The Modular DFT of the Symmetric Group

We describe the discrete Fourier transform (DFT) for a cyclic group when $p\mathrm{\mid }N$ by factoring $x^N-1$ over finite fields and constructing the Fourier transform and its inverse using Bézout’s identity for polynomials. For the symmetric group, in the modular case when $p\mathrm{\mid }n!$ we construct the Peirce decomposition using central primitive orthogonal idempotents, yielding a change-of-basis matrix which generalizes the DFT. We compute the unitary DFT for the symmetric group over number fields containing sufficiently many square roots. For $n=3$, we compute the Galois group of the splitting field of the characteristic polynomial. All constructions are implemented in SageMath.