{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:ZIXTHMA3WAPZMMBG7MXPSIUZHK","short_pith_number":"pith:ZIXTHMA3","canonical_record":{"source":{"id":"math/0611904","kind":"arxiv","version":4},"metadata":{"license":"","primary_cat":"math.RT","submitted_at":"2006-11-29T10:02:03Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b10a33be25ef12d6253a4ef553f6c30eccec7aafe09564804621728f9a92d1b2","abstract_canon_sha256":"237dd9cd46ed69e51d67674f09b32d2ca18e3a9f2941c79568befff9c52f4d7d"},"schema_version":"1.0"},"canonical_sha256":"ca2f33b01bb01f963026fb2ef922993aaa4f3ed1a4e2ca6698f02e729d8f5252","source":{"kind":"arxiv","id":"math/0611904","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0611904","created_at":"2026-07-04T15:02:13Z"},{"alias_kind":"arxiv_version","alias_value":"math/0611904v4","created_at":"2026-07-04T15:02:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0611904","created_at":"2026-07-04T15:02:13Z"},{"alias_kind":"pith_short_12","alias_value":"ZIXTHMA3WAPZ","created_at":"2026-07-04T15:02:13Z"},{"alias_kind":"pith_short_16","alias_value":"ZIXTHMA3WAPZMMBG","created_at":"2026-07-04T15:02:13Z"},{"alias_kind":"pith_short_8","alias_value":"ZIXTHMA3","created_at":"2026-07-04T15:02:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:ZIXTHMA3WAPZMMBG7MXPSIUZHK","target":"record","payload":{"canonical_record":{"source":{"id":"math/0611904","kind":"arxiv","version":4},"metadata":{"license":"","primary_cat":"math.RT","submitted_at":"2006-11-29T10:02:03Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b10a33be25ef12d6253a4ef553f6c30eccec7aafe09564804621728f9a92d1b2","abstract_canon_sha256":"237dd9cd46ed69e51d67674f09b32d2ca18e3a9f2941c79568befff9c52f4d7d"},"schema_version":"1.0"},"canonical_sha256":"ca2f33b01bb01f963026fb2ef922993aaa4f3ed1a4e2ca6698f02e729d8f5252","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:02:13.796068Z","signature_b64":"cy/uCvfY2PAMTTNmM3S65e3C9C4cCbpuz2qxu4egTte9XBdyscbSfIwQNOOMN41ibdB/qh3YfZr9Pafve3/UAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ca2f33b01bb01f963026fb2ef922993aaa4f3ed1a4e2ca6698f02e729d8f5252","last_reissued_at":"2026-07-04T15:02:13.795625Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:02:13.795625Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0611904","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T15:02:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qrCwNRe2zpLSpgWdZpWF5ZEyYAdV1Xg+HF89/pX3aZ6Sbw0Q3cpnhmUmHMUx1SP7RcHL26uFx9m+gifX8ZCCAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T10:08:50.797387Z"},"content_sha256":"7489d37417012342c1fafb746f87c51110eeac58a6811a71bb696b7e0785b92b","schema_version":"1.0","event_id":"sha256:7489d37417012342c1fafb746f87c51110eeac58a6811a71bb696b7e0785b92b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:ZIXTHMA3WAPZMMBG7MXPSIUZHK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A classification of subsystems of a root system","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RT","authors_text":"Toshio Oshima","submitted_at":"2006-11-29T10:02:03Z","abstract_excerpt":"We classify isomorphic classes of the homomorphisms of a root system $\\Xi$ to a root system $\\Sigma$ which do not change Cartan integers. We examine several types of isomorphic classes defined by the Weyl group of $\\Sigma$, that of $\\Xi$ and the automorphisms of $\\Sigma$ or $\\Xi$ etc. We also distinguish the subsystem generated by a subset of a fundamental system. We introduce the concept of the dual pair for root systems which helps to study the action of the outer automorphism of $\\Xi$ on the homomorphisms."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0611904","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0611904/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-04T15:02:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"o75eJxSbEHVcsIZimXBHWLxHgM7LcPDtk0UGqNTxH8PBvUSjbzuSYOZ5ggBNH1N5IDnMVVkoS8qGGrhVGP05Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T10:08:50.798289Z"},"content_sha256":"e140b5da2841f751f392d488410725de630213c681918caea338461401faf199","schema_version":"1.0","event_id":"sha256:e140b5da2841f751f392d488410725de630213c681918caea338461401faf199"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZIXTHMA3WAPZMMBG7MXPSIUZHK/bundle.json","state_url":"https://pith.science/pith/ZIXTHMA3