The Modular DFT of the Symmetric Group
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916351954649088 |
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| author | Walters, Jackson |
| author_facet | Walters, Jackson |
| contents | 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 Bézout'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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_05796 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Modular DFT of the Symmetric Group Walters, Jackson Representation Theory Commutative Algebra Combinatorics 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 Bézout'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. |
| title | The Modular DFT of the Symmetric Group |
| topic | Representation Theory Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2404.05796 |