{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:YDGPAULM3AOWBWFM6GU3PHT5R3","short_pith_number":"pith:YDGPAULM","schema_version":"1.0","canonical_sha256":"c0ccf0516cd81d60d8acf1a9b79e7d8ec79fb954db14d9ba78fa6d277bd22a76","source":{"kind":"arxiv","id":"1805.09893","version":2},"attestation_state":"computed","paper":{"title":"The length and depth of compact Lie 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":"2018-05-24T20:48:07Z","abstract_excerpt":"Let $G$ be a connected Lie 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 $G_{i-1}$. In this paper, we introduce the notion of the length (respectively, depth) of $G$, defined as the maximal (respectively, minimal) length of such a chain, and we establish several new results for compact groups. In particular, we compute the exact length and depth of every compact simple Lie group, and draw conclusions for arbitrary connected compact Lie groups $G$. We obtain best possible bounds on the length of"},"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":"1805.09893","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-05-24T20:48:07Z","cross_cats_sorted":[],"title_canon_sha256":"5c166f6a6ed032ddee5a26b7b70b1fb7520032aa19883196efff3bbb093b1e2d","abstract_canon_sha256":"c254f9a77e53ed99ad704f74785f1524b41c8fa9240be499717ab5bbf0982203"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:52.866222Z","signature_b64":"ZSlVPZBUmnxv3j805pLqfHlK5IICyHuOOy1fO4nGhiv2VApi5nbgHP/jba3tsRuhAM62qrWmz+qxHX8FfHVoBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c0ccf0516cd81d60d8acf1a9b79e7d8ec79fb954db14d9ba78fa6d277bd22a76","last_reissued_at":"2026-05-17T23:48:52.865664Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:52.865664Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The length and depth of compact Lie 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":"2018-05-24T20:48:07Z","abstract_excerpt":"Let $G$ be a connected Lie 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 $G_{i-1}$. In this paper, we introduce the notion of the length (respectively, depth) of $G$, defined as the maximal (respectively, minimal) length of such a chain, and we establish several new results for compact groups. In particular, we compute the exact length and depth of every compact simple Lie group, and draw conclusions for arbitrary connected compact Lie groups $G$. We obtain best possible bounds on the length of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.09893","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":"1805.09893","created_at":"2026-05-17T23:48:52.865759+00:00"},{"alias_kind":"arxiv_version","alias_value":"1805.09893v2","created_at":"2026-05-17T23:48:52.865759+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.09893","created_at":"2026-05-17T23:48:52.865759+00:00"},{"alias_kind":"pith_short_12","alias_value":"YDGPAULM3AOW","created_at":"2026-05-18T12:33:04.347982+00:00"},{"alias_kind":"pith_short_16","alias_value":"YDGPAULM3AOWBWFM","created_at":"2026-05-18T12:33:04.347982+00:00"},{"alias_kind":"pith_short_8","alias_value":"YDGPAULM","created_at":"2026-05-18T12:33:04.347982+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/YDGPAULM3AOWBWFM6GU3PHT5R3","json":"https://pith.science/pith/YDGPAULM3AOWBWFM6GU3PHT5R3.json","graph_json":"https://pith.science/api/pith-number/YDGPAULM3AOWBWFM6GU3PHT5R3/graph.json","events_json":"https://pith.science/api/pith-number/YDGPAULM3AOWBWFM6GU3PHT5R3/events.json","paper":"https://pith.science/paper/YDGPAULM"},"agent_actions":{"view_html":"https://pith.science/pith/YDGPAULM3AOWBWFM6GU3PHT5R3","download_json":"https://pith.science/pith/YDGPAULM3AOWBWFM6GU3PHT5R3.json","view_paper":"https://pith.science/paper/YDGPAULM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1805.09893&json=true","fetch_graph":"https://pith.science/api/pith-number/YDGPAULM3AOWBWFM6GU3PHT5R3/graph.json","fetch_events":"https://pith.science/api/pith-number/YDGPAULM3AOWBWFM6GU3PHT5R3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YDGPAULM3AOWBWFM6GU3PHT5R3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YDGPAULM3AOWBWFM6GU3PHT5R3/action/storage_attestation","attest_author":"https://pith.science/pith/YDGPAULM3AOWBWFM6GU3PHT5R3/action/author_attestation","sign_citation":"https://pith.science/pith/YDGPAULM3AOWBWFM6GU3PHT5R3/action/citation_signature","submit_replication":"https://pith.science/pith/YDGPAULM3AOWBWFM6GU3PHT5R3/action/replication_record"}},"created_at":"2026-05-17T23:48:52.865759+00:00","updated_at":"2026-05-17T23:48:52.865759+00:00"}