{"paper":{"title":"From green mutation to $\\mathrm{X}$-evolution: flows and foliations on cluster complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Liheng Tang, Yu Qiu","submitted_at":"2025-01-27T03:54:39Z","abstract_excerpt":"For any point $\\mathrm{X}$ in the cluster complex $\\mathrm{Cpx}(\\mathcal{C})$ of a 2-Calabi-Yau category $\\mathcal{C}$, we introduce $\\mathrm{X}$-evolution flow on $\\mathrm{Cpx}(\\mathcal{C})$. We show that such a flow induces a piecewise linear one-dimensional $\\mathrm{X}$-foliation with two singularities, the unique sink $\\mathrm{X}$ and the unique source $\\mathrm{X}[1]$. Moreover, we show that evolution flows on cluster complexes are continuous refinement/generalization of green mutations on cluster exchange graphs.\n  For the cluster category of a Dynkin or Euclidean quiver $Q$, we prove tha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.15756","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/2501.15756/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"}