Course 424

Group Representations I

Dr Timothy Murphy

G.M.B.  Friday, 22 January 1993  14:00–16:00

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? Determine all 2-dimensional representations of S_{3} (up to equivalence) from first principles.

What is meant by saying that the representation α is simple? Determine all simple representations of S_{3} 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.

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

Show that the natural n-dimensional representation of S_{n} in C^{n} (by permutation of coordinates) is the sum of 2 simple representations.

3. Define the character χ_{α} of a representation α, and show that it is a class function (constant on conjugacy classes). Define the intertwining number I(α, β) of 2 representations α, β, and show that

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

Prove that a representation α is simple if and only if I(α, α) = 1. Prove that every simple representation of an abelian group is 1-dimensional.

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

5. 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.

6. Define the representation α × β of the product-group G × H, where α is a representation of G, and β of H. Show that α × β is simple if and only if both α and β are simple; and show that every simple representation of G × H is of this form. Show that D_{6} (the symmetry group of a regular hexagon) is expressible as a product group

D_{6} = C_{2} × S_{3}.

Let γ denote the 6-dimensional representation of D_{6} defined by its action on the 6 vertices of the hexagon. Express γ in the form

γ = α_{1} × β_{1} + ⋯ + α_{r} × β_{r},

where α_{1}, …, α_{r} are simple representations of C_{2}, and β_{1}, …, α_{r} are simple representations of S_{3}.