WAPZMMBG7MXPSIUZHK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZIXTHMA3WAPZMMBG7MXPSIUZHK/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":"2026-08-18T10:08:50Z","links":{"resolver":"https://pith.science/pith/ZIXTHMA3WAPZMMBG7MXPSIUZHK","bundle":"https://pith.science/pith/ZIXTHMA3WAPZMMBG7MXPSIUZHK/bundle.json","state":"https://pith.science/pith/ZIXTHMA3WAPZMMBG7MXPSIUZHK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZIXTHMA3WAPZMMBG7MXPSIUZHK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:ZIXTHMA3WAPZMMBG7MXPSIUZHK","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":"237dd9cd46ed69e51d67674f09b32d2ca18e3a9f2941c79568befff9c52f4d7d","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.RT","submitted_at":"2006-11-29T10:02:03Z","title_canon_sha256":"b10a33be25ef12d6253a4ef553f6c30eccec7aafe09564804621728f9a92d1b2"},"schema_version":"1.0","source":{"id":"math/0611904","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0611904","created_at":"2026-07-04T15:02:13Z"},{"alias_kind":"arxiv_version","alias_value":"math/0611904v4","created_at":"2026-07-04T15:02:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0611904","created_at":"2026-07-04T15:02:13Z"},{"alias_kind":"pith_short_12","alias_value":"ZIXTHMA3WAPZ","created_at":"2026-07-04T15:02:13Z"},{"alias_kind":"pith_short_16","alias_value":"ZIXTHMA3WAPZMMBG","created_at":"2026-07-04T15:02:13Z"},{"alias_kind":"pith_short_8","alias_value":"ZIXTHMA3","created_at":"2026-07-04T15:02:13Z"}],"graph_snapshots":[{"event_id":"sha256:e140b5da2841f751f392d488410725de630213c681918caea338461401faf199","target":"graph","created_at":"2026-07-04T15:02:13Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math/0611904/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We classify isomorphic classes of the homomorphisms of a root system $\\Xi$ to a root system $\\Sigma$ which do not change Cartan integers. We examine several types of isomorphic classes defined by the Weyl group of $\\Sigma$, that of $\\Xi$ and the automorphisms of $\\Sigma$ or $\\Xi$ etc. We also distinguish the subsystem generated by a subset of a fundamental system. We introduce the concept of the dual pair for root systems which helps to study the action of the outer automorphism of $\\Xi$ on the homomorphisms.","authors_text":"Toshio Oshima","cross_cats":["math.CO"],"headline":"","license":"","primary_cat":"math.RT","submitted_at":"2006-11-29T10:02:03Z","title":"A classification of subsystems of a root system"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0611904","kind":"arxiv","version":4},"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":"sha256:7489d37417012342c1fafb746f87c51110eeac58a6811a71bb696b7e0785b92b","target":"record","created_at":"2026-07-04T15:02:13Z","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":"237dd9cd46ed69e51d67674f09b32d2ca18e3a9f2941c79568befff9c52f4d7d","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.RT","submitted_at":"2006-11-29T10:02:03Z","title_canon_sha256":"b10a33be25ef12d6253a4ef553f6c30eccec7aafe09564804621728f9a92d1b2"},"schema_version":"1.0","source":{"id":"math/0611904","kind":"arxiv","version":4}},"canonical_sha256":"ca2f33b01bb01f963026fb2ef922993aaa4f3ed1a4e2ca6698f02e729d8f5252","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ca2f33b01bb01f963026fb2ef922993aaa4f3ed1a4e2ca6698f02e729d8f5252","first_computed_at":"2026-07-04T15:02:13.795625Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:02:13.795625Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cy/uCvfY2PAMTTNmM3S65e3C9C4cCbpuz2qxu4egTte9XBdyscbSfIwQNOOMN41ibdB/qh3YfZr9Pafve3/UAg==","signature_status":"signed_v1","signed_at":"2026-07-04T15:02:13.796068Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0611904","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7489d37417012342c1fafb746f87c51110eeac58a6811a71bb696b7e0785b92b","sha256:e140b5da2841f751f392d488410725de630213c681918caea338461401faf199"],"state_sha256":"6b8098660e6f635ae490c2eef4a955d045ecf95427cff400595affcdb3f09316"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HNB3G4z5H0wP0UYXcgsqiSRg4YUZ7P+DhiiU7t0SatJJWKJPWBvGNrLPG+UPgUMkZ4mKnOGFzgmMjCnCi4niAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-18T10:08:50.804107Z","bundle_sha256":"d3ace9102cc9521bcd745df7032dfce8755f45a341067f5917b5d24fb2a4551f"}}