Let $G$ be a finite abelian group. Let $f:G\mapsto \mathbb{C}$ be a function. Now, we can view each such functions as a vector in $\mathbb{C}^{|G|}$ by taking all evaluations over $G$. We define characters on abelian group as $\chi:G\mapsto\mathbb{C}$, as a homomorphism from $G$ to $\mathbb{C}$. As, this characters, when seen as a vector in $\mathbb{C}^{|G|}$ by taking evaluations, are orthogonal, it forms a basis. A Fourier transform on $G$ is just a vector space isomorphism mapping standard basis to these character basis. Also the group of characters is isomorphic to $G$.
Now, while reading this https://www.cs.umd.edu/~amchilds/teaching/w13/l02.pdf, I came across a definition that states, $F(|x\rangle)=\sum_{y\in \hat{G}}\chi_y(x)|y\rangle$, where $x$ is a group element $F$ is the fourier transform, $|x\rangle$ is the standard basis indexed by $x$ and as per my understanding $|y\rangle$ is the character basis indexed by $y$. Here, $\hat{G}$ is the group of characters.
Can anyone help me to clarify how this definition and the one I stated in the last paragraph are the same? It appears to me that, the standard basis are getting mapped to a linear combination of character basis, instead of just mapping to a character basis.
