{"paper":{"title":"An unexpected property of $\\mathbf{g}$-vectors for rank 3 mutation-cyclic quivers","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jihyun Lee, Kyungyong Lee","submitted_at":"2024-09-01T03:32:35Z","abstract_excerpt":"Let $Q$ be a rank 3 mutation-cyclic quiver. It is known that every $\\mathbf{c}$-vector of $Q$ is a solution to a quadratic equation of the form $$\\sum_{i=1}^3 x_i^2 + \\sum_{1\\leq i<j\\leq 3} \\pm q_{ij} x_i x_j =1,$$where $q_{ij}$ is the number of arrows between the vertices $i$ and $j$ in $Q$. A similar property holds for $\\mathbf{c}$-vectors of any acyclic quiver. In this paper, we show that $\\mathbf{g}$-vectors of $Q$ enjoy an unexpected property. More precisely, every $\\mathbf{g}$-vector of $Q$ is a solution to a quadratic equation of the form $$\\sum_{i=1}^3 x_i^2 + \\sum_{1\\leq i<j\\leq 3} p_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.00599","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/2409.00599/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"}