The Modular DFT of the Symmetric Group
Abstract: 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.
- https://github.com/jacksonwalters/fourier-transform-symmetric-group
- https://github.com/jacksonwalters/dft-finite-field/blob/main/cyclic_group_dft.ipynb
- https://web.math.princeton.edu/~charchan/ModularRepresentationsSymmetricGroupSeminar.pdf
- https://github.com/jacksonwalters/dim-modular-repn-symmetric-group
- Gordon James, The Representation Theory of the Symmetric Group. 1978
- Gordon James, The Decomposition Matrices of GLn(q)𝐺subscript𝐿𝑛𝑞GL_{n}(q)italic_G italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_q ) for n ≤\leq≤ 10. 1988
- G. E. Murphy, The idempotents of the symmetric group and Nakayama’s conjecture. 1983
- https://github.com/jacksonwalters/dft-finite-field/blob/main/symmetric_group_dft.ipynb
- https://github.com/jacksonwalters/dft-finite-field/blob/main/central_primitive_orthogonal_idempotents.ipynb
- Wildon, Notes on Murphy Operators and Nakayama’s Conjecture
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