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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: 2022-23
  • 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

Additional information

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 period
A	Spring Term 2022-23

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.

Academic and graduate skills

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
Closed/in-person Exam (Centrally scheduled) Representation Theory of the Classical Groups Exam	2 hours	60
Coursework - extensions not feasible/practicable Representation Theory of the Classical Groups	N/A	40

Special assessment rules

None

Reassessment

Task	Length	% of module mark
Closed/in-person Exam (Centrally scheduled) Representation Theory of the Classical Groups Exam	2 hours	60
Coursework - extensions not feasible/practicable Representation Theory of the Classical Groups	N/A	40

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.

Indicative reading

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.