Course 424

Group Representations IIc

Dr Timothy Murphy

Mathematics 1.8 Monday, 19 April 1999 16:00–17:30

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 a measure on a compact space X? What is meant by saying that a measure on a compact group G is invariant? Sketch the proof that every compact group G carries such a measure. Prove that every representation of a compact group is semisimple.

2. Which of the following groups are (a) abelian, (b) compact, (c) 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} ?

Justify your answer in each case; no marks will be given for unsupported assertions.

3. Prove that every simple representation of a compact abelian group is 1-dimensional and unitary. Determine the simple representations of U(1). Determine also the simple representations of O(2).

4. 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}, …). Determine the representation-ring of SU(2), ie express each product D(i)D(j) as a sum of simple representations D(k).

5. 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).