Essay on Representation Theory and Simple Representations

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Course 424
Group Representations
Dr Timothy Murphy
G.M.B.

Friday, 22 May 1995

14:00–16:00

Answer as many questions as you can; all carry the same number of marks.
Unless otherwise stated, all representations are finite-dimensional over C.
1. Define a group representation. What is meant by saying that 2 representations α, β are equivalent? What is meant by saying that the representation α is simple?
Determine all simple representations of D4 (the symmetry group of a square) up to equivalence, 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.
Define the character χα of a representation α.
Define the intertwining number I(α, β) of 2 representations α, β. State without proof a formula expressing I(α, β) in terms of χα , χβ .
Show that the simple parts of a semisimple representation are unique up to order.
3. Draw up the character table of S4 .
Determine also the representation ring of S4 , ie express each product of simple representations of S4 as a sum of simple representations.
4. 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 σ1 , . . . , σs then dim2 σ1 + · · · + dim2 σs = |G|.
Determine the dimensions of the simple representations of S5 , stating clearly any results you assume.
5. Explain the division of simple representations of a finite group G over
C into real, essentially complex and quaternionic. Give an example of each (justifying your answers).
Show that if α is a simple representation with character χ then the value of χ(g 2 ) g∈G determines which of these 3 types α falls into.
6. Define a measure on a compact space. State carefully, but without proof, Haar’s Theorem on the existence of an invariant measure on a compact group. To what extent is such a measure unique?
Prove that every representation of a compact group is semisimple.
Which of the following groups are (a) compact, (b) connected:
O(n), SO(n), U(n), SU(n), GL(n, R), SL(n, R)?
(Justify your answer in each
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