Markus Schmidmeier
Mathematical Sciences
Florida Atlantic University

# Groups and Representations

Fall 2016

Welcome to my course on Groups and Representations (CRN: 14175, MAS 9396, 3 credits). We meet Monday, Wednesday and Friday, 9:00 - 9:50 a.m. in SE 215.

In this course we treat several topics from the theory of groups and their representations. After an introduction to mostly finite groups, we study the structure of the general linear group with coefficients in a finite field. We provide two approches to group representations: First, we introduce semisimple algebras and classify their modules. Then, we use characters and character tables to understand the decomposition of a group representation.

Prerequisite

Introductory Abstract Algebra (MAS 5311) and Linear Algebra (e.g. MAS 5145).

Textbook and Topics

Textbook: J. L. Alperin and R. B. Bell, Groups and Representations, Springer, Gratuate Texts in Mathematics 162, ISBN 0-387-94526-1, (1995).

Topics:

 Introduction to groups Sections 2, 3(3 weeks) Group automorphisms, the action of a group on a set. The General Linear Group Sections 4-6(3+ weeks) Bruhat decomposition of the general linear group, parabolic subgroups, special linear group. Composition Series Section 10(1+ weeks) Composition series of groups, Jordan-Hölder theorem. Semisimple Algebras Sections 12, 13(3 weeks) The theorems by Maschke and Wedderburn Characters Sections 14, 15(as time permits) Characters and the character table.

Objectives

• Recognize actions of finite groups on complex or real vector spaces,
• Develop familiarity with important groups,
• Use Maschke's theorem to distinguish the modular from the semisimple case,
• Understand representations of semisimple algebras, and
• Use characters to decompose spaces into irreducible representations,

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.

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).

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

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.

Credit

Homework:  I will assign seven sets of homework problems. The best six (midterm: 3) will count for 60 % of the grade.

Presentations:  Two presentations (midterm: 1 presentation before 10-7) of about 10 minutes each, typically about a problem from the "Further Exercises", together count for 40 % of the grade.

Final Exam Time:  We will have presentations during the final exam time: Friday, December 9, 7:45 - 10:15 a.m.

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:  MWRF, 12 noon - 1 p.m. in SE 272.