Course 424

Group Representations II

Dr Timothy Murphy

Arts Block A5039 Friday, 8 March 1991 15:00–17:00

Answer as many questions as you can; all carry the same number of marks. All representations are finite-dimensional over ℂ.

1. Define a measure on a compact space. State carefully, but without proof, Haar’s Theorem on the existence of an invariant measure on a compact group. To what extent is such a measure unique? 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.) Prove that every representation of a compact group is semisimple.

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(n).

Prove that SU(2) has just one simple representation of each dimension 1, 2, …; and determine the character of this representation. If D(j) denotes the simple representation of SU(2) of dimension 2j +1, for j = 0, 1/2, 1, …, express the product D(j)D(k) as a sum of D(j)’s.

4. Determine the conjugacy classes in SO(n).

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