Course 424

Group Representations I

Dr Timothy Murphy

Exam Hall  Friday, 8 March 1991  15:00–17:00

Answer as many questions as you can; all carry the same number of marks. Unless otherwise stated, all groups are finite, and all representations are finite-dimensional 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 finite-dimensional representation α of a finite group over ℂ is semisimple.

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

Define the intertwining number I(α, β) of 2 representations α, β. State and prove a formula expressing I(α, β) in terms of χ_{α}, χ_{β}.

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

3. Explain how a representation β of a subgroup H ⊂ G induces a representation β^{G} of G.

Show that

\frac{ḡ}{|G|}χ_{β^{G}}(ḡ) = ∑_{h̄⊂ḡ} \frac{h̄}{|H|}χ_{β}(h̄).

Determine the characters of S_{4} induced by the simple characters of S_{3}, and so draw up the character table of S_{4}.

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

Show also that if these representations are σ_{1}, …, σ_{s} then

dim^{2} σ_{1} + ⋯ + dim^{2} σ_{s} = |G|.

Determine the dimensions of the simple representations of S_{5}, stating clearly any results you assume.

5. Draw up the character table of the alternating group A_{4} (the subgroup of S_{4} formed by even permutations).

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

6. Explain the division of simple representations (over ℂ) into real, essentially complex and quaternionic. Give an example of each (justifying your answers).

Show that if α is a simple representation with character χ then the value of

∑_{g∈G} χ(g^{2})

determines which of these 3 types α falls into.