# Math 473: Group Representation Theory.

### Title

Group Representation Theory.

(3:3:0)

F

### Description

FG-modules; Maschke's theorem; Shur's lemma; characters of groups; orthogonality relations of characters; induced, lifted, and restricted characters; construction of character tables; Burnside's theorem.

## Desired Learning Outcomes

This course is aimed at undergraduate mathematics majors. It is a second course in abstract algebra, and covers the representation theory of finite groups. Representation theory is an important topic in mathematics, as well as having applications in physics and chemistry.

### Prerequisites

Math 371 is a prerequisite for this course. In particular, students should be familiar with the concepts of groups, rings, and fields, and they should understand quotient groups, quotient rings, and homomorphisms. By this point in their mathematical career, students should be comfortable proving theorems by themselves.

### Minimal learning outcomes

Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the below concepts, related to, but not identical to, statements proven by the text or instructor.

1. FG-modules
• Groups and homomorphisms
• Vector spaces and linear transformations
• Group representations
• Group Algebras
• Definition of FG-modules
• FG-Homomorphisms
2. Reduciblity of modules
• Maschke’s Theorem
• Schur’s Lemma
• Irreducible modules
3. Character Theory
• Definition of Characters
• Inner products of characters
• Conjugacy classes
• The number of irreducible characters
• Orthogonality relations
• Character Tables
4. Operations on Characters
• Normal subgroups and lifted characters
• Tensor products
• Restriction to a subgroup
• Induced modules and characters
5. Properties of Characters
• Real representations
• Divisibility properties
• Properties of character tables
6. Applications of characters to group theory
• Characters of groups of order pq
• Characters of some p-groups
• Burnside’s pq theorem
7. Representations of symmetric groups
• Young tableaux
• Frobenius formula

Some textbooks that treat all (or almost all) of the minimal learning objectives at an appropriate level are Representations and Characters of Groups by Gordon James and Martin Liebeck, and Character theory of finite groups, by M. Isaacs.