Unit information: Representation Theory in 2019/20

Unit name	Representation Theory
Unit code	MATHM4600
Credit points	20
Level of study	M/7
Teaching block(s)	Teaching Block 1 (weeks 1 - 12)
Unit director	Professor. Tim Burness
Open unit status	Not open
Pre-requisites	MATH21100 Linear Algebra 2 and MATH33300 Group Theory
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Unit Aims

To develop the basic theory of linear representations of groups, especially of finite groups over the complex numbers. To develop techniques for constructing characters and character tables. To explore applications of the theory.

Unit Description

Representation theory studies the linear actions of a group G on a vector space V defined over a field. By fixing a basis for V, such an action yields a map from G to a group of invertible matrices, so we can "represent" the elements of G in a very concrete form. Moreover, this viewpoint allows us to apply techniques and tools from linear algebra to study groups, and this turns out to be a very powerful idea.

In this course, we will develop the basic theory of linear representations of groups, with a particular focus on finite groups and representations defined over the complex numbers. We will also introduce the theory of characters as a tool for studying representations and we will develop techniques for constructing characters and character tables. We will also describe some important applications of the theory, including Burnside's famous theorem on the solubility of finite groups of order p^aq^b.

Relation to Other Units

This is one of three Level 7 units which develop abstract algebra in various directions. The others are Galois Theory and Algebraic Topology.

Intended Learning Outcomes

Learning Objectives

After taking this unit, students should:

• know the standard general properties of the character table of a finite group, and have an understanding of why these properties hold.
• be able to apply a variety of methods for constructing characters.
• be able to deduce properties of a group from its character table.

Transferable Skills

The application of abstract ideas to concrete calculations. The ability to tackle problems by making a sensible choice from among a variety of available techniques.

Teaching Information

Lectures, exercises to be done by the students. (If there is not sufficient demand this unit may be given as a directed reading course, or not at all).

Assessment Information

90% Examination. 10% Coursework.

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

Reading and References

Recommended

• Gordon James and M.W. Liebeck, Representations and Characters of Groups, 2nd Edition Cambridge University Press, 2001
• Walter Ledermann, Introduction to Group Characters, Cambridge University Press, 1977
• Jean-Pierre Serre, Linear Representations of Finite Groups, Springer, 1977
• Charles B. Thomas, Representations of Finite and Lie Groups, Imperial College Press, 2004