Representation Theory

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

After setting up the basics of the general theory of representations of groups, this unit will concentrate on representations of finite groups over the complex numbers. The theoretical properties of the character table of a group will be studied in detail, together with practical methods of calculating the character tables of particular groups, and several applications of the theory will be given.

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:

• know the standard general properties of the character table of a finite group, and have an understanding of why these properties hold.
• be able to apply a variety of methods for constructing characters.
• be able to deduce properties of a group from its character table.

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.

Syllabus

Review of group actions.

Representations; 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.

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.

Unit code: MATHM4600
Level of study: M/7
Credit points: 20
Teaching block (weeks): 1 (1-12)
Lecturer: Dr Tim Burness

Pre-requisites

MATH21100 Linear Algebra 2; MATH3330 Group Theory

None

Methods of teaching

Lectures, exercises to be done by the students.

Methods of Assessment

The pass mark for this unit is 50.

The final mark is calculated as follows:

• 90% from a 2 hour 30 minute exam
• 10% from assigned homework questions

NOTE: Calculators are not allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.