Let $k$ be a field, $G$ be a finite abelian group such that $\operatorname{ch}=0$ or $\operatorname{ch}(k)\nmid \#G$.
Then, the dual group $\hat{G}$ is defined by
$\hat{G}=\{\chi\mid {\rm group~homomorphism}~ \chi:G\to k^{*}\}$.
Consider $\#G$-dimensional $k$-vector space $k^G=\oplus_{g\in G}k$ and $k$-linear map $\phi: k^G\to k^{\hat{G}}$ s
$\phi((a_g)_{g\in G})=(\sum_{g\in G}\chi(g)a_g)_{\chi\in \hat{G}}$
I want to know the property of $\phi$ such as determinant, trace, inverse map...
Does this map have a name? If you know of any good books or papers, please let me know.