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Given representations $\rho_1 : G \to \mathrm{GL}(V)$ and $\rho_2 : G \to \mathrm{GL}(W)$, how can we define explicitly the representation of $G$ over $\mathrm{Hom}(V, W) \cong V^* \otimes W$?

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up vote 8 down vote accepted

The isomorphism of $\mathrm{Hom}(V,W)$ with $V^*\otimes W$ only holds when $V$ is finite-dimensional. I'll assume both $V$ and $W$ are finite-dimensional. If we write our group actions on the right, if $\phi\in\mathrm{Hom}(V,W)$ then $$\phi g:v\mapsto\phi(vg^{-1})g.$$

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@user314: See, for example, page 4 in Fulton & Harris:… – Hans Lundmark Oct 11 '10 at 6:48

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