Course 424

Group Representations II

Dr Timothy Murphy

Exam Hall Tuesday, 4 April 1997 15:15–16:45

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

1. What is meant by an invariant measure on a compact group G? Sketch the proof that every compact group G carries such a measure. To what extent is this measure unique?

2. Prove that every simple representation of a compact abelian group is 1-dimensional and unitary.

Determine the simple representations of U(1).

3. Determine the conjugacy classes in SU(2). Prove that SU(2) has just one simple representation of each dimension m = 1, 2, 3, …; and determine the character of this representation.

4. Show that there exists a surjective homomorphism

Θ : SU(2) → SO(3)

with finite kernel.

Hence or otherwise determine all simple representations of SO(3).