Course 424

Group Representations I

Dr Timothy Murphy

Walton Theatre Friday, 10 January 2003 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_{5} 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 that the natural representation of S_{n} in ℂ^{n} (by permutation of coordinates) splits into 2 simple parts, for any n > 1.

Continued overleaf 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 S_{3}, expressing each induced character as a sum of simple parts.

7. Determine the character table of the quaternion group

Q_{8} = {±1, ±i, ±j, ±k}.

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