Fourier sampling is used in the hidden subgroup problem in Shor's algorithm for $\mathbb{Z}/N\mathbb{Z}$ in the abelian case and the symmetric group $S_n$ for attempts at graph isomorphism in the nonabelian case.
Representations are over $\mathbb{C}$ in both cases. Is there a generalization to $\text{char}(k)=p > 0$?
There are some gaps in knowledge for the modular representations of $S_n$. The modular Fourier transform isn't quite developed. However, for finite abelian groups it's straightforward as long as $p \nmid n$.
