) We review tensor products of vector spaces, exterior and symmetric powers, and duality. Part I(Lectures 1-4) Group Representations(5 weeks) We study actions of a finite group on a finite dimensional complex or real vector space, in particular in the case of symmetric groups. Part II(Lectures 7-10) From Lie Groups to Lie Algebras(5 weeks) The action of a Lie group on a manifold is best understood by considering representations of the corresponding Lie algebra. We give a rough classification of Lie algebras. Parts II-III(Lectures 11-14) Representations of sl2 and sl3(3 weeks) We analyze the irreducible representations for the Lie algebras sl2 and sl3.

Objectives

• Recognize actions of finite groups on complex or real vector spaces,
• decompose spaces into irreducible representations,
• understand the interplay between Lie groups and Lie algebras,
• classify the irreducible representations for sl2 and sl3

Credit

Homework:  I will assign several sets of homework problems. They will count for 60 % of the grade.

Presentations:  Two presentations together count for 40 % of the grade.

Further Information

For the Disability Policy, the Make-Up Policy, the Code of Academic Integrity, Religious Accommodation, my Grading Scale and Financial Assistance Opportunities please visit Infos for all my courses.

Contact Me

Office hours:  Mondays and Wednesdays, 3 - 5 p.m. in SE 230.