| 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. Rickard |
| Open unit status | Not open |
| Pre-requisites | |
| Co-requisites |
None |
| School/department | School of Mathematics |
| Faculty | Faculty of Science |
The basic idea of representation theory is to describe or represent some unfamiliar algebraic system - like a large finite group or an algebra - by a more familiar system - often a group or collection of matrices defined over the complex numbers. At present this is one of the main research areas in pure mathematics. The unit will introduce the basic concepts, prove some elementary results about characters, and apply them to derive a few important results in pure group theory, including Burnsides theorem on solubility of finite groups.
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.
Syllabus
Review of group actions.
Representations and modules; equivalence of the two ideas; subrepresentations and homomorphisms; irreducible representations.
Schur's Lemma and Maschke's Theorem.
Characters; inner product of characters; character tables; orthogonality relations.
Tensor products, induction and restriction of representations and characters.
Examples of the construction of character tables.
Burnside's theorem and other applications to group theory.
[If time allows, the rudiments of the representation theory of compact topological groups, stressing the analogy with finite groups, or another further topic (non-examinable).]
Relation to Other Units
This is one of three Level 4 units which develop abstract algebra in various directions. The others are Galois Theory and Algebraic Topology.
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.
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).
The final assessment mark for Representation Theory is calculated from a 2½-hour written examination in April consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted in this examination.
G. James and M. Liebeck, Representations and characters of groups, 2nd Edition C.U.P., 2001.
W.Ledermann, Introduction to group characters, C.U.P., 1977.
J.-P.Serre, Linear representations of finite groups, Springer, 1977
C.B. Thomas, Representations of Finite and Lie Groups, Imperial College Press, 2004
James and Liebeck is the recommended book. Ledermann covers similar material, but in a little less detail. Serre is concise and elegant, and may be more useful for consolidating ideas than for a first treatment.
