{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:3544D6JFRYZMZLOEOBKKTM66SW","short_pith_number":"pith:3544D6JF","canonical_record":{"source":{"id":"1808.01136","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-08-03T09:48:18Z","cross_cats_sorted":["math.CO","math.MG","math.NT"],"title_canon_sha256":"ce9eaae18b9f7c7d5b2c046a72c109b64b54b7b8cd38783bb95555f153574a6d","abstract_canon_sha256":"ca0f7cb5aaa7fe0a2e3277e908e23d209bf74a428085fdfe3a9aca37750d16e1"},"schema_version":"1.0"},"canonical_sha256":"df79c1f9258e32ccadc47054a9b3de959f0856f99ebb468ee5eacd2fc1636656","source":{"kind":"arxiv","id":"1808.01136","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.01136","created_at":"2026-05-17T23:39:32Z"},{"alias_kind":"arxiv_version","alias_value":"1808.01136v3","created_at":"2026-05-17T23:39:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.01136","created_at":"2026-05-17T23:39:32Z"},{"alias_kind":"pith_short_12","alias_value":"3544D6JFRYZM","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_16","alias_value":"3544D6JFRYZMZLOE","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_8","alias_value":"3544D6JF","created_at":"2026-05-18T12:32:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:3544D6JFRYZMZLOEOBKKTM66SW","target":"record","payload":{"canonical_record":{"source":{"id":"1808.01136","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-08-03T09:48:18Z","cross_cats_sorted":["math.CO","math.MG","math.NT"],"title_canon_sha256":"ce9eaae18b9f7c7d5b2c046a72c109b64b54b7b8cd38783bb95555f153574a6d","abstract_canon_sha256":"ca0f7cb5aaa7fe0a2e3277e908e23d209bf74a428085fdfe3a9aca37750d16e1"},"schema_version":"1.0"},"canonical_sha256":"df79c1f9258e32ccadc47054a9b3de959f0856f99ebb468ee5eacd2fc1636656","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:32.282602Z","signature_b64":"shd8zs/DadeHPcKD8ldByIqeEONW8XfZvFjoysKm39NLt09YxMjGOFKrBvabYJgxhIAxf3HhDwUC694asx21CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"df79c1f9258e32ccadc47054a9b3de959f0856f99ebb468ee5eacd2fc1636656","last_reissued_at":"2026-05-17T23:39:32.282075Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:32.282075Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1808.01136","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:39:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J1opEB5FN2Hk9A56TJp6x6taiiLb4YU7hG0dmUI/pjG/Ya2352QGP5ULm1n5WcWCqa9xMK7smwOYagVkVQ60DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T02:36:15.726377Z"},"content_sha256":"f69d680982d012ab632739a34a410b1f0229ae7efd18b4fedec624150998c72d","schema_version":"1.0","event_id":"sha256:f69d680982d012ab632739a34a410b1f0229ae7efd18b4fedec624150998c72d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:3544D6JFRYZMZLOEOBKKTM66SW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Root systems in number fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG","math.NT"],"primary_cat":"math.GR","authors_text":"Vladimir L. Popov, Yuri G. Zarhin","submitted_at":"2018-08-03T09:48:18Z","abstract_excerpt":"We classify the types of root systems $R$ in the rings of integers of number fields $K$ such that the Weyl group $W(R)$ lies in the group $\\mathcal L(K)$ generated by ${\\rm Aut} (K)$ and multiplications by the elements of $K^*$. We also classify the Weyl groups of roots systems of rank $n$ which are isomorphic to a subgroup of $\\mathcal L(K)$ for a number field $K$ of degree $n$ over $\\mathbb Q$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.01136","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:39:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nIyM5gisA6KD5Tx0BQ0IWpAfGNSVvl6qnAORwc365rS/0Jq7CbhroJDZ3Rozuiue0RCRnyu01Hf7WBb+u9ebBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T02:36:15.727028Z"},"content_sha256":"f7df999a504beebb317d7ff4cb30b34e2dae7a2f73ef016746a860a6b8711105","schema_version":"1.0","event_id":"sha256:f7df999a504beebb317d7ff4cb30b34e2dae7a2f73ef016746a860a6b8711105"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3544D6JFRYZMZLOEOBKKTM66SW/bundle.json","state_url":"https://pith.science/pith