 Solution

ID: past-exam-of-the-lie-groups-and-differentiable-manifolds-course-of-the-university-of-oxford/2008/a/4/b/solution

Schur orthogonality for irreducible complex characters of a compact group, with normalized Haar measure, is

⟨ χ_{α}, χ_{β} ⟩ := \int_{G} χ_{α} (g) \overline{χ_{β} (g)} d μ (g) = δ_{α β} .

Equivalent representations clearly have equal characters, since trace is invariant under conjugation. Conversely complete reducibility gives

χ_{V} = \sum_{α} m_{α} χ_{α}

, where the nonnegative integer multiplicity is recovered by

m_{α} = ⟨ χ_{V}, χ_{α} ⟩

. Equal characters thus give equal multiplicities of every irreducible constituent, hence equivalent representations.

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