MAS8709 : Representation theory

• Offered for Year: 2023/24
• Module Leader(s): Dr Martina Balagovic
• Owning School: Mathematics, Statistics and Physics
• Teaching Location: Newcastle City Campus
Semesters
 Semester 2 Credit Value: 10 ECTS Credits: 5

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 (subrepresentations, morphisms of representations). Maschke’s Theorem. Schur’s Lemma. Characters and orthogonality relations. Regular representation. Projection formulas. Representations of the symmetric group.

Teaching Methods

Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision Lectures
Scheduled Learning And Teaching ActivitiesLecture201:0020:00Formal Lectures
Scheduled Learning And Teaching ActivitiesLecture51:005:00Problem Classes
Guided Independent StudyIndependent study581:0058:00Preparation time for lectures, background reading, coursework review
Guided Independent StudyIndependent study151:0015:00Completion of in course assessments
Total100:00
Jointly Taught With
Code Title
MAS3709Representation theory
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 Methods

The format of resits will be determined by the Board of Examiners

Exams
Description Length Semester When Set Percentage Comment
Written Examination1202A80N/A
Exam Pairings
Module Code Module Title Semester Comment
MAS3709Representation theory2Taught together
Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises2M5Problem-solving exercises assessment
Prob solv exercises2M5Problem-solving exercises assessment
Prob solv exercises2M5Problem-solving exercises assessment
Prob solv exercises2M5Problem-solving exercises assessment
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 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