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 S_{3} 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) = \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.

4. Define the character χ_{α} of a representation α.

State without proof a formula expressing the intertwining number I(α, β) in terms of the characters χ_{α}, χ_{β}.

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 S_{4}.

6. Draw up the character table of D_{5} (the symmetry group of a regular pentagon).

Determine also the representation ring of D_{5}, ie express each product of simple representations of D_{5} as a sum of simple representations.

7. Show that the number of simple representations of a finite group G is equal to the number s of conjugacy classes in G.

8. Show that if the simple representations of the finite group G are σ_{1}, …, σ_{s}

then

(deg σ_{1})^{2} + ⋯ + (deg σ_{s})^{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}.