I'm looking for some (short of) non-abelian generalization of the following result:
Let $G$ be a finite abelian group and let $f$ be a function on $G$ with values in some field of characteristic $0$. If $\sum_\sigma f(\sigma)=0$, then:
$$\det(f(\sigma \tau^{-1}))_{\sigma, \tau \neq 1}=\frac{1}{|G|}\prod_{\chi \neq 1} \sum_{\sigma \in G} \chi (\sigma) f (\sigma)$$
What can we say if $G$ in non-abelian (but still finite), under the same conditions?