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# Representation Theory of the Classical Groups - MAT00084M

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• Department: Mathematics
• Module co-ordinator: Prof. Michael Bate
• Credit value: 10 credits
• Credit level: M
• Academic year of delivery: 2020-21
• See module specification for other years: 2021-22

## Module summary

To introduce students to the modern Representation Theory by working with the classical examples: the general linear, orthogonal and symplectic groups.

## Related modules

• None

### Prohibited combinations

• None

Pre-requisite knowledge for MSc students: familiarity with and maturity in algebraic structures such as groups, and in analytic structures such as metric spaces.

## Module will run

Occurrence Teaching cycle
A Spring Term 2020-21

## Module aims

To introduce students to the modern Representation Theory by working with the classical examples: the general linear, orthogonal and symplectic groups.

## Module learning outcomes

Subject content

• Basic topological and analytic properties of the classical groups.
• General properties of the finite-dimensional representations of compact groups.
• Finite-dimensional representations of "small" classical groups.

Develop an appreciation of the power of combining algebraic with analytic tools learnt in the Mathematics programme, and their applicability. This course gives an essential first taste of representations to those who will go on to further study, and also an “exit level” module for those who wish to appreciate the area of Representation Theory, which is one of the major themes of modern Mathematics.

## Module content

Syllabus

• Topological groups and classical groups.
• General properties of group representations.
• Invariant measure on a compact topological group.
• General properties of group characters.
• Invariant integration over the unitary groups.
• Irreducible representations of the unitary groups.
• Irreducible characters of the classical Lie groups

## Assessment

Task Length % of module mark
Essay/coursework
Coursework
N/A 40
Online Exam
Representation Theory of the Classical Groups
N/A 60

None

### Reassessment

Task Length % of module mark
Essay/coursework
Coursework
N/A 40
Online Exam
Representation Theory of the Classical Groups
N/A 60

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

A. Barut and R. Raczka, Theory of group representations and applications, World Scientific, 1986.

W. Fulton and J. Harris, Representation theory: a first course, Springer, 1991.

R. Goodman and N. Wallach, Representations and invariants of the classical groups, CUP, 1998.

H. Weyl, The classical groups: their invariants and representations, Princeton University Press, 1997.

The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.