Course 424

Group Representations I

Dr Timothy Murphy

Exam Hall Monday, 13 January 1997 14:0016:00

Attempt 6 questions. (If you attempt more, only the best 6 willbe counted.) All questions carry the same number of marks.Unless otherwise stated, all groups are finite, and all representa-tions are of finite degree over C.

1. Define a group representation. What is meant by saying that 2 repre-sentations , are equivalent? Determine all representations of S3 ofdegree 2 (up to equivalence) from first principles.

What is meant by saying that the representation is simple? Deter-mine all simple representations of S3 from first principles.

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

Prove that every representation of a finite group G (of finite degreeover C) is semisimple.Show that the natural n-dimensional representation of Sn in C

n (bypermutation of coordinates) is the sum of 2 simple representations.

3. Define the character of a representation .

Define the intertwining number I(, ) of 2 representations , . Stateand prove a formula expressing I(, ) in terms of , .

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

4. Explain how a representation of a subgroup H G induces a repre-sentation G of G.

Show thatg

|G|G(g) =

hg

h

|H|(h).

Determine the characters of S4 induced by the simple characters of S3,and hence or otherwise draw up the character table of S4.

5. Show that the number of simple representations of a finite group G isequal to the number s of conjugacy classes in G.

Show also that if these representations are 1, . . . , s then

dim2 1 + + dim2 s = |G|.

6. Draw up the character table of the dihedral group D5 (the symmetrygroup of a regular pentagon).

Determine also the representation ring of D5, ie express each productof simple representations of D5 as a sum of simple representations.

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

Show that is simple if and only if both and are simple; andshow that every simple representation of GH 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.

8. By considering the eigenvalues of 5-cycles, or otherwise, show that Snhas no simple representation of degree 2 if n > 4.