Papers › The Modular DFT of the Symmetric Group
The Modular DFT of the Symmetric Group
Jackson Walters
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We describe the discrete Fourier transform (DFT) for a cyclic group when p|N by factoring xᴺ-1 over finite fields and constructing the Fourier transform and its inverse using B\'{e}zout's identity for polynomials. For the symmetric group, in the modular case when p|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.
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