On the Cycle Structure of the Metacommutation Map
António Leite, António Machiavelo
TL;DR
The paper analyzes the cycle structure of the metacommutation map $\tau_{\xi,p}$ on the $p+1$ left-classes of Hurwitz primes above an odd prime $p$. It translates the action to a linear operator $\phi_{\xi,p}$ on the conic $C_p$ over $\mathbb{F}_p$ and leverages cyclotomic polynomials $\Phi_t$ to determine non-trivial cycle lengths via the gcd with $f_{\xi,p}(x)=x^2+(2-\frac{\mathrm{Tr}(\xi)^2}{q})x+1$. A key result is the criterion that $\ell_{\xi,p}=t$ if and only if $(f_{\xi,p},\Phi_t)\neq 1$, which yields a complete description of possible cycle lengths and fixed points, including constructions with a single $p$-cycle and explicit cases for small lengths. The work also connects fixed points to common left-right divisors from quaternionic arithmetic, giving concrete congruence conditions and showing that multiple distinct $\xi$ can yield the same cycle length while fixing different primes. Overall, the results fuse finite-field cyclotomic theory with quaternionic metacommutation to precisely characterize cycle lengths and fixed-point sets.
Abstract
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.