Course 424

Group Representations II

Dr Timothy Murphy

EELT3  Tuesday, 13 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. 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 SO(2). Determine also the simple representations of O(2).

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}, …). Determine the representation-ring of SU(2), ie 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 over ℂ into real, essentially complex and quaternionic. Give an example of each (justifying your answers). Show that if α is a simple representation with character χ then the value of

∫_{G} χ(g^{2}) dg

determines which of these three types α falls into.

Continued overleaf