{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:PPXBURIFFTCFGAQTS5NLLD7RXC","short_pith_number":"pith:PPXBURIF","canonical_record":{"source":{"id":"1903.04683","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2019-03-12T01:19:31Z","cross_cats_sorted":[],"title_canon_sha256":"fd309a5de4673357c6e371296d4fade9b4b59a458f5b6793e0ee01d3d6274155","abstract_canon_sha256":"2efe48751eb98bd126d558a2ff0c5a4591378f4eac0cd3d01bdbecece584c8de"},"schema_version":"1.0"},"canonical_sha256":"7bee1a45052cc4530213975ab58ff1b8ab3ecaf7632e4788aac583468b66e03a","source":{"kind":"arxiv","id":"1903.04683","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.04683","created_at":"2026-07-05T01:16:08Z"},{"alias_kind":"arxiv_version","alias_value":"1903.04683v1","created_at":"2026-07-05T01:16:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.04683","created_at":"2026-07-05T01:16:08Z"},{"alias_kind":"pith_short_12","alias_value":"PPXBURIFFTCF","created_at":"2026-07-05T01:16:08Z"},{"alias_kind":"pith_short_16","alias_value":"PPXBURIFFTCFGAQT","created_at":"2026-07-05T01:16:08Z"},{"alias_kind":"pith_short_8","alias_value":"PPXBURIF","created_at":"2026-07-05T01:16:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:PPXBURIFFTCFGAQTS5NLLD7RXC","target":"record","payload":{"canonical_record":{"source":{"id":"1903.04683","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2019-03-12T01:19:31Z","cross_cats_sorted":[],"title_canon_sha256":"fd309a5de4673357c6e371296d4fade9b4b59a458f5b6793e0ee01d3d6274155","abstract_canon_sha256":"2efe48751eb98bd126d558a2ff0c5a4591378f4eac0cd3d01bdbecece584c8de"},"schema_version":"1.0"},"canonical_sha256":"7bee1a45052cc4530213975ab58ff1b8ab3ecaf7632e4788aac583468b66e03a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:16:08.247506Z","signature_b64":"9eDyuq0Tr+4MHsHVX2Z05aH3rerIFdJlOkqeJdW5OolBdV5AFwr7Qg3YaSWKstVph8OM9twut7i+pXwkl42gDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7bee1a45052cc4530213975ab58ff1b8ab3ecaf7632e4788aac583468b66e03a","last_reissued_at":"2026-07-05T01:16:08.247042Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:16:08.247042Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1903.04683","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":"2026-07-05T01:16:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U6QhlR62ME9lRBYWaiMn7/7HfzZ8O5AtZ5o+xo9KJdRq//QcJG6gW/hYXS7RLT1Qll8ciskHwPKt+Fzw6XmDDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-15T08:42:41.105571Z"},"content_sha256":"94de677edfa04ce1165814bdc221e54453e221c70599f32098f570bf39454e55","schema_version":"1.0","event_id":"sha256:94de677edfa04ce1165814bdc221e54453e221c70599f32098f570bf39454e55"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:PPXBURIFFTCFGAQTS5NLLD7RXC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Odd singular vector formula for general linear Lie superalgebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Jie Liu, Li Luo, Weiqiang Wang","submitted_at":"2019-03-12T01:19:31Z","abstract_excerpt":"We establish a closed formula for a singular vector of weight $\\lambda-\\beta$ in the Verma module of highest weight $\\lambda$ for Lie superalgebra $\\mathfrak{gl}(m|n)$ when $\\lambda$ is atypical with respect to an odd positive root $\\beta$. It is further shown that this vector is unique up to a scalar multiple, and it descends to a singular vector, again unique up to a scalar multiple, in the corresponding Kac module when both $\\lambda$ and $\\lambda-\\beta$ are dominant integral."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.04683","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1903.04683/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":"2026-07-05T01:16:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hr8yhA7Pe5MD3cZhRauyxIorYXAwuQ82v15uQQF+anWYm24pJFlJ02+Oy5qZhm0rVstcv+MLCsJUL/OtsFzVAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-15T08:42:41.105948Z"},"content_sha256":"9c530db01770823f1e11b33acb95a6c1fdd1c7194ad8e904931ecf6b53d0fbc7","schema_version":"1.0","event_id":"sha256:9c530db01770823f1e11b33acb95a6c1fdd1c7194ad8e904931ecf6b53d0fbc7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PPXBURIFFTCFGAQTS5NLLD7RXC/bundle.json","state_url":"https://pith.science