Solution

The induced representation is

Ind_{H}^{G} (V) = C [G] \otimes_{C [H]} V,

with

G

acting by left multiplication on the first tensor factor. If

R

is a set of representatives for the left cosets

G / H

, then as a vector space it is

⨁_{r \in R} r \otimes V

The operator

g

permutes these summands. A summand indexed by

x H

contributes to the trace exactly when

gx H = x H

, equivalently when

x^{- 1} gx \in H

, and its contribution is

χ_{V} (x^{- 1} gx)

. Accounting for the

∣ H ∣

representatives of each coset gives

χ_{Ind_{H}^{G} V} (g) = \frac{1}{∣ H ∣} x \in G x^{- 1} gx \in H \sum χ_{V} (x^{- 1} gx) .

For a

G

-representation

W

, restrict

W

H

in the left-hand induced representation. Define

Φ : C [G] \otimes_{C [H]} (W \otimes V) ⟶ W \otimes (C [G] \otimes_{C [H]} V)

Φ (x \otimes (w \otimes v)) = x w \otimes (x \otimes v) .

It is well-defined because both representatives of the balancing relation give

x h w \otimes (x \otimes h v) .

It is

G

-equivariant, and its inverse is

w \otimes (x \otimes v) ⟼ x \otimes (x^{- 1} w \otimes v) .

Consequently

Ind_{H}^{G} (Res_{H}^{G} W \otimes V) ≅ W \otimes Ind_{H}^{G} (V) .

Solved by gpt-5.6-sol high.

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