{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:7ZWJDPQGW3O6SLNIVWY7JAYL4G","short_pith_number":"pith:7ZWJDPQG","canonical_record":{"source":{"id":"1905.10151","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-05-24T11:17:31Z","cross_cats_sorted":[],"title_canon_sha256":"3a179c30160e381f1c9c561729b4e78e6db2a3975c8c2a0445aa2bab3f7a3829","abstract_canon_sha256":"c257beeef59a4d96e3cbed731093390466eefa9d1d7543724ae6e251a770ada4"},"schema_version":"1.0"},"canonical_sha256":"fe6c91be06b6dde92da8adb1f4830be1afa5e74dd96026a745356e72c4d4dec2","source":{"kind":"arxiv","id":"1905.10151","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1905.10151","created_at":"2026-07-05T00:45:43Z"},{"alias_kind":"arxiv_version","alias_value":"1905.10151v2","created_at":"2026-07-05T00:45:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.10151","created_at":"2026-07-05T00:45:43Z"},{"alias_kind":"pith_short_12","alias_value":"7ZWJDPQGW3O6","created_at":"2026-07-05T00:45:43Z"},{"alias_kind":"pith_short_16","alias_value":"7ZWJDPQGW3O6SLNI","created_at":"2026-07-05T00:45:43Z"},{"alias_kind":"pith_short_8","alias_value":"7ZWJDPQG","created_at":"2026-07-05T00:45:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:7ZWJDPQGW3O6SLNIVWY7JAYL4G","target":"record","payload":{"canonical_record":{"source":{"id":"1905.10151","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-05-24T11:17:31Z","cross_cats_sorted":[],"title_canon_sha256":"3a179c30160e381f1c9c561729b4e78e6db2a3975c8c2a0445aa2bab3f7a3829","abstract_canon_sha256":"c257beeef59a4d96e3cbed731093390466eefa9d1d7543724ae6e251a770ada4"},"schema_version":"1.0"},"canonical_sha256":"fe6c91be06b6dde92da8adb1f4830be1afa5e74dd96026a745356e72c4d4dec2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:45:43.292712Z","signature_b64":"2sySa+xxXtHSzza/f9Qb/INIKmWC1l9BVNhPPKVmVJN2RJBp78Yh3MmBG3IpE8/B4DiQk1yB3KJd8R/tLFEnCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe6c91be06b6dde92da8adb1f4830be1afa5e74dd96026a745356e72c4d4dec2","last_reissued_at":"2026-07-05T00:45:43.292270Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:45:43.292270Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1905.10151","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:45:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vkbmSPdh+HVaBkHZSZovBB3NmOFMJejKtslmEWejHTqhzYtJermjcAQmrZj+vYi6pZZ0rnWFHp3/0xUdhTgXDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T16:22:48.468515Z"},"content_sha256":"7004702c7b90688292de38163b64bf1af45a774cc1dba90aeb49b9d2b67842f4","schema_version":"1.0","event_id":"sha256:7004702c7b90688292de38163b64bf1af45a774cc1dba90aeb49b9d2b67842f4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:7ZWJDPQGW3O6SLNIVWY7JAYL4G","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Vector Bundles on Flag varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Rong Du, Xinyi Fang, Yun Gao","submitted_at":"2019-05-24T11:17:31Z","abstract_excerpt":"We study vector bundles on flag varieties over an algebraically closed field $k$. In the first part, we suppose $G=G_k(d,n)$ $(2\\le d\\leq n-d)$ to be the Grassmannian manifold parameterizing linear subspaces of dimension $d$ in $k^n$, where $k$ is an algebraically closed field of characteristic $p>0$. Let $E$ be a uniform vector bundle over $G$ of rank $r\\le d$. We show that $E$ is either a direct sum of line bundles or a twist of a pull back of the universal bundle $H_d$ or its dual $H_d^{\\vee}$ by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.10151","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1905.10151/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":"2026-07-05T00:45:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cZkUD1E5/LhmVT9bWjxEOBZwC2lBoqz5pcSX9yMEHKm76G/w4mA798OdtPxY1bpl+yoo8AorWWokpRNgthX+Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T16:22:48.468984Z"},"content_sha256":"9eedd4b3ae2aa46287f6e5458686326f376f5b58b0740984ed55ab0c32f2a675","schema_version":"1.0","event_id":"sha256:9eedd4b3ae2aa46287f6e5458686326f376f5b58b0740984ed55ab0c32f2a675"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7ZWJDPQGW3O6SLNIVWY7JAYL4G/bundle.json","state_url":"https://pith.science/pith/7ZWJDPQGW3O6SLNIVWY7JAYL4G/