Group Representations

Dr Timothy Murphy

GMB ??

Friday, ?? 2005

??:00–??:00

Attempt 7 questions. (If you attempt more, only the best 7 will be counted.) All questions carry the same number of marks.

Unless otherwise stated, all groups are compact (or finite),, and all representations are of finite degree over C.

1. Define a group representation. What is meant by saying that 2 representations α, β are equivalent? Find all representations of S3 of degree

2 (up to equivalence).

What is meant by saying that a representation α is simple? Find all simple representations of D4 from first principles.

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

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.

3. Define the character χα of a representation α, and show that it is a class function (ie it is constant on conjugacy classes).

1

Define the intertwining number I(α, β) of 2 representations α, β of a group G, and show that if G is compact then

I(α, β) =

χα (g)χβ (g) dg.

G

Prove that a representation α is simple if and only if I(α, α) = 1.

4. Draw up the character table for S4 .

Determine also the representation-ring for this group, ie express the product αβ of each pair of simple representation as a sum of simple representations. 5. 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.

6. Determine the conjugacy classes in SU(2); and prove that this group has just one simple representation of each dimension.

Find the character of the representation D(j) of dimensions 2j + 1

(where j = 0, 12 , 1, 23 , . . . ).

Express each product D(i)D(j) as a sum of simple representations

D(k).

7. Define the exponential eX of a…