REPRESENTATION THEORY - 2017/8
Module code: MATM035
MCORIST J Dr (Maths)
Number of Credits
FHEQ Level 7
Module cap (Maximum number of students)
Overall student workload
Lecture Hours: 34
|Assessment type||Unit of assessment||Weighting|
|School-timetabled exam/test||IN-SEMESTER TEST||20%|
Prerequisites / Co-requisites
Symmetries are a powerful method for easily understanding properties of otherwise complicated mathematical and physical objects. Group theory is a branch of mathematics developed to understand symmetries, however it often leads to complicated abstract quantities. Group representation theory turns such abstract algebraic concepts into linear transformations of vector spaces, a much easier system to solve. In doing, representation theory can unveil deep symmetry properties of physical systems as well as leading to powerful and compact solutions to otherwise difficult and intractable problems.
Introduce the concept of a group representation.
Construct representations of finite groups and some simple examples of Lie groups and algebras.
Develop the concept of Dynkin diagrams and it to construct representations of Lie algebras.
Illustrate these concepts in physical systems.
|Understand the concepts, theorems and techniques of group representation theory.||K|
|Have a clear understanding of how to construct irreducible representations of groups.||CT|
|Be able to explicitly apply the theory to small order groups.||CT|
C - Cognitive/analytical
K - Subject knowledge
T - Transferable skills
P - Professional/Practical skills
1. Symmetry and finite groups: review of group theory definitions, subgroups, cyclic groups, cosets, conjugacy elements and classes.
2. Representation theory basics: definitions, Schur's lemma, reducible and irreducible representations, orthogonality theorems, decomposition of reducible representations. Example representations of finite groups.
3. Lie groups and algebras: generators, Jacobi identity, Lie algebra, adjoint representation, exponentiation
4. SU(2): definition, representations, raising and lowering operators
5. Roots and weights: weights, roots, raising and lowering, su(3)
6. Dynkin diagrams and simple roots: positive weights, simple roots, Dynkin diagrams
7. Physical applications: quantum mechanics.
Methods of Teaching / Learning
The learning and teaching strategy is designed to provide:
A detailed introduction to the relevant group theory and representation theory
Experience the methods used to interpret, understand and solve concrete problems, especially for small order groups
The learning and teaching methods include:
3 x 1 hour lectures per week x 11 weeks, with black/whiteboard written notes to supplement the module notes.
The assessment strategy is designed to provide students with the opportunity to demonstrate:
· Understanding of and ability to interpret and manipulate mathematical statements.
· Subject knowledge through the recall of key definitions, theorems and their proofs.
· Analytical ability through the solution of unseen problems in the test and exam.
Thus, the summative assessment for this module consists of:
· One two-hour examination at the end of the Semester; worth 80% of the module mark.
· An in-semester test worth 20% of the module mark when combined together.
Formative assessment and feedback
Students receive written feedback via a number of un-assessed coursework assignments over the 11-week period. Students are then encouraged to arrange meetings with the module convener for verbal feedback.
Reading list for REPRESENTATION THEORY : http://aspire.surrey.ac.uk/modules/matm035
Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2017/8 academic year.