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Representation Theory (MATH M4600)

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Administrative Information

  1. Unit number and title: MATH M4600 Representation Theory
  2. Level: M/7
  3. Credit point value: 20 credit points
  4. Year: 07/08
  5. First Given in this form: 1997-98
  6. Unit Organiser: Jeremy Rickard
  7. Lecturer: Prof J Rickard
  8. Teaching block: 1
  9. Prerequisites: MATH 21100 Linear Algebra 2; MATH 33300 Group Theory (may be taken concurrently).

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.

General Description of the Unit

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 4 units which develop abstract algebra in various directions. The others are Galois Theory and Algebraic Topology.

Teaching Methods

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).

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.

Assessment Methods

The 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.

Award of Credit Points

Credit points are gained by either passing the unit or by getting an examination mark of 30 or over and also handing in satisfactory attempts to at least 5 homework assignments.

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.

Texts

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 strongly recommended, and much of the course will follow this book quite closely. 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.

Syllabus

Review of group actions.

Representations and FG-modules; equivalence of the two ideas; submodules and homomorphisms; irreducible modules.

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 (non-examinable).