{"paper":{"title":"Extending $\\pi$-systems to bases of root systems","license":"","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Helmer Aslaksen, Mong Lung Lang","submitted_at":"2004-10-15T20:40:39Z","abstract_excerpt":"Let $R$ be an indecomposable root system. It is well known that any root is part of a basis $B$ of $R$. But when can you extend a set of two or more roots to a basis $B$ of $R$? A $\\pi$-system is a linearly independent set of roots, $C$, such that if $\\alpha$ and $\\beta$ are in $C$, then $\\alpha - \\beta$ is not a root. We will use results of Dynkin and Bourbaki to show that with two exceptions, $A_3 \\subset B_n$ and $A_7 \\subset E_8$, an indecomposable $\\pi$-system whose Dynkin diagram is a subdiagram of the Dynkin diagram of $R$ can always be extended to a basis of $R$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0410357","kind":"arxiv","version":3},"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"}