/pith/PPXBURIFFTCFGAQTS5NLLD7RXC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PPXBURIFFTCFGAQTS5NLLD7RXC/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":"2026-07-15T08:42:41Z","links":{"resolver":"https://pith.science/pith/PPXBURIFFTCFGAQTS5NLLD7RXC","bundle":"https://pith.science/pith/PPXBURIFFTCFGAQTS5NLLD7RXC/bundle.json","state":"https://pith.science/pith/PPXBURIFFTCFGAQTS5NLLD7RXC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PPXBURIFFTCFGAQTS5NLLD7RXC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:PPXBURIFFTCFGAQTS5NLLD7RXC","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":"2efe48751eb98bd126d558a2ff0c5a4591378f4eac0cd3d01bdbecece584c8de","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2019-03-12T01:19:31Z","title_canon_sha256":"fd309a5de4673357c6e371296d4fade9b4b59a458f5b6793e0ee01d3d6274155"},"schema_version":"1.0","source":{"id":"1903.04683","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.04683","created_at":"2026-07-05T01:16:08Z"},{"alias_kind":"arxiv_version","alias_value":"1903.04683v1","created_at":"2026-07-05T01:16:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.04683","created_at":"2026-07-05T01:16:08Z"},{"alias_kind":"pith_short_12","alias_value":"PPXBURIFFTCF","created_at":"2026-07-05T01:16:08Z"},{"alias_kind":"pith_short_16","alias_value":"PPXBURIFFTCFGAQT","created_at":"2026-07-05T01:16:08Z"},{"alias_kind":"pith_short_8","alias_value":"PPXBURIF","created_at":"2026-07-05T01:16:08Z"}],"graph_snapshots":[{"event_id":"sha256:9c530db01770823f1e11b33acb95a6c1fdd1c7194ad8e904931ecf6b53d0fbc7","target":"graph","created_at":"2026-07-05T01:16:08Z","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/1903.04683/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We establish a closed formula for a singular vector of weight $\\lambda-\\beta$ in the Verma module of highest weight $\\lambda$ for Lie superalgebra $\\mathfrak{gl}(m|n)$ when $\\lambda$ is atypical with respect to an odd positive root $\\beta$. It is further shown that this vector is unique up to a scalar multiple, and it descends to a singular vector, again unique up to a scalar multiple, in the corresponding Kac module when both $\\lambda$ and $\\lambda-\\beta$ are dominant integral.","authors_text":"Jie Liu, Li Luo, Weiqiang Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2019-03-12T01:19:31Z","title":"Odd singular vector formula for general linear Lie superalgebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.04683","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":"sha256:94de677edfa04ce1165814bdc221e54453e221c70599f32098f570bf39454e55","target":"record","created_at":"2026-07-05T01:16:08Z","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":"2efe48751eb98bd126d558a2ff0c5a4591378f4eac0cd3d01bdbecece584c8de","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2019-03-12T01:19:31Z","title_canon_sha256":"fd309a5de4673357c6e371296d4fade9b4b59a458f5b6793e0ee01d3d6274155"},"schema_version":"1.0","source":{"id":"1903.04683","kind":"arxiv","version":1}},"canonical_sha256":"7bee1a45052cc4530213975ab58ff1b8ab3ecaf7632e4788aac583468b66e03a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7bee1a45052cc4530213975ab58ff1b8ab3ecaf7632e4788aac583468b66e03a","first_computed_at":"2026-07-05T01:16:08.247042Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:16:08.247042Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9eDyuq0Tr+4MHsHVX2Z05aH3rerIFdJlOkqeJdW5OolBdV5AFwr7Qg3YaSWKstVph8OM9twut7i+pXwkl42gDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T01:16:08.247506Z","signed_message":"canonical_sha256_bytes"},"source_id":"1903.04683","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:94de677edfa04ce1165814bdc221e54453e221c70599f32098f570bf39454e55","sha256:9c530db01770823f1e11b33acb95a6c1fdd1c7194ad8e904931ecf6b53d0fbc7"],"state_sha256":"0d1e790e11dba6058afddbebea8fbe035fece7d110f9eb888659ffaa6918c103"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PMNYdRulG9oCASq6Kq7ul46aLZReyw1Q7dyoenjMe7mKWScfUxzkC9thflyIIFHQP03wEPDazn5/vKomGeMmAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-15T08:42:41.108088Z","bundle_sha256":"bc9f967fd9e47665e977009ebad002fad62ceca80ea5d57707955c0859f962ee"}}