{"paper":{"title":"A conjecture on $C$-matrices of cluster algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.RT"],"primary_cat":"math.RA","authors_text":"Fang Li, Min Huang, Peigen Cao","submitted_at":"2017-02-04T01:41:31Z","abstract_excerpt":"For a skew-symmetrizable cluster algebra $\\mathcal A_{t_0}$ with principal coefficients at $t_0$, we prove that each seed $\\Sigma_t$ of $\\mathcal A_{t_0}$ is uniquely determined by its {\\bf C-matrix}, which was proposed by Fomin and Zelevinsky in \\cite{FZ3} as a conjecture. Our proof is based on the fact that the positivity of cluster variables and sign-coherence of $c$-vectors hold for $\\mathcal A_{t_0}$, which was actually verified in \\cite{GHKK}. More discussion is given in the sign-skew-symmetric case so as to obtain a conclusion as weak version of the conjecture in this general case."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.01221","kind":"arxiv","version":2},"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/1702.01221/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"}