Modules and Representations

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• Understand the notion of a module over a ring, and the basic properties of a free module;
• calculate the irreducible representations and character tables of some small finite groups
• understand the definitions and basic properties of the classical linear groups;
• Understand and apply the classification of finitely generated modules over a principal ideal domain.

• Modules: definitions, first examples; basic properties; submodules; factor modules; isomorphism theorems; correspondence theorem.
• Free modules; rank; universal property; free modules over integral domains; the torsion submodule.
• Modules over a principal ideal domain; The classification of finitely generated modules over a principal ideal domain.
• The classification of finitely generated abelian groups.
• The Jordan normal form of matrices over the complex numbers.
• Matrix groups; general and special linear, orthogonal, symplectic and unitary groups; possibly a survey of crystallographic groups in dimensions 2 and 3.
• Representation theory for finite groups over the complex numbers; Schur’s Lemma, Maschke’s Theorem, character theory and examples of character tables in small examples.
• Burnside’s p-q Theorem

Teaching and learning methods

Lectures, printed notes, private study

Resources & Reading list

Textbooks

ELLIOTT J P & DAWBER P G. Symmetry in Physics, vol. 1. MacMillan.

GORDON, J & Liebeck, M. Representations and Characters of Finite Groups.

CAMERON, P J. Introduction to algebra. OUP.

SERRE J-P. Linear Representations of Finite Groups. Springer.

LANG S. Algebra. Springer.

FULTON W & HARRIS J,. Representation Theory. Springer.

CURTIS M L. Matrix Groups. Springer.

ALPERIN J L & BELL R B. Groups and representations. Springer.

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat Information

Repeat type: Internal & External