Course 424

Group Representations IIa

Dr Timothy Murphy

Mathematics 1.8 Friday, 9 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. Determine the conjugacy classes in SU(2). Prove that SU(2) has just one simple representation of each dimension 1, 2, 3, …; and determine the character of this representation.

3. 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).

4. Explain the division of simple representations of a compact group G into real, essentially complex and quaternionic. Show that the representations of SO(3) are all real.

5. Define the exterior product λ^{r}α of a representation α.

Show that the character of λ^{2}α is given by

χ_{λ^{2}α}(g) = \frac{1}{2} (χ_{α}(g)^{2} − χ_{α}(g^{2})) .

Hence or otherwise show that if D(j) is the representation of SU(2) of dimension 2j + 1 then

λ^{2}D(j) = D(2j − 1) + D(2j − 3) + ⋯ + \begin{cases}D(1) if j is integral \\ D(0) if j is half-integral \end{cases} .