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If $ \rho: G \to GL_{\mathbb{C}}(V)$ and $ \sigma: G \to GL_{\mathbb{C}}(W)$ are irreducible representations, is it necessarily true that the induced representation $G \to GL_{\mathbb{C}}(Hom(V,W))$ is also irreducible?

I have found a counterexample for $ \rho \otimes \sigma: G \to GL_{\mathbb{C}}(V \otimes W)$. I also know that $V \otimes W^* \cong Hom(V,W)$. I am not sure how to further develop this first step, though.

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up vote 5 down vote accepted

The action of $G$ on $\def\Hom{\operatorname{Hom}}\Hom(V,W)$ is given by $g\cdot f=\sigma(g)\circ f\circ \rho(g^{-1})\in\Hom(V,W)$. (Comparing this with what you wrote, you can see that you put the asterisk on the wrong vector space: it should be $\Hom(V,W)\cong V^*\otimes W$, as you can check by taking $W=\Bbb C$.) Now you can find easy examples where $\Hom(V,W)$ is reducible: take $V=W$ and $\rho=\sigma$, then the $1$-dimensional subspace of $\Hom(V,V)$ of multiples of the identity map is $G$-invariant, so in this case $\Hom(V,V)$ is reducible as soon as $\dim V>1$, regardless of the representation $\rho$.

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