{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:464I22AOLNQUMF6M5AJ7HW3FDR","short_pith_number":"pith:464I22AO","schema_version":"1.0","canonical_sha256":"e7b88d680e5b614617cce813f3db651c50fb4c9852ae984eb1f248fd187812f1","source":{"kind":"arxiv","id":"1712.08214","version":3},"attestation_state":"computed","paper":{"title":"The length and depth of algebraic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Aner Shalev, Martin W. Liebeck, Timothy C. Burness","submitted_at":"2017-12-21T21:18:33Z","abstract_excerpt":"Let $G$ be a connected algebraic group. An unrefinable chain of $G$ is a chain of subgroups $G = G_0 > G_1 > \\cdots > G_t = 1$, where each $G_i$ is a maximal connected subgroup of $G_{i-1}$. We introduce the notion of the length (respectively, depth) of $G$, defined as the maximal (respectively, minimal) length of such a chain. Working over an algebraically closed field, we calculate the length of a connected group $G$ in terms of the dimension of its unipotent radical $R_u(G)$ and the dimension of a Borel subgroup $B$ of the reductive quotient $G/R_u(G)$. In particular, a simple algebraic gro"},"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":"1712.08214","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-12-21T21:18:33Z","cross_cats_sorted":[],"title_canon_sha256":"de3185a7a8505acf2536c4dcc1155a0cb8c3324ae86e42f79ff776111d00d6c2","abstract_canon_sha256":"8341e83eda4c6ffe4bd8e3ca66c88142154055c6cce9dba42da5038fa9ef921b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:59.655846Z","signature_b64":"L/g8hGMTmQaowI0uGgGcQZcc5C4LW/XSrSLbK+3DJYF92YKhoJ6+zD2lZZGX2Dr5iF2vA+EZZTS6ri6Rt5oPDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e7b88d680e5b614617cce813f3db651c50fb4c9852ae984eb1f248fd187812f1","last_reissued_at":"2026-05-18T00:14:59.655091Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:59.655091Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The length and depth of algebraic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Aner Shalev, Martin W. Liebeck, Timothy C. Burness","submitted_at":"2017-12-21T21:18:33Z","abstract_excerpt":"Let $G$ be a connected algebraic group. An unrefinable chain of $G$ is a chain of subgroups $G = G_0 > G_1 > \\cdots > G_t = 1$, where each $G_i$ is a maximal connected subgroup of $G_{i-1}$. We introduce the notion of the length (respectively, depth) of $G$, defined as the maximal (respectively, minimal) length of such a chain. Working over an algebraically closed field, we calculate the length of a connected group $G$ in terms of the dimension of its unipotent radical $R_u(G)$ and the dimension of a Borel subgroup $B$ of the reductive quotient $G/R_u(G)$. In particular, a simple algebraic gro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.08214","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1712.08214","created_at":"2026-05-18T00:14:59.655237+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.08214v3","created_at":"2026-05-18T00:14:59.655237+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.08214","created_at":"2026-05-18T00:14:59.655237+00:00"},{"alias_kind":"pith_short_12","alias_value":"464I22AOLNQU","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_16","alias_value":"464I22AOLNQUMF6M","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_8","alias_value":"464I22AO","created_at":"2026-05-18T12:30:58.224056+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/464I22AOLNQUMF6M5AJ7HW3FDR","json":"https://pith.science/pith/464I22AOLNQUMF6M5AJ7HW3FDR.json","graph_json":"https://pith.science/api/pith-number/464I22AOLNQUMF6M5AJ7HW3FDR/graph.json","events_json":"https://pith.science/api/pith-number/464I22AOLNQUMF6M5AJ7HW3FDR/events.json","paper":"https://pith.science/paper/464I22AO"},"agent_actions":{"view_html":"https://pith.science/pith/464I22AOLNQUMF6M5AJ7HW3FDR","download_json":"https://pith.science/pith/464I22AOLNQUMF6M5AJ7HW3FDR.json","view_paper":"https://pith.science/paper/464I22AO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.08214&json=true","fetch_graph":"https://pith.science/api/pith-number/464I22AOLNQUMF6M5AJ7HW3FDR/graph.json","fetch_events":"https://pith.science/api/pith-number/464I22AOLNQUMF6M5AJ7HW3FDR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/464I22AOLNQUMF6M5AJ7HW3FDR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/464I22AOLNQUMF6M5AJ7HW3FDR/action/storage_attestation","attest_author":"https://pith.science/pith/464I22AOLNQUMF6M5AJ7HW3FDR/action/author_attestation","sign_citation":"https://pith.science/pith/464I22AOLNQUMF6M5AJ7HW3FDR/action/citation_signature","submit_replication":"https://pith.science/pith/464I22AOLNQUMF6M5AJ7HW3FDR/action/replication_record"}},"created_at":"2026-05-18T00:14:59.655237+00:00","updated_at":"2026-05-18T00:14:59.655237+00:00"}