# REPRESENTATION THEORY - 2024/5

Module code: MATM035

## Module Overview

Group theory is a branch of mathematics developed to understand symmetries, which are powerful tools for understanding the properties of complex mathematical problems and physical systems. Group representations map abstract group elements to linear transformations on vector spaces. Representation theory allows us to better understand the properties of symmetry groups, and leads to powerful and compact solutions of otherwise difficult and intractable problems.

This module builds on groups theory in MAT1031 Algebra and MAT2048 Groups & Rings.

### Module provider

Mathematics & Physics

PRINSLOO Andrea (Maths & Phys)

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

Independent Learning Hours: 64

Lecture Hours: 33

Tutorial Hours: 5

Guided Learning: 15

Captured Content: 33

Semester 2

None

## Module content

Indicative content includes:

• Basic group theory (a review): Definitions and examples of groups. Subgroups and cosets. Conjugacy classes. Homomorphism and isomorphisms.

• Representation theory: Definitions and examples of group representations. Reducible and irreducible representations. Unitarity and reducibility of groups. Schur's Lemmas. Group algebra. Orthogonality relations. Characters and character tables. Tensor product representations.

• Lie groups and Lie algebras: Definitions and examples of Lie groups. The matrix exponential. The Lie algebra of a Lie group, and examples of Lie algebras. Lie subalgebras. Simple and semi-simple Lie algebras. Structure constants.

• Representations of Lie algebras: The adjoint representation. Lie algebra homomorphisms. Lie algebra representations. Reducible and irreducible representations of Lie algebras. Skew-adjoint representations. Complexification of a real Lie algebra, and representations of real and complexified Lie algebras.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-Semester Test (50 mins) 20
Examination End-of-Semester Examination (2 hours) 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding of subject knowledge, and recall of key definitions and theorems in representation theory.

• The ability to construct simple proofs similar to those in the module.

• The ability to identify and use appropriate techniques to solve problems relating to finite groups, Lie groups and Lie algebras.

Thus, the summative assessment for this module consists of:

• One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1, 2 and 5.

• A synoptic examination (2 hours), worth 80% of the module mark, corresponding to all Learning Outcomes 1 to 5.

Formative assessment

There are two formative unassessed courseworks over an eleven week period, designed to consolidate student learning.

Feedback

Students will receive individual written feedback on both the formative unassessed courseworks and the in-semester test. The feedback is timed such that feedback from the first coursework will assist students with preparation for the in-semester test. The feedback from both courseworks and the in-semester test will assist students with preparation for the synoptic examination. Students also receive verbal feedback in tutorials and office hours.

## Module aims

• Provide students with an introduction to the basic definitions and theorems of representation theory.
• Develop students' understanding of representations of finite groups.
• Develop students' understanding of Lie groups and Lie algebras, and their representations.
• Enable students to apply techniques from representation theory to better understand symmetries in mathematical problems and physical systems.

## Learning outcomes

 Attributes Developed 001 Students will understand the definitions, theorems and techniques of representation theory. K 002 Students will be able to construct character tables and irreducible representations of finite groups. KC 003 Students will recognise standard examples of matrix Lie groups and will be able to construct the associated Lie algebras. KC 004 Students will be able to construct representations of Lie groups and Lie algebras, including the trivial, adjoint and defining representations of matrix Lie groups and Lie algebras. KC 005 Students will be able to construct simple proofs similar to those encountered in the module. KCT

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 students with an introduction to the representation theory of finite groups, Lie groups and Lie algebras.

• Provide students with experience of techniques in representation theory used to interpret, understand and solve mathematical problems and study physical systems with underlying symmetry groups.

The learning and teaching methods include:

• Three one-hour lectures for eleven weeks, with module notes provided to complement the lectures. These lectures provide a structured learning environment and opportunities for students to ask questions and to practice methods taught.

• Five biweekly one-hour tutorials per semester. These tutorials provide an opportunity for students to gain feedback and assistance with the exercises which complement the module notes.

• Two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive feedback on these courseworks as guidance on their progress and understanding.

• Lectures may be recorded or equivalent recordings of lecture material provided. These recordings are intended to give students an opportunity to review parts of lectures which they may not fully have understood and should not be seen as an alternative to attending lectures.

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.

Upon accessing the reading list, please search for the module using the module code: MATM035

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourceness and Resilience, and Sustainability. This module is designed to allow students to develop knowledge, skills and capabilities in the following areas:

Digital Capabilities: The SurreyLearn page for MATM035 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing.

Employability: The module MATM035 equips students with skills which significantly enhance their employability. The mathematical proficiency gained will hone their critical thinking and problem-solving abilities. Students will learn to evaluate complex problems relating to abstract algebra and symmetries groups of physical systems, break them into manageable components, and apply representation theory and logical reasoning to arrive at solutions. These are highly sought after skills in many professions.

Global and Cultural Capabilities: Students enrolled in MATM035 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problem-solving teaching activities in tutorials and lectures, which naturally facilitates the sharing of different cultures.

Resourcefulness and Resilience: MATM035 is a module which demands a rigorous approach to abstract algebra and representation theory, to which students will learn to adapt. They will gain skills in analysing problems in abstract algebra relating to symmetry groups of physical systems using lateral thinking. Students will complete assessments which challenge them and build resilience.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Statistics MMath 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics MMath 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics and Physics MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics and Physics MMath 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics MSc 2 Optional A weighted aggregate mark of 50% is required to pass the module

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 2024/5 academic year.