{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:WJ46HKYKBLLLCD43D27VFNRQRN","short_pith_number":"pith:WJ46HKYK","canonical_record":{"source":{"id":"1011.4835","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-11-22T15:04:25Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"99867d7684e00ee86e91c8b8b69aa9505cd036d9714005920c223dbf6aa3b2c3","abstract_canon_sha256":"e01acf5467151d6f83f3ba7dc4deacc5a87e2bc9bf33f332ed959ab8ad56b9a5"},"schema_version":"1.0"},"canonical_sha256":"b279e3ab0a0ad6b10f9b1ebf52b6308b63b93455c725939e5655d8eeedbb6e78","source":{"kind":"arxiv","id":"1011.4835","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.4835","created_at":"2026-05-18T04:35:03Z"},{"alias_kind":"arxiv_version","alias_value":"1011.4835v1","created_at":"2026-05-18T04:35:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.4835","created_at":"2026-05-18T04:35:03Z"},{"alias_kind":"pith_short_12","alias_value":"WJ46HKYKBLLL","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"WJ46HKYKBLLLCD43","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"WJ46HKYK","created_at":"2026-05-18T12:26:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:WJ46HKYKBLLLCD43D27VFNRQRN","target":"record","payload":{"canonical_record":{"source":{"id":"1011.4835","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-11-22T15:04:25Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"99867d7684e00ee86e91c8b8b69aa9505cd036d9714005920c223dbf6aa3b2c3","abstract_canon_sha256":"e01acf5467151d6f83f3ba7dc4deacc5a87e2bc9bf33f332ed959ab8ad56b9a5"},"schema_version":"1.0"},"canonical_sha256":"b279e3ab0a0ad6b10f9b1ebf52b6308b63b93455c725939e5655d8eeedbb6e78","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:35:03.916602Z","signature_b64":"tTJzYCHwklNOryIZvzdu6I4OqmCE1eKSBp76EG4ec+uIxK4h6iJ3W0502Khp83PxndxEwydPHCz8ffDZUh1BCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b279e3ab0a0ad6b10f9b1ebf52b6308b63b93455c725939e5655d8eeedbb6e78","last_reissued_at":"2026-05-18T04:35:03.916185Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:35:03.916185Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1011.4835","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:35:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8GTvfbqzhr3OsoTEOy28JU1Vz0PFm0Tqs+F69qiO9qV6HzfsWio+SrrK5cE84qq6p1vG53spLwx3MI2cP3mXCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T09:48:21.002884Z"},"content_sha256":"21ea678fb23ac2ea03968472b8f46c3bfc882f31e6e94836aafcce7080ec0870","schema_version":"1.0","event_id":"sha256:21ea678fb23ac2ea03968472b8f46c3bfc882f31e6e94836aafcce7080ec0870"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:WJ46HKYKBLLLCD43D27VFNRQRN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"G-complete reducibility and the exceptional algebraic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.GR","authors_text":"David I. Stewart","submitted_at":"2010-11-22T15:04:25Z","abstract_excerpt":"Let $G=G(K)$ be a simple algebraic group defined over an algebraically closed field $K$ of characteristic $p>0$. A subgroup $X$ of $G$ is said to be $G$-completely reducible if, whenever it is contained in a parabolic subgroup of $G$, it is contained in a Levi subgroup of that parabolic. A subgroup $X$ of $G$ is said to be $G$-irreducible if $X$ is in no parabolic subgroup of $G$; and $G$-reducible if it is in some parabolic of $G$. In this thesis, we consider the case that $G$ is of exceptional type. When $G$ is of type $G_2$ we find all conjugacy classes of closed, connected, reductive subgr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.4835","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:35:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DTYyF/553q7E1BbJgXrF9h9OWnrhhfoixnEknWSJS8I/mzIa0lqU93KH8jg2kwiXXv+FD4a//7y/E4ONJBplAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T09:48:21.003217Z"},"content_sha256":"8f9e8a8fb6cb7e8b811fe83fa4c3187a20e2254f0679c7fecefd2e3fdfad11fe","schema_version":"1.0","event_id":"sha256:8f9e8a8fb6cb7e8b811fe83fa4c3187a20e2254f0679c7fecefd2e3fdfad11fe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WJ46HKYKBLLLCD43D27VFNRQRN/bundle.json","state_url":"https://pith.science/pith/WJ46HKYKBLLLCD43D27VFNRQRN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WJ46HKYKBLLLCD43D27VFNRQRN/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":"2026-06-30T09:48:21Z","links":{"resolver":"https://pith.science/pith/WJ46HKYKBLLLCD43D27VFNRQRN","bundle":"https://pith.science/pith/WJ46HKYKBLLLCD43D27VFNRQRN/bundle.json","state":"https://pith.science/pith/WJ46HKYKBLLLCD43D27VFNRQRN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WJ46HKYKBLLLCD43D27VFNRQRN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:WJ46HKYKBLLLCD43D27VFNRQRN","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":"e01acf5467151d6f83f3ba7dc4deacc5a87e2bc9bf33f332ed959ab8ad56b9a5","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-11-22T15:04:25Z","title_canon_sha256":"99867d7684e00ee86e91c8b8b69aa9505cd036d9714005920c223dbf6aa3b2c3"},"schema_version":"1.0","source":{"id":"1011.4835","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.4835","created_at":"2026-05-18T04:35:03Z"},{"alias_kind":"arxiv_version","alias_value":"1011.4835v1","created_at":"2026-05-18T04:35:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.4835","created_at":"2026-05-18T04:35:03Z"},{"alias_kind":"pith_short_12","alias_value":"WJ46HKYKBLLL","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"WJ46HKYKBLLLCD43","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"WJ46HKYK","created_at":"2026-05-18T12:26:15Z"}],"graph_snapshots":[{"event_id":"sha256:8f9e8a8fb6cb7e8b811fe83fa4c3187a20e2254f0679c7fecefd2e3fdfad11fe","target":"graph","created_at":"2026-05-18T04:35:03Z","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":"Let $G=G(K)$ be a simple algebraic group defined over an algebraically closed field $K$ of characteristic $p>0$. A subgroup $X$ of $G$ is said to be $G$-completely reducible if, whenever it is contained in a parabolic subgroup of $G$, it is contained in a Levi subgroup of that parabolic. A subgroup $X$ of $G$ is said to be $G$-irreducible if $X$ is in no parabolic subgroup of $G$; and $G$-reducible if it is in some parabolic of $G$. In this thesis, we consider the case that $G$ is of exceptional type. When $G$ is of type $G_2$ we find all conjugacy classes of closed, connected, reductive subgr","authors_text":"David I. Stewart","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-11-22T15:04:25Z","title":"G-complete reducibility and the exceptional algebraic groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.4835","kind":"arxiv","version":1},"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":"sha256:21ea678fb23ac2ea03968472b8f46c3bfc882f31e6e94836aafcce7080ec0870","target":"record","created_at":"2026-05-18T04:35:03Z","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":"e01acf5467151d6f83f3ba7dc4deacc5a87e2bc9bf33f332ed959ab8ad56b9a5","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-11-22T15:04:25Z","title_canon_sha256":"99867d7684e00ee86e91c8b8b69aa9505cd036d9714005920c223dbf6aa3b2c3"},"schema_version":"1.0","source":{"id":"1011.4835","kind":"arxiv","version":1}},"canonical_sha256":"b279e3ab0a0ad6b10f9b1ebf52b6308b63b93455c725939e5655d8eeedbb6e78","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b279e3ab0a0ad6b10f9b1ebf52b6308b63b93455c725939e5655d8eeedbb6e78","first_computed_at":"2026-05-18T04:35:03.916185Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:35:03.916185Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tTJzYCHwklNOryIZvzdu6I4OqmCE1eKSBp76EG4ec+uIxK4h6iJ3W0502Khp83PxndxEwydPHCz8ffDZUh1BCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:35:03.916602Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.4835","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:21ea678fb23ac2ea03968472b8f46c3bfc882f31e6e94836aafcce7080ec0870","sha256:8f9e8a8fb6cb7e8b811fe83fa4c3187a20e2254f0679c7fecefd2e3fdfad11fe"],"state_sha256":"a5be1fdceed410f57a0c2b3949d1f35641f7b28777da830283c4f060d7dea8ff"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zg5oGILb/NZFUlkYKApuUBv18yNxl7HVyjNH14s7ct3m7XBUZeEDZ/AjaKCAz5iacmnckGVrUdB/HhX5f0LNAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T09:48:21.004958Z","bundle_sha256":"b66ed495004c35a162794c4b3d9022a33cc05fd69f5907606694382bb74560fb"}}