| Unit name | Representation Theory |
|---|---|
| Unit code | MATHM4600 |
| Credit points | 20 |
| Level of study | M/7 |
| Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |
| Unit director | Professor. Tim Dokchitser |
| Open unit status | Not open |
| Units you must take before you take this one (pre-requisite units) |
MATH33300 Group Theory |
| Units you must take alongside this one (co-requisite units) |
None |
| Units you may not take alongside this one |
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 how groups can act on vector spaces, in other words how elements of a group can be "represented" by matrices (which is where the name comes from). It is a bit of a blend of 80% group theory, 10% algebra and 10% linear algebra. Most examples of finite groups come either from permutation groups (acting on sets) or groups of symmetries (acting on vector spaces or things like polyhedra inside vector spaces), and representation theory studies the latter. This is a classical and very pretty subject with neat and clear results, at least if one to sticks to finite groups acting on complex vector spaces.
We will introduce the theory of characters as a tool for studying representations and we will develop techniques for constructing characters and character tables, that encode all representation of a given group. 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, and look at some groups of small order
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. It would be desirable, if students taking this unit have done Linear Algebra 2 before.
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 selection of lectures, online materials, independent activities such as problem sheets and other exercises, problem classes and office hours.
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.
When assessment does not go to plan
The Board of Examiners will consider all cases where students have failed or not completed the assessments required for credit. The Board considers each student's outcomes across all the units which contribute to each year's programme of study. If you have self-certificated your absence from an assessment (or, in certain scenarios, if you had failed the assessment), you will normally be required to complete it in the reassessment period. Please refer to your official results for the details of your reassessments. The Board of Examiners will take into account any exceptional circumstances and operates within the Regulations and Code of Practice for Taught Programmes.
If this unit has a Resource List, you will normally find a link to it in the Blackboard area for the unit. Sometimes there will be a separate link for each weekly topic.
If you are unable to access a list through Blackboard, you can also find it via the Resource Lists homepage. Search for the list by the unit name or code (e.g. MATHM4600).
How much time the unit requires
Each credit equates to 10 hours of total student input. For example a 20 credit unit will take you 200 hours
of study to complete. Your total learning time is made up of contact time, directed learning tasks,
independent learning and assessment activity.
See the University Workload statement relating to this unit for more information.
Assessment
The assessment methods listed in this unit specification are designed to enable students to demonstrate the named learning outcomes (LOs). Where a disability prevents a student from undertaking a specific method of assessment, schools will make reasonable adjustments to support a student to demonstrate the LO by an alternative method or with additional resources.
The Board of Examiners will consider all cases where students have failed or not completed the assessments required for credit.
The Board considers each student's outcomes across all the units which contribute to each year's programme of study. For appropriate assessments, if you have self-certificated your absence, you will normally be required to complete it the next time it runs (for assessments at the end of TB1 and TB2 this is usually in the next re-assessment period).
The Board of Examiners will take into account any exceptional circumstances and operates
within the Regulations and Code of Practice for Taught Programmes.
