Prove that every representation of a compact group is semisimple.
3. Define the character χα of a representation α, and show that it is a
class function (ie it is constant on conjugacy classes).
1
Define the intertwining number I(α, β) of 2 representations α, β of a
group G, and show that if G is compact then
I(α, β) =
χα (g)χβ (g) dg.
G
Prove that a representation α is simple if and only if I(α, α) = 1.
4. Draw up the character table for S4 .…

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