{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:JJWAQN5YV7LOCPREFXW6JDAYLL","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":"8087ff53b4d4d0af2290790cdbc614d9603e2a3800e41ca33313a4b58a8ab5c4","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2026-04-28T03:43:53Z","title_canon_sha256":"2d94206f8cfb7715051514f5ded3d2265d96c522ebc73164080e4a53f5e86599"},"schema_version":"1.0","source":{"id":"2604.25185","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.25185","created_at":"2026-06-19T16:12:20Z"},{"alias_kind":"arxiv_version","alias_value":"2604.25185v2","created_at":"2026-06-19T16:12:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.25185","created_at":"2026-06-19T16:12:20Z"},{"alias_kind":"pith_short_12","alias_value":"JJWAQN5YV7LO","created_at":"2026-06-19T16:12:20Z"},{"alias_kind":"pith_short_16","alias_value":"JJWAQN5YV7LOCPRE","created_at":"2026-06-19T16:12:20Z"},{"alias_kind":"pith_short_8","alias_value":"JJWAQN5Y","created_at":"2026-06-19T16:12:20Z"}],"graph_snapshots":[{"event_id":"sha256:e25bbe32849ad04c69f4898376513966fb220e5aad0ce8fb2c70361a4b9bedcd","target":"graph","created_at":"2026-06-19T16:12:20Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"Each block Ω^{~S_2}_a of the category of (A_2, bar S_2)-Whittaker modules with finite-dimensional Whittaker vector spaces is equivalent to the finite-dimensional module category over the parabolic subalgebra bar S_2^{≥0}; all simple Whittaker bar S_2-modules with finite-dimensional Whittaker vector spaces are classified using gl_2-modules; and Ω^{bar S_2}_1 is equivalent to H_1-fmod."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The Whittaker vector spaces are finite-dimensional; this restriction is essential for the block decomposition, the equivalences to parabolic and H_1 modules, and the classification via gl_2 to hold as stated."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Blocks of Whittaker modules over bar S_2 with finite-dimensional Whittaker spaces are equivalent to finite-dimensional modules over a parabolic subalgebra, with simples classified via gl_2-modules and one block equivalent to H_1-fmod."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Each block of Whittaker modules over the Cartan-type Lie algebra bar S_2 with finite-dimensional Whittaker vector spaces is equivalent to the finite-dimensional modules over its parabolic subalgebra bar S_2 to the non-negative part."}],"snapshot_sha256":"95da618fb97507ea1f99f648f2ad1d95d711f572e1d8956f303ebe3ff8111ef1"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-21T05:37:50.593149Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-19T21:23:52.894546Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2604.25185/integrity.json","findings":[],"snapshot_sha256":"80491cd52432514d4662e430db53c76c5327d0f7425e29f8fcbff2101ac561bc","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Lie algebra $\\bar{S}_2$ of polynomial vector fields on $\\mathbb{C}^2$ with constant divergence is an important Cartan type Lie algebra. In this paper, we study Whittaker $\\bar{S}_2$-modules that are locally finite\n  over $\\text{span}\\{\\frac{\\partial}{\\partial t_1}, \\frac{\\partial}{\\partial t_2}\\}$. We first show that each block $\\Omega^{\\widetilde{S}_2}_{\\mathbf{a}}$ of the category of $(A_2, \\bar{S}_2)$-Whittaker modules with finite-dimensional Whittaker vector spaces is equivalent to the category of finite-dimensional modules over the parabolic subalgebra $\\bar{S}_2^{\\geq 0}$. Then we cl","authors_text":"Genqiang Liu, Xiaoyao Zheng, Yufang Zhao","cross_cats":[],"headline":"Each block of Whittaker modules over the Cartan-type Lie algebra bar S_2 with finite-dimensional Whittaker vector spaces is equivalent to the finite-dimensional modules over its parabolic subalgebra bar S_2 to the non-negative part.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2026-04-28T03:43:53Z","title":"The category of Whittaker modules over the Cartan Type Lie algebra $\\bar{S}_2$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.25185","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-07T14:40:02.406308Z","id":"f3efa1b9-ea4b-4841-aac6-e9c2033472f3","model_set":{"reader":"grok-4.3"},"one_line_summary":"Blocks of Whittaker modules over bar S_2 with finite-dimensional Whittaker spaces are equivalent to finite-dimensional modules over a parabolic subalgebra, with simples classified via gl_2-modules and one block equivalent to H_1-fmod.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Each block of Whittaker modules over the Cartan-type Lie algebra bar S_2 with finite-dimensional Whittaker vector spaces is equivalent to the finite-dimensional modules over its parabolic subalgebra bar S_2 to the non-negative part.","strongest_claim":"Each block Ω^{~S_2}_a of the category of (A_2, bar S_2)-Whittaker modules with finite-dimensional Whittaker vector spaces is equivalent to the finite-dimensional module category over the parabolic subalgebra bar S_2^{≥0}; all simple Whittaker bar S_2-modules with finite-dimensional Whittaker vector spaces are classified using gl_2-modules; and Ω^{bar S_2}_1 is equivalent to H_1-fmod.","weakest_assumption":"The Whittaker vector spaces are finite-dimensional; this restriction is essential for the block decomposition, the equivalences to parabolic and H_1 modules, and the classification via gl_2 to hold as stated."}},"verdict_id":"f3efa1b9-ea4b-4841-aac6-e9c2033472f3"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:370e2d31ac494e6d253c161151bb5051dc8b526d04e98d6883c5496b162dddfb","target":"record","created_at":"2026-06-19T16:12:20Z","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":"8087ff53b4d4d0af2290790cdbc614d9603e2a3800e41ca33313a4b58a8ab5c4","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2026-04-28T03:43:53Z","title_canon_sha256":"2d94206f8cfb7715051514f5ded3d2265d96c522ebc73164080e4a53f5e86599"},"schema_version":"1.0","source":{"id":"2604.25185","kind":"arxiv","version":2}},"canonical_sha256":"4a6c0837b8afd6e13e242dede48c185adc5e4c012698e051ce6bd7a9e2250b6b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4a6c0837b8afd6e13e242dede48c185adc5e4c012698e051ce6bd7a9e2250b6b","first_computed_at":"2026-06-19T16:12:20.385051Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-19T16:12:20.385051Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/ASCcVssrLTrhnuf2u6CXGM8xyw8XCdDW3+h9DuHZii9EGQUslChwLr1Cg3Zd/K9nTuexNVONYzapPMDJbTlDg==","signature_status":"signed_v1","signed_at":"2026-06-19T16:12:20.385511Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.25185","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:370e2d31ac494e6d253c161151bb5051dc8b526d04e98d6883c5496b162dddfb","sha256:e25bbe32849ad04c69f4898376513966fb220e5aad0ce8fb2c70361a4b9bedcd"],"state_sha256":"8f3143bcb6e28791398bd3a3f8e3efaf72c79794a8ccd44a56704a0e9a7b55d2"}