{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:FNSO7WZB2YDKKCG4HTVYO6DCIS","short_pith_number":"pith:FNSO7WZB","schema_version":"1.0","canonical_sha256":"2b64efdb21d606a508dc3ceb87786244af50f63f15ca654041f505917921ecc5","source":{"kind":"arxiv","id":"1606.02247","version":1},"attestation_state":"computed","paper":{"title":"On the dimension of matrix embeddings of torsion-free nilpotent groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Armin Wei{\\ss}, Funda Gul","submitted_at":"2016-06-07T18:27:44Z","abstract_excerpt":"Since the work of Jennings (1955), it is well-known that any finitely generated torsion-free nilpotent group can be embedded into unitriangular integer matrices $UT_N(Z)$ for some $N$. In 2006, Nickel proposed an algorithm to calculate such embeddings. In this work, we show that if $UT_n(Z)$ is embedded into $UT_N(Z)$ using Nickel's algorithm, then $N \\geq 2^{n/2 -2}$ if the standard ordering of the Mal'cev basis (as in Nickel's original paper) is used. In particular, we establish an exponential worst-case running time of Nickel's algorithm.\n  On the other hand, we also prove a general exponen"},"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":"1606.02247","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-06-07T18:27:44Z","cross_cats_sorted":[],"title_canon_sha256":"cd8586b8674e9c5b007214480ff55bc2050cc25c0b4a41ce288180b9f1aaeab9","abstract_canon_sha256":"541759e2ecf8565c3a7776d4de7bd685cecfd2c9227d499a20dfa60fcacb4ab5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:44.395095Z","signature_b64":"adSL2AwjYIoo8s+8Oc13ZQxODvgn2Vgk309NxUB9SDIq7hHbc07fJx9nvgkJpuV3VRWPzguRZEZIeiEcwKh4Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b64efdb21d606a508dc3ceb87786244af50f63f15ca654041f505917921ecc5","last_reissued_at":"2026-05-18T01:12:44.394742Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:44.394742Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the dimension of matrix embeddings of torsion-free nilpotent groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Armin Wei{\\ss}, Funda Gul","submitted_at":"2016-06-07T18:27:44Z","abstract_excerpt":"Since the work of Jennings (1955), it is well-known that any finitely generated torsion-free nilpotent group can be embedded into unitriangular integer matrices $UT_N(Z)$ for some $N$. In 2006, Nickel proposed an algorithm to calculate such embeddings. In this work, we show that if $UT_n(Z)$ is embedded into $UT_N(Z)$ using Nickel's algorithm, then $N \\geq 2^{n/2 -2}$ if the standard ordering of the Mal'cev basis (as in Nickel's original paper) is used. In particular, we establish an exponential worst-case running time of Nickel's algorithm.\n  On the other hand, we also prove a general exponen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02247","kind":"arxiv","version":1},"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":"1606.02247","created_at":"2026-05-18T01:12:44.394799+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.02247v1","created_at":"2026-05-18T01:12:44.394799+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.02247","created_at":"2026-05-18T01:12:44.394799+00:00"},{"alias_kind":"pith_short_12","alias_value":"FNSO7WZB2YDK","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_16","alias_value":"FNSO7WZB2YDKKCG4","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_8","alias_value":"FNSO7WZB","created_at":"2026-05-18T12:30:15.759754+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/FNSO7WZB2YDKKCG4HTVYO6DCIS","json":"https://pith.science/pith/FNSO7WZB2YDKKCG4HTVYO6DCIS.json","graph_json":"https://pith.science/api/pith-number/FNSO7WZB2YDKKCG4HTVYO6DCIS/graph.json","events_json":"https://pith.science/api/pith-number/FNSO7WZB2YDKKCG4HTVYO6DCIS/events.json","paper":"https://pith.science/paper/FNSO7WZB"},"agent_actions":{"view_html":"https://pith.science/pith/FNSO7WZB2YDKKCG4HTVYO6DCIS","download_json":"https://pith.science/pith/FNSO7WZB2YDKKCG4HTVYO6DCIS.json","view_paper":"https://pith.science/paper/FNSO7WZB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.02247&json=true","fetch_graph":"https://pith.science/api/pith-number/FNSO7WZB2YDKKCG4HTVYO6DCIS/graph.json","fetch_events":"https://pith.science/api/pith-number/FNSO7WZB2YDKKCG4HTVYO6DCIS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FNSO7WZB2YDKKCG4HTVYO6DCIS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FNSO7WZB2YDKKCG4HTVYO6DCIS/action/storage_attestation","attest_author":"https://pith.science/pith/FNSO7WZB2YDKKCG4HTVYO6DCIS/action/author_attestation","sign_citation":"https://pith.science/pith/FNSO7WZB2YDKKCG4HTVYO6DCIS/action/citation_signature","submit_replication":"https://pith.science/pith/FNSO7WZB2YDKKCG4HTVYO6DCIS/action/replication_record"}},"created_at":"2026-05-18T01:12:44.394799+00:00","updated_at":"2026-05-18T01:12:44.394799+00:00"}