{"paper":{"title":"Local and global patterns of rank 3 $G$-fans of totally-infinite type","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tomoki Nakanishi","submitted_at":"2024-11-25T11:03:47Z","abstract_excerpt":"We focus on the $G$-fans associated with cluster patterns whose initial exchange matrices are of infinite type. We study the asymptotic behavior of the $g$-vectors around the initial $G$-cone under the alternating mutations for two indices of infinite type. In the rank 3 case, we classify them into several patterns. As an application, the incompleteness of the $G$-fans of infinite type is proved. We observed that the local pattern of a rank 3 $G$-fan of totally-infinite type classified by the above types correlates with its global pattern. Following the classification of the local patterns (to"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.16283","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/2411.16283/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"}