Course 424

Group Representations I

Dr Timothy Murphy

Joly Theatre Friday, 19 January 2001 16:0018:00

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 2-dimensional representations of D4 up to equivalence,from first principles.

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

Determine all simple representations of S3, from first principles.

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

Prove that every finite-dimensional representation of a finite groupover C is semisimple.Show that the natural representation of Sn in Cn (by permutation ofcoordinates) splits into 2 simple parts, for any n > 1.

4. 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.

5. Draw up the character table of S4, explaining your reasoning through-out.

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

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

State (without proof) a formula for the character of G in terms of thatof .

Determine the characters of S4 induced by the simple characters of theViergruppe V4, expressing each induced character as a sum of simpleparts.

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. Show that a finite group G has only a finite number of simple repre-sentations (up to equivalence), say 1, . . . , r; and show that

(deg 1)2 + + (deg r)2 = G.

Show that the number of simple representations of Sn of degree d iseven if d is odd. Hence or otherwise determine the dimensions of thesimple representations of S5.