{"paper":{"title":"Conjugation invariants determine the metacommutation permutation only up to relabelling","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"cs.GT","authors_text":"Matthew Fried","submitted_at":"2026-08-03T02:35:51Z","abstract_excerpt":"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 perm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.01610","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.01610/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}