Conjugation invariants determine the metacommutation permutation only up to relabelling

[2608.01610] Conjugation invariants determine the metacommutation permutation only up to relabelling Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Computer Science > Computer Science and Game Theory arXiv:2608.01610 (cs) [Submitted on 3 Aug 2026 ( v1 ), last revised 14 Aug 2026 (this version, v2)] Title: Conjugation invariants determine the metacommutation permutation only up to relabelling Authors: Matthew Fried View a PDF of the paper titled Conjugation invariants determine the metacommutation permutation only up to relabelling, by Matthew Fried View PDF HTML (experimental) Abstract: Let $\mathcal{H}$ be the Hurwitz quaternions, $p$ an odd prime, and $Q \in \mathcal{H}$ a prime of norm $q \neq p$. Metacommutation $PQ = Q'P'$ induces a permutation $\pi_Q$ of the $p+1$ left-associate classes of primes of norm $p$. Cohn and Kumar compute its sign and fixed-point count, and Leite and Machiavelo its full cycle structure, by formulas depending only on conjugation-invariant data of $Q$ (namely $q$ and $\mathrm{tr}\,Q$). We prove this is exactly the boundary of what such invariants can carry: no quantity $I(Q)$ invariant under unit conjugation determines $\pi_Q$ as a labelled permutation of the intrinsic class set, or even the image of a single specified class. The proof combines an equivariance identity $\pi_{uQu^{-1}} = \rho_u \pi_Q \rho_u^{-1}$ with a minimal, fully explicit witness at $(p,q) = (3,5)$: the four primes $2+i$, $2+j$, $2+k$, $2-i$ form a single unit-conjugacy orbit, hence agree under every conjugation-invariant function, yet induce four pairwise distinct $4$-cycles of the same four classes. We further observe that isomorphisms $\mathcal{H}/p\mathcal{H} \to M_2(\mathbb{F}_p)$ form a torsor under $\mathrm{PGL}_2(\mathbb{F}_p)$, so no projective labelling of the classes is canonical, and that by orbit-stabilizer the datum of one destination $\pi_Q(C)$ is exactly a coset $g_Q G_C$ in $\mathrm{PGL}_2(\mathbb{F}_p)/G_C$. All sixteen refactorizations in the witness are listed in the appendix and have been verified by machine along two independent routes. Comments: 7 pages Subjects: Computer Science and Game Theory (cs.GT) ; Group Theory (math.GR) Cite as: arXiv:2608.01610 [cs.GT] (or arXiv:2608.01610v2 [cs.GT] for this version) https://doi.org/10.48550/arXiv.2608.01610 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Matthew Fried [ view email ] [v1] Mon, 3 Aug 2026 02:35:51 UTC (95 KB) [v2] Fri, 14 Aug 2026 20:18:13 UTC (98 KB) Full-text links: Access Paper: View a PDF of the paper titled Conjugation invariants determine the metacommutation permutation only up to relabelling, by Matthew Fried View PDF HTML (experimental) TeX Source view license Ancillary-file links: Ancillary files ( details ) : verify_witness.py Current browse context: cs.GT < prev | next > new | recent | 2026-08 Change to browse by: cs math math.GR References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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