{"paper":{"title":"Minimal dimensional representations of reduced enveloping algebras for $\\mathfrak{gl}_n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.RT","authors_text":"Lewis Topley, Simon M. Goodwin","submitted_at":"2018-05-03T14:33:56Z","abstract_excerpt":"Let $\\mathfrak g = \\mathfrak{gl}_N(k)$, where $k$ is an algebraically closed field of characteristic $p > 0$, and $N \\in \\mathbb Z_{\\ge 1}$. Let $\\chi \\in \\mathfrak g^*$ and denote by $U_\\chi(\\mathfrak g)$ the corresponding reduced enveloping algebra. The Kac--Weisfeiler conjecture, which was proved by Premet, asserts that any finite dimensional $U_\\chi(\\mathfrak g)$-module has dimension divisible by $p^{d_\\chi}$, where $d_\\chi$ is half the dimension of the coadjoint orbit of $\\chi$. Our main theorem gives a classification of $U_\\chi(\\mathfrak g)$-modules of dimension $p^{d_\\chi}$. As a conseq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.01327","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"}