Today I came across two possible canonical maps in the context of representation theory from reading stackexchange and a friend's email. However, I don't really know what they are (and have thus far not found satisfactory information about them): 1) $hom_k(V \otimes V,k ) \rightarrow hom_k(V, V^*)$ 2) $(V^*)^* \otimes V \rightarrow hom_k(V, V^*)$
In both cases $V$ is a finite dimensional representation of some (finite) group $G$. In 2) $k$ maybe assumed algebraically closed. Perhaps 1) is a result of exterior derivatives (my guess). Any pointer is welcomed.