Course 424

Group Representations II

Dr Timothy Murphy

Joly Theatre Friday, 6 April 2001 16:00–17:30

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 representation of a compact group is semisimple. Give an example of a representation of the additive group ℝ which is not semisimple.

4. Prove that every simple representation of a compact abelian group is 1-dimensional. Determine the simple representations of U(1).

5. Determine the conjugacy classes in SU(2).

Prove that SU(2) has just one simple representation of each dimension 1, 2, …; and determine the character of this representation.

6. Determine the conjugacy classes in SO(3).

Prove that SO(3) has just one simple representation of each odd dimension 1, 3, 5, . . ..