Course 424

Group Representations II

Dr Timothy Murphy

Walton Theatre Friday, 4 April 2003 11:15–12:45

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. 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 D_{6} (the symmetry group of a regular hexagon) is expressible as a product group

D_{6} = C_{2} × S_{3}.

Let γ denote the 3-dimensional representation of D_{6} defined by its action on the 3 diagonals of the hexagon. 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_{3}.

2. Prove that the only finite-dimensional division-algebras (or skew-fields) over ℝ are ℝ, ℂ and ℍ.

Show that the endomorphism-ring End(β) of a simple representation β over ℝ is a finite-dimensional division-algebra over ℝ; and give examples of three such representations with End(β) = ℝ, ℂ, ℍ.

3. 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.

Prove that every representation of a compact group is semisimple.

4. 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), Sp(n), E(n)?

(Justify your answer in each case. E(n) is the group of isometries of n-dimensional Euclidean space.)

5. Prove that every simple representation of a compact abelian group is 1-dimensional.

Determine the simple representations of SO(2).

Determine the simple representations of O(2).

6. 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.

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

Prove that SO(3) has just one simple representation of each odd dimension 1, 3, 5, …; and determine the character of this representation.