Group Representation Theory (0540514) 10 credits

Representation Theory of the Classical Lie Groups

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Aims

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

This course is very demanding in terms of effort, yet very rewarding in terms of gained knowledge.

Objectives

By the end of the course students should know:

• basic topological and analytic properties of classical Lie groups,
• general properties of the finite-dimensional representations of compact groups,
• finite-dimensional representations of "small" Lie groups.

Syllabus

1. Topological groups and classical Lie groups.
2. General properties of group representations.
3. Existence of invariant measure on a compact topological group.
4. Averaging over compact topological groups.
5. General properties of group characters.
6. Invariant integration over the unitary groups.
7. Irreducible representations of the unitary groups.
8. The unitary trick of Hermann Weyl.
9. Irreducible characters of the classical Lie groups.

Prerequisites

• Groups and Rings (0590019)
• Differential Geometry (0590302)
• Metric Spaces (0590313)
• Matrices and Forms (0590314)
• Topology (0590317)
• Galois Theory (0590203)
• Lie Algebras (0590312)
• Advanced Algebra (0590406 )

Recommended texts

1. A Barut and R Raczka, Theory of group representations and applications, World Scientific, 1986 (S 2.86 BAR) *
2. W Fulton and J Harris, Representation theory: a first course, Springer, 1991 (S 2.86 FUL) ***
3. R Goodman and N Wallach, Representations and invariants of the classical groups, Cambridge University Press, 1998 (S 2.86 GOO) *
4. H Weyl, The classical groups: their invariants and representations, Princeton University Press, 1997 (S 2.86 WEY) **

Teaching and support teaching

• 2 lectures per week
• problems classes by request

Assessment

• Examination in Week 1 of Spring Term (90%)
• Coursework (10%)

Elective Information

Not available

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Revised 5 March 2009