# REPRESENTATION THEORY - 2022/3

Module code: MATM035

## Module Overview

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.

### Module provider

Mathematics

NIETO GARCIA Juan (Maths)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 106

Lecture Hours: 11

Seminar Hours: 11

Guided Learning: 11

Captured Content: 11

Semester 1

None

## Module content

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.

## Assessment pattern

Assessment type Unit of assessment Weighting
Online Scheduled Summative Class Test ONLINE TEST 20
Examination Online ONLINE EXAM 80

N/A

## Assessment Strategy

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.

## Module aims

• 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.

## Learning outcomes

 Attributes Developed 001 Understand the concepts, theorems and techniques of group representation theory. K 002 Have a clear understanding of how to construct irreducible representations of groups. CT 003 Be able to explicitly apply the theory to small order groups. CT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## 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.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.