Course 424

Group Representations I

Dr Timothy Murphy

Synge Theatre Monday, 22 January 1999 16:00–18:00

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 of G are equivalent? Determine all representations of S_{3} of degree 2 (up to equivalence), from first principles.

Show that if α, β are equivalent representations of G then α, β have the same eigenvalues for each g ∈ G. Is the converse true, ie if α(g), β(g) have the same eigenvalues for each g, does it follow that α, β are equivalent?

2. What is meant by saying that the representation α is simple? Determine all simple representations of D_{4} from first principles.

Show that a simple representation of G necessarily has degree ≤ ∥G∥.

3. 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.

4. Define the intertwining number I(α, β) of 2 representations α, β.

Show that if α, β are simple then

I(α, β) = \begin{cases}1 & if α = β \\ 0 & if α ≠ β. \end{cases}

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

5. Define the character χ_{α} of a representation α. Outline the proof that

I(α, β) = \frac{1}{∥G∥} ∑_{g∈G} \overline{χ_{α}(g)}χ_{β}(g).

Show that two representations α, β of G are equivalent if and only if they have the same character.

6. Show that a simple representation of an abelian group is necessarily of degree 1.

Prove conversely that if every simple representation of G is of degree 1 then G must be abelian.

7. Draw up the character table of S_{4}.

Determine also the representation ring of S_{4}, ie express each product of simple representations of S_{4} as a sum of simple representations.

8. Show that a finite group G has only a finite number of simple representations (up to equivalence), say σ_{1}, …, σ_{r}; and show that

(deg σ_{1})^{2} + ⋯ + (deg σ_{r})^{2} = ∥G∥.

Show that the number of simple representations of S_{n} of degree d is even if d is odd. Hence or otherwise determine the dimensions of the simple representations of S_{5}.