Module Catalogue 2026/27

Pre Requisite Comment

MAS2708 would be acceptable in place of MAS2703

Co Requisite Comment

N/A

Aims

To introduce the basic ideas of studying groups through their representations as matrices, and to describe finite dimensional complex representations of finite groups.

Module Summary

The module presents the theory of finite dimensional complex representations of finite groups including the discussion of important classes of examples. Starting from the motivating question how a group can act linearly on a vector space, students will see an instance of a complete mathematical theory. While of major importance for the study of finite groups, this setup also forms a starting point for more general representation theory.

During the module it will be explained how relaxing any of the assumptions on the objects studied (finite, groups, complex, finite dimensional) leads to open questions in an active research area, which the students will be able to understand.

Outline Of Syllabus

Review of group theory, general linear group. Review of linear algebra. (Complex) representations of groups (sub-representations, morphisms of representations). Maschke’s Theorem. Schur’s Lemma. Characters and orthogonality relations. Regular representation. Projection formulas. Representations of the symmetric group.

Intended Knowledge Outcomes

Students will be able to demonstrate a reasonable understanding of the theory of representations of finite groups. They will be able to reproduce, explain, and apply concepts and results of representation theory as listed in the syllabus.

Intended Skill Outcomes

Upon successful completion of the module students will be able to: perform elementary mathematical arguments involving representation theory of finite groups (see knowledge outcomes), determine character tables for small groups, use them to decompose representations into irreducible sub-representations, and manipulate Young diagrams/tableaux and interpret them in terms of representations of the symmetric group.

Students will develop skills across the cognitive domain (Bloom’s taxonomy, 2001 revised edition): remember, understand, apply, analyse, evaluate and create.

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking.

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

Note: the exam for MAS8709 is more challenging than the exam for MAS3709.

General Notes

N/A