Course 424

Group Representations IIa

Dr Timothy Murphy

Room WR3.8 Wednesday, 30 April 2003 15:15–16:45

Attempt 5 questions. (If you attempt more, only the best 5 will be counted.) All questions carry the same number of marks. All representations are finite-dimensional over ℂ, unless otherwise stated.

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.

6. Define the representation α × β of the product-group G × H, where α is a representation of G, and β of H.

Show that if G and H are finite then α × β is simple if and only if both α and β are simple; and show that every simple representation of G × H is of this form.

Show that the symmetry group G of a cube is expressible as a productgroup

G = C_{2} × S_{4}.

Let γ denote the 4-dimensional representation of G defined by its action on the 4 diagonals of the cube. Express γ in the form

γ = α_{1} × β_{1} + ⋯ + α_{r} × β_{r},

where α_{1}, …, α_{r} are simple representations of C_{2}, and β_{1}, …, β_{r} are simple representations of S_{4}.