Group Representations

Dr Timothy Murphy

G.M.B.

Friday, 22 May 1995

14:00–16:00

Answer as many questions as you can; all carry the same number of marks.

Unless otherwise stated, all representations are finite-dimensional over C.

1. Define a group representation. What is meant by saying that 2 representations α, β are equivalent? What is meant by saying that the representation α is simple?

Determine all simple representations of D4 (the symmetry group of a square) up to equivalence, from first principles.

2. What is meant by saying that the representation α is semisimple?

Prove that every finite-dimensional representation α of a finite group over C is semisimple.

Define the character χα of a representation α.

Define the intertwining number I(α, β) of 2 representations α, β. State without proof a formula expressing I(α, β) in terms of χα , χβ .

Show that the simple parts of a semisimple representation are unique up to order.

3. Draw up the character table of S4 .

Determine also the representation ring of S4 , ie express each product of simple representations of S4 as a sum of simple representations.

4. Show that the number of simple representations of a finite group G is equal to the number s of conjugacy classes in G.

Show also that if these representations are σ1 , . . . , σs then dim2 σ1 + · · · + dim2 σs = |G|.

Determine the dimensions of the simple representations of S5 , stating clearly any results you assume.

5. Explain the division of simple representations of a finite group G over

C into real, essentially complex and quaternionic. Give an example of each (justifying your answers).

Show that if α is a simple representation with character χ then the value of χ(g 2 ) g∈G determines which of these 3 types α falls into.

6. 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?

Prove that every representation of a compact group is semisimple.

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

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

(Justify your answer in each…