/3544D6JFRYZMZLOEOBKKTM66SW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3544D6JFRYZMZLOEOBKKTM66SW/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-01T02:36:15Z","links":{"resolver":"https://pith.science/pith/3544D6JFRYZMZLOEOBKKTM66SW","bundle":"https://pith.science/pith/3544D6JFRYZMZLOEOBKKTM66SW/bundle.json","state":"https://pith.science/pith/3544D6JFRYZMZLOEOBKKTM66SW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3544D6JFRYZMZLOEOBKKTM66SW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:3544D6JFRYZMZLOEOBKKTM66SW","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ca0f7cb5aaa7fe0a2e3277e908e23d209bf74a428085fdfe3a9aca37750d16e1","cross_cats_sorted":["math.CO","math.MG","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-08-03T09:48:18Z","title_canon_sha256":"ce9eaae18b9f7c7d5b2c046a72c109b64b54b7b8cd38783bb95555f153574a6d"},"schema_version":"1.0","source":{"id":"1808.01136","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.01136","created_at":"2026-05-17T23:39:32Z"},{"alias_kind":"arxiv_version","alias_value":"1808.01136v3","created_at":"2026-05-17T23:39:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.01136","created_at":"2026-05-17T23:39:32Z"},{"alias_kind":"pith_short_12","alias_value":"3544D6JFRYZM","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_16","alias_value":"3544D6JFRYZMZLOE","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_8","alias_value":"3544D6JF","created_at":"2026-05-18T12:32:02Z"}],"graph_snapshots":[{"event_id":"sha256:f7df999a504beebb317d7ff4cb30b34e2dae7a2f73ef016746a860a6b8711105","target":"graph","created_at":"2026-05-17T23:39:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We classify the types of root systems $R$ in the rings of integers of number fields $K$ such that the Weyl group $W(R)$ lies in the group $\\mathcal L(K)$ generated by ${\\rm Aut} (K)$ and multiplications by the elements of $K^*$. We also classify the Weyl groups of roots systems of rank $n$ which are isomorphic to a subgroup of $\\mathcal L(K)$ for a number field $K$ of degree $n$ over $\\mathbb Q$.","authors_text":"Vladimir L. Popov, Yuri G. Zarhin","cross_cats":["math.CO","math.MG","math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-08-03T09:48:18Z","title":"Root systems in number fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.01136","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f69d680982d012ab632739a34a410b1f0229ae7efd18b4fedec624150998c72d","target":"record","created_at":"2026-05-17T23:39:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ca0f7cb5aaa7fe0a2e3277e908e23d209bf74a428085fdfe3a9aca37750d16e1","cross_cats_sorted":["math.CO","math.MG","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-08-03T09:48:18Z","title_canon_sha256":"ce9eaae18b9f7c7d5b2c046a72c109b64b54b7b8cd38783bb95555f153574a6d"},"schema_version":"1.0","source":{"id":"1808.01136","kind":"arxiv","version":3}},"canonical_sha256":"df79c1f9258e32ccadc47054a9b3de959f0856f99ebb468ee5eacd2fc1636656","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"df79c1f9258e32ccadc47054a9b3de959f0856f99ebb468ee5eacd2fc1636656","first_computed_at":"2026-05-17T23:39:32.282075Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:39:32.282075Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"shd8zs/DadeHPcKD8ldByIqeEONW8XfZvFjoysKm39NLt09YxMjGOFKrBvabYJgxhIAxf3HhDwUC694asx21CQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:39:32.282602Z","signed_message":"canonical_sha256_bytes"},"source_id":"1808.01136","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f69d680982d012ab632739a34a410b1f0229ae7efd18b4fedec624150998c72d","sha256:f7df999a504beebb317d7ff4cb30b34e2dae7a2f73ef016746a860a6b8711105"],"state_sha256":"5ee31596cd245af722f92c72e2e3050f2bb84d153dc301def8512f319764d621"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X8P2kaBpwbHNcXFA68ine31YGLN+LONPkmX6rT1B+Agzt3pNDZLxiLW9ohnvhWpjENTwtusJWvSQ/VxwN06yDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T02:36:15.730575Z","bundle_sha256":"efcb15f2d69049277ad28771627ee538a35be67da465c3f3f314145b9706e2a7"}}