{"paper":{"title":"A graph-theoretic approach for comparing dimensions of components in simply-graded algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RA","authors_text":"Ofir Schnabel, Yuval Ginosar","submitted_at":"2013-05-21T08:06:43Z","abstract_excerpt":"Any simple group-grading of a finite dimensional complex algebra induces a natural family of digraphs. We prove that $|E\\circ E^{\\text{op}}\\cup E^{\\text{op}}\\circ E|\\geq |E|$ for any digraph $\\Gamma =(V,E)$ without parallel edges, and deduce that for any simple group-grading, the dimension of the trivial component is maximal."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.4749","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"}