{"paper":{"title":"Birational Weyl Group Action on the Symplectic Groupoid and Cluster Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.RT"],"primary_cat":"math.QA","authors_text":"Woojin Choi","submitted_at":"2026-01-26T16:09:21Z","abstract_excerpt":"A. Bondal's symplectic groupoid of triangular bilinear forms induces a Poisson structure on the space $\\mathcal{A}_n$ of $n \\times n$ unipotent upper-triangular matrices. It is governed by the classical $\\mathfrak{so}(n)$ reflection equation. L. Chekhov and M. Shapiro described log-canonical coordinates on this groupoid via the $\\mathcal{A}_n$-quiver.\n  We introduce a birational Weyl group action on this symplectic groupoid, generated by cluster transformations associated with certain cycles of the quiver. We prove that the Poisson subalgebra of Weyl group invariants is a finite central extens"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.18636","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/2601.18636/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"}