Course 424

Group Representations II

Dr Timothy Murphy

Arts Block A2039 Friday, 20 January 1989 15.45–17.45

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.

2. What is meant by saying that the representation α is simple?

Determine all simple representations of D_{4}, 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 from first principles 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. Prove that every simple representation of an abelian group is 1-dimensional.

Is the converse true, ie if every simple representation of a finite group G is 1-dimensional, is G necessarily abelian? (Justify your answer.)

6. 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.

7. 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.

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