{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/what-is-the-modular-fourier-transform","title":"The Modular DFT of the Symmetric Group","arxiv_id":"2404.05796","date":"2024-04-08","proceeding":null,"authors":["Jackson Walters"],"abstract":"We describe the discrete Fourier transform (DFT) for a cyclic group when $p|N$ by factoring $x^N-1$ 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.","url_abs":"https://arxiv.org/abs/2404.05796v3","url_pdf":"https://arxiv.org/pdf/2404.05796v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"what-is-the-modular-fourier-transform","repo_url":"https://github.com/jacksonwalters/dft-finite-groups","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}