MATH6156 Modules and Representations
This module provides an introduction to the theory of modules over a principal ideal domain and the representation theory of finite groups, two basic tools in advanced mathematics.
Aims and Objectives
This is a compulsory lecture module for MMath students in their fourth year, building on topics studied earlier in group theory and applied mathematics. The aim of this modules is to extend the successful theory of vector spaces to the case of modules over principal ideal domains, and provide a structure theory for such modules with applications to finitely generated abelian groups and matrix normal forms. From there, the students will be introduced to various matrix groups, in order to furnish them with a variety of examples. If time permits, there will also be a discussion of crystallographic groups in dimensions 2 and 3. The notion of representing an abstract group by matrices leads on to representation theory and character theory, and the last part of the module will develop this theory for finite groups over the complex field, introducing characters, orthogonality relations with the aim of reaching Burnside’s celebrated p-q Theorem, as a demonstration of the power of the theory
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;
- Understand and apply the classification of finitely generated modules over a principal ideal domain.
- understand the definitions and basic properties of the classical linear groups;
- calculate the irreducible representations and character tables of some small finite groups
• 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
Learning and Teaching
Teaching and learning methods
Lectures, printed notes, private study
|Preparation for scheduled sessions||24|
|Wider reading or practice||12|
|Completion of assessment task||24|
|Total study time||150|
Resources & Reading list
SERRE J-P. Linear Representations of Finite Groups.
ALPERIN J L & BELL R B. Groups and representations.
FULTON W & HARRIS J,. Representation Theory.
CAMERON, P J. Introduction to algebra.
CURTIS M L. Matrix Groups.
LANG S. Algebra.
ELLIOTT J P & DAWBER P G. Symmetry in Physics, vol. 1.
GORDON, J & Liebeck, M. Representations and Characters of Finite Groups.
Repeat type: Internal & External
To study this module, you will need to have studied the following module(s):