{"paper":{"title":"Real $C$-, $G$-structures and sign-coherence of cluster algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.RA"],"primary_cat":"math.RT","authors_text":"Ryota Akagi, Zhichao Chen","submitted_at":"2025-09-08T09:49:23Z","abstract_excerpt":"We generalize the theory of integer $C$-, $G$-matrices in cluster algebras to the real case. By a skew-symmetrizing method, we can reduce the problem of skew-symmetrizable patterns to skew-symmetric patterns. In this sense, the sign-coherence of a more general real class called of quasi-integer type can be inherited directly from that of integer $C$-, $G$-matrices proved by Gross-Hacking-Keel-Kontsevich. However, the sign-coherence of real $C$-, $G$-matrices does not always hold in general. For this purpose, we classify all the rank $2$ case and the finite type case via the Coxeter diagrams. W"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.06486","kind":"arxiv","version":3},"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/2509.06486/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"}