{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:JOCG5FL4WGXKWXSLYR2KLR7VLG","short_pith_number":"pith:JOCG5FL4","schema_version":"1.0","canonical_sha256":"4b846e957cb1aeab5e4bc474a5c7f5598fba6f4acb62cf3d6eca00979781afc2","source":{"kind":"arxiv","id":"1707.01935","version":2},"attestation_state":"computed","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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1707.01935","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-07-06T18:48:45Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"92027de441e1fbc8d3e4da2acca5d630bc17664cae663b0f817c5dc64bafad30","abstract_canon_sha256":"fe6a5fadab87db0a74f7894b2735f498965d4c0d3eab1fd906d60e48a6e91a00"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:34.598756Z","signature_b64":"h2bxOzMAoZukkN+xoz9kSg2LJLP2bqIXoUU52h7ZA4NonUwVyrUE9E+GdvOZeFyocJNyl8cawfysHKa8mTzdBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4b846e957cb1aeab5e4bc474a5c7f5598fba6f4acb62cf3d6eca00979781afc2","last_reissued_at":"2026-05-17T23:51:34.598094Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:34.598094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1707.01935","created_at":"2026-05-17T23:51:34.598191+00:00"},{"alias_kind":"arxiv_version","alias_value":"1707.01935v2","created_at":"2026-05-17T23:51:34.598191+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.01935","created_at":"2026-05-17T23:51:34.598191+00:00"},{"alias_kind":"pith_short_12","alias_value":"JOCG5FL4WGXK","created_at":"2026-05-18T12:31:24.725408+00:00"},{"alias_kind":"pith_short_16","alias_value":"JOCG5FL4WGXKWXSL","created_at":"2026-05-18T12:31:24.725408+00:00"},{"alias_kind":"pith_short_8","alias_value":"JOCG5FL4","created_at":"2026-05-18T12:31:24.725408+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JOCG5FL4WGXKWXSLYR2KLR7VLG","json":"https://pith.science/pith/JOCG5FL4WGXKWXSLYR2KLR7VLG.json","graph_json":"https://pith.science/api/pith-number/JOCG5FL4WGXKWXSLYR2KLR7VLG/graph.json","events_json":"https://pith.science/api/pith-number/JOCG5FL4WGXKWXSLYR2KLR7VLG/events.json","paper":"https://pith.science/paper/JOCG5FL4"},"agent_actions":{"view_html":"https://pith.science/pith/JOCG5FL4WGXKWXSLYR2KLR7VLG","download_json":"https://pith.science/pith/JOCG5FL4WGXKWXSLYR2KLR7VLG.json","view_paper":"https://pith.science/paper/JOCG5FL4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1707.01935&json=true","fetch_graph":"https://pith.science/api/pith-number/JOCG5FL4WGXKWXSLYR2KLR7VLG/graph.json","fetch_events":"https://pith.science/api/pith-number/JOCG5FL4WGXKWXSLYR2KLR7VLG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JOCG5FL4WGXKWXSLYR2KLR7VLG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JOCG5FL4WGXKWXSLYR2KLR7VLG/action/storage_attestation","attest_author":"https://pith.science/pith/JOCG5FL4WGXKWXSLYR2KLR7VLG/action/author_attestation","sign_citation":"https://pith.science/pith/JOCG5FL4WGXKWXSLYR2KLR7VLG/action/citation_signature","submit_replication":"https://pith.science/pith/JOCG5FL4WGXKWXSLYR2KLR7VLG/action/replication_record"}},"created_at":"2026-05-17T23:51:34.598191+00:00","updated_at":"2026-05-17T23:51:34.598191+00:00"}