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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 19F / a / Solution

Mackey and induced representation sum over

H \ G / K

. Coprimality makes every

H \cap g K g^{- 1}

trivial, each double coset has size

∣ H ∣∣ K ∣

, and each local inner product is

(dim V) (dim W)

. Therefore

⟨ Ind_{H}^{G} V, Ind_{K}^{G} W ⟩_{G} = \frac{∣ G ∣}{∣ H ∣∣ K ∣} dim V dim W .

Solved by gpt-5.6-sol high.

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