Course 424

Group Representations II

Dr Timothy Murphy

School of Mathematics Tuesday, 30 April 1991 16:00–18:00

Answer as many questions as you can; all carry the same number of marks. All representations are finite-dimensional over ℂ.

1. Define a measure on a compact space. State carefully, and outline the main steps in the proof of, Haar’s Theorem on the existence of an invariant measure on a compact group. Prove that every representation of a compact group is semisimple.

2. Which of the following groups are (a) compact, (b) connected:

O(n), SO(n), U(n), SU(n), GL(n, ℝ), SL(n, ℝ), GL(n, ℂ), SL(n, ℂ)?

(Note: You must justify your answer in each case; no marks will be given for unsupported assertions)

3. Determine the conjugacy classes in SU(2); and prove that this group has just one simple representation of each dimension. Find the character of this representation. Determine the representation-ring of SU(2).

4. Show that there exists a surjective homomorphism

Θ : SU(2) → SO(3)

having finite kernel. Hence or otherwise determine all simple representations of SO(3). Determine also all simple representations of O(3).