Identity involving characters $\prod\langle f,\psi\rangle = \det\left(f\left(gh^{-1}\right)\right)$

Question

I want to show that for a finite abelian group $G$ and a function $f : G \to \mathbb{C}$, the following identity holds $$ \prod_{\psi\in\hat{G}} \langle f,\psi \rangle = \det_{g,h \in G} \left(f\left(gh^{-1}\right)\right). $$

This is Exercise 1 in Chapter 3 of the book by Iwaniec and Kowalski, and is apparently due to Dedekind.

I have tried to simply expand both sides and compare them but got nowhere, and I am thinking that there must be a better proof. Any help is appreciated.

PS: I am not sure how to title this, so I am also open to suggestions on this front.

Answer 1

Look at $\mathbb{C}G$ in two different ways. The RHS corresponds to the determinant of the map "multiplying by $\sum_g f(g)g$". The LHS corresponds to decomposing $\mathbb{C}G$ into the direct sum of $1$-dimensional irreps. Note that multiplication by $\sum_g\psi(g)g$ is $0$ except on the submodule $\mathbb{C}\sum_g\psi(g)g$.