Course 424

Group Representations I

Dr Timothy Murphy

Joly Theatre Friday, 19 January 2001 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 are equivalent? Determine all 2-dimensional representations of D_{4} up to equivalence, from first principles.

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

3. 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. Show that the natural representation of S_{n} in ℂ^{n} (by permutation of coordinates) splits into 2 simple parts, for any n > 1.

4. 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.

5. Draw up the character table of S_{4}, explaining your reasoning throughout.

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

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

State (without proof) a formula for the character of β^{G} in terms of that of β.

Determine the characters of S_{4} induced by the simple characters of the Viergruppe V_{4}, expressing each induced character as a sum of simple parts.

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. 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}.