{"paper":{"title":"${\\rm SL}_3$-laminations as bases for ${\\rm PGL}_3$ cluster varieties for surfaces","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math-ph","math.GT","math.MP","math.QA"],"primary_cat":"math.AG","authors_text":"Hyun Kyu Kim","submitted_at":"2020-11-30T13:25:25Z","abstract_excerpt":"In this paper we partially settle Fock-Goncharov's duality conjecture for cluster varieties associated to their moduli spaces of ${\\rm G}$-local systems on a punctured surface $\\frak{S}$ with boundary data, when ${\\rm G}$ is a group of type $A_2$, namely ${\\rm SL}_3$ and ${\\rm PGL}_3$. Based on Kuperberg's ${\\rm SL}_3$-webs, we introduce the notion of ${\\rm SL}_3$-laminations on $\\frak{S}$ defined as certain ${\\rm SL}_3$-webs with integer weights. We introduce coordinate systems for ${\\rm SL}_3$-laminations, and show that ${\\rm SL}_3$-laminations satisfying a congruence property are geometric "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2011.14765","kind":"arxiv","version":4},"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/2011.14765/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"}