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Let $\pi$ resp. $\chi$ be a finite dimensional representation resp. 1-diml. rep. of a finite group.

We define $\chi \otimes \pi (g) = \chi(g) \pi(g)$ as a rep of $G$.

For $N$ subgroup, does hold $$Res_N \chi \otimes \pi = Res_N \chi \otimes Res_N \pi?$$

If not, what happens, if $N$ is normal inside $G$?

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    What do you mean by $\chi\otimes\pi$? One tensors representations with representations and multiplies characters with characters, but one does not mix the two.2012-01-27
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    What do you mean? $\mathbb{C}$ acts on every complex vector space.2012-01-27
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    @late_learner: you should just verify the left hand side and the right hand side are equal at every element of N. However, you should be careful that *π* is actually a *linear* character of *G*, not just the character of a representation, otherwise your χ⊗π will not be a homomorphism.2012-01-27

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