Course 424
Group Representations Ia
Dr Timothy Murphy
Seminar Room

Thursday, 17 June 1999

16:30–18:30

Attempt 6 questions. (If you attempt more, only the best 6 will be counted.) All questions carry the same number of marks.
Unless otherwise stated, all groups are 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? What is meant by saying that the representation α is simple?
Determine all simple representations of S3 up to equivalence, from first principles. 2. What is meant by saying that the representation α is semisimple?
Prove that every representation α of a finite group G (of finite degree over C) is semisimple.
3. Define the intertwining number I(α, β) of 2 representations α, β.
Show that if α, β are simple then
I(α, β) =

1 if α = β
0 if α = β.

Hence or otherwise show that the simple parts of a semisimple representation are unique up to order.

4. Define the character χα of a representation α.
State without proof a formula expressing the intertwining number I(α, β) in terms of the characters χα , χβ .
Show that two representations α, β of G are equivalent if and only if they have the same character.
5. Draw up the character table of S4 .
6. Draw up the character table of D5 (the symmetry group of a regular pentagon). Determine also the representation ring of D5 , ie express each product of simple representations of D5 as a sum of simple representations.