Course 424

Group Representations IIa

Dr Timothy Murphy

Joly Theatre Tuesday, 17 April 2001 14:00–15:30

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.

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.

2. Which of the following groups are (a) compact, (b) connected:

O(n), SO(n), U(n), SU(n), GL(n, ℝ), SL(n, ℝ), GL(n, ℂ), SL(n, ℂ)?

(Justify your answer in each case.)

3. 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).

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

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).

6. Show that every function f(g) on a finite group G is expressible as a linear combination of the functions

χ(ag)

as χ runs through the simple characters of G, and a runs through the elements of G.