{"paper":{"title":"Cluster Variables on Certain Double Bruhat Cells of Type $(u,e)$ and Monomial Realizations of Crystal Bases of Type A","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.QA","authors_text":"Toshiki Nakashima, Yuki Kanakubo","submitted_at":"2014-09-30T16:36:02Z","abstract_excerpt":"Let $G$ be a simply connected simple algebraic group over $\\mathbb{C}$, $B$ and $B_-$ be two opposite Borel subgroups in $G$ and $W$ be the Weyl group. For $u$, $v\\in W$, it is known that the coordinate ring ${\\mathbb C}[G^{u,v}]$ of the double Bruhat cell $G^{u,v}=BuB\\cap B_-vB_-$ is isomorphic to an upper cluster algebra $\\bar{{\\mathcal A}}({\\bf i})_{{\\mathbb C}}$ and the generalized minors $\\{\\Delta(k;{\\bf i})\\}$ are the cluster variables belonging to a given initial seed in ${\\mathbb C}[G^{u,v}]$ [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52, math.RT/0305434]. In "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.8622","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":""},"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"}