Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 160 / 2 / c

Table of contents
- Solution c

Solution

Use the character formula for an induced representation. For

g \in G

,

χ (g) (ϕ ↑_{H}^{G}) (g) = \frac{1}{∣ H ∣} x \in G x^{- 1} gx \in H \sum χ (g) ϕ (x^{- 1} gx) = \frac{1}{∣ H ∣} x \in G x^{- 1} gx \in H \sum χ (x^{- 1} gx) ϕ (x^{- 1} gx) = ((χ ↓_{H}) ϕ) ↑_{H}^{G} (g) .

The middle equality uses that a character of a representation is constant on conjugacy classes. This proves the tensor identity for an induced character.

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