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:
- MA132 Foundations or MA138 Sets and Numbers
- MA136 Introduction to Abstract Algebra
- MA106 Linear Algebra
- MA251 Algebra I: Advanced Linear Algebra
- MA249 Algebra II: Groups and Rings
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:
Many books explain the theory well and give different examples:
James and Liebeck - Representations and Characters of Groups;
Steinberg - Representation Theory of Finite Groups; An Introductory Approach;
Sagan - The Symmetric Group; Representations, Combinatorial Algorithms and Symmetric Functions;
Fulton and Harris - Representation Theory; A First Course;
Serre - Linear Representations of Finite Groups.