Course 424

Group Representations II

Dr Timothy Murphy

G.M.B.  Tuesday, 13 April 1993  14:00–16:00

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. 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:

SO(2), O(2), U(2), SU(2), Sp(2), GL(2, ℝ), SL(2, ℝ), GL(2, ℂ), SL(2, ℂ), 𝕋^{2} ?

(Note: 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 the representation D(j) of dimensions 2j + 1 (where j = 0, \frac{1}{2}, 1, \frac{3}{2}, …). Express each product D(i)D(j) as a sum of simple representations D(k).

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). Determine also all simple representations of O(3).

5. Explain the division of simple representations of a finite or compact group G into real, essentially complex and quaternionic. Determine into which of these 3 categories the representations D(j) of SU(2) fall.