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 | |
Co-requisites |
None |
School/department | School of Mathematics |
Faculty | Faculty of Science |
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.
Learning Objectives
After taking this unit, students should:
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.
The unit will be taught through a combination of
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.
If you fail this unit and are required to resit, reassessment is by a written examination in the August/September Resit and Supplementary exam period.
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