On the Cycle Structure of the Metacommutation Map
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866912321439268864 |
|---|---|
| author | Leite, António Machiavelo, António |
| author_facet | Leite, António Machiavelo, António |
| contents | Cohn and Kumar showed that the permutation on the set of the classes of left associated Hurwitz primes above an odd prime $p$ induced through metacommutation by a Hurwitz prime $ξ$ of norm $q$ has either $0$, $1$ or $2$ fixed points, and that the permutation $τ_{ξ,p}$ induced on the non-fixed points splits into cycles of the same length. Here we show how to find the length of those cycles, in terms of $p$ and $ξ$, using cyclotomic polynomials over $\mathbb{F}_p$. We then show that, given an odd prime $p$, there is always a prime quaternion $ξ$ such that the permutation $τ_{ξ,p}$ has only one non-trivial cycle of length $p$. Finally, we give conditions for a prime $π$ of norm $p$ to be a fixed point of the aforementioned metacommutation map. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_08709 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Cycle Structure of the Metacommutation Map Leite, António Machiavelo, António Number Theory Cohn and Kumar showed that the permutation on the set of the classes of left associated Hurwitz primes above an odd prime $p$ induced through metacommutation by a Hurwitz prime $ξ$ of norm $q$ has either $0$, $1$ or $2$ fixed points, and that the permutation $τ_{ξ,p}$ induced on the non-fixed points splits into cycles of the same length. Here we show how to find the length of those cycles, in terms of $p$ and $ξ$, using cyclotomic polynomials over $\mathbb{F}_p$. We then show that, given an odd prime $p$, there is always a prime quaternion $ξ$ such that the permutation $τ_{ξ,p}$ has only one non-trivial cycle of length $p$. Finally, we give conditions for a prime $π$ of norm $p$ to be a fixed point of the aforementioned metacommutation map. |
| title | On the Cycle Structure of the Metacommutation Map |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.08709 |