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 other- wise 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 .

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Course 424

Group Representations II

Dr Timothy Murphy

Walton Theatre Friday, 4 April 2003 11:1512:45

Attempt 5 questions. (If you attempt more, only the best 5 willbe counted.) All questions carry the same number of marks.All representations are finite-dimensional over C, unless other-wise stated.

1. Define the representation of the product-group GH, where is a representation of G, and of H.

Show that if G and H are finite then is simple if and only ifboth and are simple; and show that every simple representation ofGH is of this form.Show that D6 (the symmetry group of a regular hexagon) is expressibleas a product group

D6 = C2 S3.

Let denote the 3-dimensional representation of D6 defined by itsaction on the 3 diagonals of the hexagon. Express in the form

= 1 1 + + r r,

where 1, . . . , r are simple representations of C2, and 1, . . . , r aresimple representations of S3.
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() of a simple representation over R is a finite-dimensional division-algebra over R; and give exam-ples of three such representations with End() = R,C,H.

3. Define a measure on a compact space. State carefully, and outlinethe main steps in the proof of, Haars Theorem on the existence of aninvariant 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 ofn-dimensional Euclidean space.)

5. Prove that every simple representation of a compact abelian group is1-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 dimension1, 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 di-mension 1, 3, 5, . . . ; and determine the character of this representation.