Course 424

Group Representations II

Dr Timothy Murphy

Arts Block A2039 Friday, 20 January 1989 15.45–17.45

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

1. Explain carefully and fully what is meant by the statement that there exists a unique left-invariant measure μ on every compact group G; and sketch the proof of this statement. Determine the measure in SU(2) of the set of all matrices having eigenvalues e^{±iθ}, where 0 ≤ θ ≤ π/4.

2. Prove that every simple representation of a compact abelian group A is 1-dimensional. 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 1, 2, …; and determine the character of this representation.

4. Determine the conjugacy classes in SO(3).

Show that there exists a covering Θ : SU(2) → SO(3), ie a surjective homomorphism with finite kernel. Hence or otherwise determine the simple characters of SO(3). Show that all the representations of SO(3) are real.