|Module Title||GROUP REPRESENTATION THEORY|
|Co-ordinator||Professor A O Morris|
|Semester||Intended For Use In Future Years|
|Next year offered||N/A|
|Next semester offered||N/A|
|Course delivery||Lecture||20 x 1hour lectures|
|Seminars / Tutorials||7 x 1hour seminars|
|Resit assessment||2 hour written examination||100%|
Ever since the basic work of Frobenius (1896) and Schur (1901), representing groups by groups of matrices has been regarded as an invaluable tool in the study of groups. In more modern times there have been important applications not only in other branches of mathematics but also in physics and chemistry. This module contains all the background ideas with an emphasis on matrix representations and their characters and also on explicit methods to calculate these in important particular examples such as symmetric groups.
To introduce students to the concepts and the general theory of the representation of groups by matrices, to familiarise students with practical methods for constructing representations and character tables of groups and to enable them to obtain information about the structure of groups from their representations and character tables.
A student who successfully completes this course should be able to:
1. MATRIX REPRESENTATION OF GROUPS
Examples for cyclic groups, symmetric groups, permutation groups. Permutation and regular representations. Equivalence of representations.
2. IRREDUCIBLE REPRESENTATIONS
Reducible and completely reducible representations. Mashke's Theorem.
3. GROUPS CHARACTERS
Permutation and regular character. Examples. Condition of irreducibility and equivalence in terms of characters. Number of irreducible characters equal to number of classes of conjugate elements. Character tables. Schur's lemma and relations. Orthogonality relations.
4. CONSTRUCTION OF REPRESENTATIONS
Direct product of representations. Induced representations and characters. Lifting of representations.
5. CALCULATION OF CHARACTER TABLES
Programme for claculating character tables. Many examples, cyclic groups, abelian groups, metacyclic groups, symmetric and alternating groups.
6. APPLICATIONS TO GROUP THEORY
Normal subgroups. Simple groups. Centres of groups. Burnside's Theorem.
7. PROJECTIVE REPRESENTATIONS OF GROUPS
** Supplementary Text
W Ledermann. Introduction to Group Characters. CUP
** Essential Reading
G D James & M Liebeck. Representation of Characters and Groups. Cambridge Mathematical Textbooks, CUP