# MA3E1 Groups & Representations

Lecturer: Christian Ikenmeyer

Term(s): Term 2

Status for Mathematics students: List A

Commitment: 30 one-hour lectures

Assessment: Homework 15%, 3 hour written exam 85%

Formal registration prerequisites: None

Assumed knowledge:

Useful background: Only the above

Synergies: This module goes well with other third year algebra modules, particularly Group Theory.

Leads to: The following modules have this module listed as assumed knowledge or useful background:

Content: The concept of a group is defined abstractly (as set with an associative binary operation, a neutral element, and a unary operation of inversion) but is better understood through concrete examples, for instance:

• Permutation groups
• Matrix groups
• Groups defined by generators and relations.

All these concrete forms can be investigated with computers. In this module we will study groups by:

• Finding matrix groups to represent them
• Using matrix arithmetic to uncover new properties. In particular, we will study the irreducible characters of a group and the square table of complex numbers they define. Character tables have a tightly-constrained structure and contain a great deal of information about a group in condensed form. The emphasis of this module will be on the interplay of theory with calculation and examples.

Aims: To introduce representation theory of finite groups in a hands-on fashion.

Objectives: To enable students to:

• Understand matrix and linear representations of groups and their associated modules
• Compute representations and character tables of groups
• Know the statements and understand the proofs of theorems about groups and representations covered in this module.

Books:

We will work through printed notes written by the lecturer.
A nice book that we shall not use is:
G James & M Liebeck, Representations and Characters of Groups, Cambridge University Press, 1993. Second edition, 2001. (IBSN: 052100392X).