Course 424

Group Representations I

Dr Timothy Murphy

Exam Hall Monday, 13 January 1997 14:00–16:00

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 are equivalent? Determine all representations of S_{3} of degree 2 (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 representation α of a finite group G (of finite degree over ℂ) is semisimple. 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.

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.

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 the simple characters of S_{3}, and hence or otherwise 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|.

6. Draw up the character table of the dihedral group 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. 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 3-dimensional representation of D_{6} defined by its action on the 3 diagonals 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}, …, β_{r} are simple representations of S_{3}.

8. By considering the eigenvalues of 5-cycles, or otherwise, show that S_{n} has no simple representation of degree 2 if n > 4.