{"paper":{"title":"Cluster algebras of finite type via a Coxeter element and Demazure Crystals of type A","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.QA","authors_text":"Toshiki Nakashima, Yuki Kanakubo","submitted_at":"2017-03-24T09:31:39Z","abstract_excerpt":"Let $G$ be a simply connected simple algebraic group over $\\mathbb{C}$, $B$ and $B_-$ be its two opposite Borel subgroups. For two elements $u$, $v$ of the Weyl group $W$, 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 a cluster algebra $\\mathcal{A}(\\textbf{i})_{{\\mathbb C}}$ [arXiv:math/0305434, arXiv:1602.00498]. In the case $u=e$, $v=c^2$ ($c$ is a Coxeter element), the algebra ${\\mathbb C}[G^{e,c^2}]$ has only finitely many cluster variables. In this article, for $G={\\rm SL}_{r+1}(\\mathbb{C})$, we obtain expl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.08323","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"}