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7ZWJDPQGW3O6SLNIVWY7JAYL4G/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":"2026-08-04T16:22:48Z","links":{"resolver":"https://pith.science/pith/7ZWJDPQGW3O6SLNIVWY7JAYL4G","bundle":"https://pith.science/pith/7ZWJDPQGW3O6SLNIVWY7JAYL4G/bundle.json","state":"https://pith.science/pith/7ZWJDPQGW3O6SLNIVWY7JAYL4G/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7ZWJDPQGW3O6SLNIVWY7JAYL4G/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:7ZWJDPQGW3O6SLNIVWY7JAYL4G","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":"c257beeef59a4d96e3cbed731093390466eefa9d1d7543724ae6e251a770ada4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-05-24T11:17:31Z","title_canon_sha256":"3a179c30160e381f1c9c561729b4e78e6db2a3975c8c2a0445aa2bab3f7a3829"},"schema_version":"1.0","source":{"id":"1905.10151","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1905.10151","created_at":"2026-07-05T00:45:43Z"},{"alias_kind":"arxiv_version","alias_value":"1905.10151v2","created_at":"2026-07-05T00:45:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.10151","created_at":"2026-07-05T00:45:43Z"},{"alias_kind":"pith_short_12","alias_value":"7ZWJDPQGW3O6","created_at":"2026-07-05T00:45:43Z"},{"alias_kind":"pith_short_16","alias_value":"7ZWJDPQGW3O6SLNI","created_at":"2026-07-05T00:45:43Z"},{"alias_kind":"pith_short_8","alias_value":"7ZWJDPQG","created_at":"2026-07-05T00:45:43Z"}],"graph_snapshots":[{"event_id":"sha256:9eedd4b3ae2aa46287f6e5458686326f376f5b58b0740984ed55ab0c32f2a675","target":"graph","created_at":"2026-07-05T00:45:43Z","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/1905.10151/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study vector bundles on flag varieties over an algebraically closed field $k$. In the first part, we suppose $G=G_k(d,n)$ $(2\\le d\\leq n-d)$ to be the Grassmannian manifold parameterizing linear subspaces of dimension $d$ in $k^n$, where $k$ is an algebraically closed field of characteristic $p>0$. Let $E$ be a uniform vector bundle over $G$ of rank $r\\le d$. We show that $E$ is either a direct sum of line bundles or a twist of a pull back of the universal bundle $H_d$ or its dual $H_d^{\\vee}$ by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles","authors_text":"Rong Du, Xinyi Fang, Yun Gao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-05-24T11:17:31Z","title":"Vector Bundles on Flag varieties"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.10151","kind":"arxiv","version":2},"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":"sha256:7004702c7b90688292de38163b64bf1af45a774cc1dba90aeb49b9d2b67842f4","target":"record","created_at":"2026-07-05T00:45:43Z","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":"c257beeef59a4d96e3cbed731093390466eefa9d1d7543724ae6e251a770ada4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-05-24T11:17:31Z","title_canon_sha256":"3a179c30160e381f1c9c561729b4e78e6db2a3975c8c2a0445aa2bab3f7a3829"},"schema_version":"1.0","source":{"id":"1905.10151","kind":"arxiv","version":2}},"canonical_sha256":"fe6c91be06b6dde92da8adb1f4830be1afa5e74dd96026a745356e72c4d4dec2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fe6c91be06b6dde92da8adb1f4830be1afa5e74dd96026a745356e72c4d4dec2","first_computed_at":"2026-07-05T00:45:43.292270Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:45:43.292270Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2sySa+xxXtHSzza/f9Qb/INIKmWC1l9BVNhPPKVmVJN2RJBp78Yh3MmBG3IpE8/B4DiQk1yB3KJd8R/tLFEnCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T00:45:43.292712Z","signed_message":"canonical_sha256_bytes"},"source_id":"1905.10151","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7004702c7b90688292de38163b64bf1af45a774cc1dba90aeb49b9d2b67842f4","sha256:9eedd4b3ae2aa46287f6e5458686326f376f5b58b0740984ed55ab0c32f2a675"],"state_sha256":"81cc9bf49b8a276866d5e57cbf67d86532efa676a83df7acb4d2a41397b8bf37"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pS9BBxPNIfM4ie1El8rZ8Ian0Ycgxxvy+JUe4W+QwSQ7VXsiS4CSD/7Vc3pTsUzbEIjJQr+dHboAITvunHbkCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T16:22:48.489976Z","bundle_sha256":"f6b490ca99641d6f2a73c3c9d029b30085102cc7ef615db33ccad45fd6abbfac"}}