{"paper":{"title":"A path model for MV polytopes in type A_n","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Orientations of the fundamental alcove produce paths that exactly label the cluster variables in the initial seed of C[N].","cross_cats":[],"primary_cat":"math.RT","authors_text":"Zijun Li","submitted_at":"2026-03-30T16:18:12Z","abstract_excerpt":"We introduce a one-skeleton path model for Mirkovic-Vilonen polytopes in type A_n. We prove that the Minkowski sum of (MV) polytopes corresponds to the concatenation of one-skeleton paths of this model. We show that MV polytopes induced by fundamental one-skeleton paths are Harder-Narasimhan polytopes. The paths given by an orientation of the fundamental alcove parameterize precisely the cluster variables in the initial seed of the coordinate ring C[N].\n  We also establish a correspondence between fundamental one-skeleton paths and folded galleries representing maximal faces of subword complex"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The paths given by an orientation of the fundamental alcove parameterize precisely the cluster variables in the initial seed of the coordinate ring C[N].","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the newly defined one-skeleton paths faithfully capture the combinatorial and geometric structure of MV polytopes so that Minkowski sum equals path concatenation and the alcove orientations exactly match the cluster variables.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Introduces a one-skeleton path model for MV polytopes in type A_n, proving Minkowski sums correspond to path concatenations and that alcove-oriented paths parameterize cluster variables in the initial seed of C[N].","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Orientations of the fundamental alcove produce paths that exactly label the cluster variables in the initial seed of C[N].","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"44621fc3d2090c5315849836967ff4dd883e46db9d3a472f288cf3fd86ec1c17"},"source":{"id":"2603.28634","kind":"arxiv","version":2},"verdict":{"id":"9128949a-db14-4116-a31c-b49580c598d2","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T16:55:55.790684Z","strongest_claim":"The paths given by an orientation of the fundamental alcove parameterize precisely the cluster variables in the initial seed of the coordinate ring C[N].","one_line_summary":"Introduces a one-skeleton path model for MV polytopes in type A_n, proving Minkowski sums correspond to path concatenations and that alcove-oriented paths parameterize cluster variables in the initial seed of C[N].","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the newly defined one-skeleton paths faithfully capture the combinatorial and geometric structure of MV polytopes so that Minkowski sum equals path concatenation and the alcove orientations exactly match the cluster variables.","pith_extraction_headline":"Orientations of the fundamental alcove produce paths that exactly label the cluster variables in the initial seed of C[N]."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.28634/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":2,"snapshot_sha256":"349f06f0a528b22b84505db77884cd0bf52a0d264253678685069aaab1d852d4"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}