 Solution

ID: past-exam-of-the-algebra-course-of-the-university-of-oxford/2010/b/6/b/solution

The row orthogonality relations for a character table assert that distinct complex irreducible characters are orthogonal and each has squared norm one:

⟨ χ_{i}, χ_{j} ⟩ = δ_{ij} .

⟨ ψ, ψ ⟩ = \sum_{i, j} λ_{i} \overline{λ_{j}} ⟨ χ_{i}, χ_{j} ⟩ = \sum_{i} ∣ λ_{i} ∣^{2} .

Since the multiplicities are nonnegative integers, this is

⟨ ψ, ψ ⟩ = i \sum λ_{i}^{2} .

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