Groups and Representations - MA576

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Module delivery information

Location Term Level1 Credits (ECTS)2 Current Convenor3 2020 to 2021
Canterbury
Spring 6 15 (7.5) DR C Bowman

Overview

Groups arise naturally in many areas of mathematics as well as in chemistry and physics. A concrete way to approach groups is by representing them as a group of matrices, in which explicit computations are easy. This approach has been very fruitful in developing our understanding of groups over the last century. It also helps students to understand aspects of their mathematical education in a broader context, in particular concepts from earlier modules (From Geometry to Algebra/Groups and Symmetries and Linear Algebra) have been amalgamated into more general and powerful tools.

This module will provide a rigorous introduction to the main ideas and notions of groups and representations. It will also have a strong computational strand: a large part of the module will be devoted to explicit computations of representations and character tables (a table of complex numbers associated to any finite group).

38 hours

Method of assessment

80% examination and 20% coursework.

We will not follow a single text, and the lecture notes will cover the entire syllabus. Nevertheless
G.D. James and M. Liebeck, Representations and characters of groups, CUP (2001)
J.P. Serre, Linear representations of finite groups, Springer GTM (1977)
J.L. Alperin and R.B. Bell, Groups and Representations, Springer GTM (1995)
contain a large amount of the material.

See the library reading list for this module (Canterbury)

Learning outcomes

The intended subject specific learning outcomes.
On successfully completing the level 6 module students will be able to:
1 demonstrate systematic understanding of key aspects of the theory and practice of groups (with examples including permutation groups and matrix groups, and the combinatorics of the symmetric group), of linear algebra, and of representations and characters of groups.
2 demonstrate the capability to deploy established approaches accurately to analyse and solve problems using a reasonable level of skill in calculation and manipulation of the material in the following areas: calculations within permutation groups and matrix groups; computations of the character tables of small groups; derivation of structural information about a group from its character table; formulation and proof of simple statements about groups and representations in precise abstract algebraic language; breaking up representations into smaller simpler objects.
3 apply key aspects of group theory and representation theory in well-defined contexts, showing judgement in the selection and application of tools and techniques.

The intended generic learning outcomes.
On successfully completing the level 6 module students will be able to:
1 manage their own learning and make use of appropriate resources.
2 understand logical arguments, identifying the assumptions made and the conclusions drawn
3 communicate straightforward arguments and conclusions reasonably accurately and clearly
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working
5 solve problems relating to qualitative and quantitative information
6 make competent use of information technology skills such as online resources (moodle), internet communication.
7 communicate technical material competently
8 demonstrate an increased level of skill in numeracy and computation
9 demonstrate the acquisition of the study skills needed for continuing professional development.

Notes

1. Credit level 6. Higher level module usually taken in Stage 3 of an undergraduate degree.
2. ECTS credits are recognised throughout the EU and allow you to transfer credit easily from one university to another.
3. The named convenor is the convenor for the current academic session.