{"paper":{"title":"Fock-Goncharov conjecture and polyhedral cones for $U \\subset SL_n$ and base affine space $SL_n /U$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.AG","authors_text":"Timothy Magee","submitted_at":"2015-02-12T18:49:23Z","abstract_excerpt":"I prove several conjectures of \\cite{GHKK} on the cluster structure of $SL_n$, which in particular imply the full Fock-Goncharov conjecture for the open double Bruhat cell $\\mathcal{A} \\subset SL_n/U$, for $U \\subset SL_n$ a maximal unipotent subgroup. This endows the mirror cluster variety $\\mathcal{X}$ with a canonical potential function $W$, and determines a canonical cone $W^T \\geq 0 \\subset \\mathcal{X}\\left(\\mathbb{R}^T\\right)$ of the mirror tropical space, whose integer points parametrize a basis of $H^0\\left(SL_n/U,\\mathcal{O}_{SL_n/U}\\right)$, canonically determined by the open subset "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.03769","kind":"arxiv","version":1},"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"}