{"paper":{"title":"Root data with group actions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.RT","authors_text":"Jeffrey D. Adler, Joshua M. Lansky","submitted_at":"2017-07-06T18:48:45Z","abstract_excerpt":"Suppose $k$ is a field, $G$ is a connected reductive algebraic $k$-group, $T$ is a maximal $k$-torus in $G$, and $\\Gamma$ is a finite group that acts on $(G,T)$. From the above, one obtains a root datum $\\Psi$ on which $\\text{Gal}(k)\\times\\Gamma$ acts. Provided that $\\Gamma$ preserves a positive system in $\\Psi$, not necessarily invariant under $\\text{Gal}(k)$, we construct an inverse to this process. That is, given a root datum on which $\\text{Gal}(k)\\times\\Gamma$ acts appropriately, we show how to construct a pair $(G,T)$, on which $\\Gamma$ acts as above.\n  Although the pair $(G,T)$ and the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.01935","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"}