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We prove that $H^d(Z,\\mathcal{L})$ is the simple $\\operatorname{GL}_{d+1}$-module corresponding to the partition $\\lambda_0 = (p-1+f,p-1,f+1)$. When $f= 0$, using the first author's description of $H^d(Z,\\mathcal{L})$ and Jantzen's sum formula, we obtain as a by-product that the sum of the monomial symmetric functions $m_\\lambda"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1908.08432","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2019-08-22T15:08:35Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0278903a1e9001f454f5f978fa365356887362b3f21dd7a7c03b5c817cca5c2c","abstract_canon_sha256":"129ce4c95010fd71dcf50b5423e1a27d6c1278e7d0a08b589ef88c38135be84d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:41:28.456507Z","signature_b64":"5qs5ZJIkPTyr7GOViP9+EckeeOlgy1tj7nSZv0+XOoaWQmKmYhzluTE4wkp3xrxZcOLz/3hTRmM3/L3UG1UFCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e649456169fa1bfb425d2edf1baf242d28e2a3bb9e51033f6bcfe305850b418b","last_reissued_at":"2026-07-05T01:41:28.455987Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:41:28.455987Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the cohomology of line bundles over certain flag schemes II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RT","authors_text":"Linyuan Liu, Patrick Polo","submitted_at":"2019-08-22T15:08:35Z","abstract_excerpt":"Over a field $K$ of characteristic $p$, let $Z$ be the incidence variety in $\\mathbb{P}^d \\times (\\mathbb{P}^d)^*$ and let $\\mathcal{L}$ be the restriction to $Z$ of the line bundle $\\mathcal{O}(-n-d) \\boxtimes \\mathcal{O}(n)$, where $n = p+f$ with $0 \\leq f \\leq p-2$. We prove that $H^d(Z,\\mathcal{L})$ is the simple $\\operatorname{GL}_{d+1}$-module corresponding to the partition $\\lambda_0 = (p-1+f,p-1,f+1)$. When $f= 0$, using the first author's description of $H^d(Z,\\mathcal{L})$ and Jantzen's sum formula, we obtain as a by-product that the sum of the monomial symmetric functions $m_\\lambda"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.08432","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/1908.08432/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1908.08432","created_at":"2026-07-05T01:41:28.456051+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.08432v4","created_at":"2026-07-05T01:41:28.456051+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.08432","created_at":"2026-07-05T01:41:28.456051+00:00"},{"alias_kind":"pith_short_12","alias_value":"4ZEUKYLJ7IN7","created_at":"2026-07-05T01:41:28.456051+00:00"},{"alias_kind":"pith_short_16","alias_value":"4ZEUKYLJ7IN7WQS5","created_at":"2026-07-05T01:41:28.456051+00:00"},{"alias_kind":"pith_short_8","alias_value":"4ZEUKYLJ","created_at":"2026-07-05T01:41:28.456051+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4ZEUKYLJ7IN7WQS5F3PRXLZEFU","json":"https://pith.science/pith/4ZEUKYLJ7IN7WQS5F3PRXLZEFU.json","graph_json":"https://pith.science/api/pith-number/4ZEUKYLJ7IN7WQS5F3PRXLZEFU/graph.json","events_json":"https://pith.science/api/pith-number/4ZEUKYLJ7IN7WQS5F3PRXLZEFU/events.json","paper":"https://pith.science/paper/4ZEUKYLJ"},"agent_actions":{"view_html":"https://pith.science/pith/4ZEUKYLJ7IN7WQS5F3PRXLZEFU","download_json":"https://pith.science/pith/4ZEUKYLJ7IN7WQS5F3PRXLZEFU.json","view_paper":"https://pith.science/paper/4ZEUKYLJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.08432&json=true","fetch_graph":"https://pith.science/api/pith-number/4ZEUKYLJ7IN7WQS5F3PRXLZEFU/graph.json","fetch_events":"https://pith.science/api/pith-number/4ZEUKYLJ7IN7WQS5F3PRXLZEFU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4ZEUKYLJ7IN7WQS5F3PRXLZEFU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4ZEUKYLJ7IN7WQS5F3PRXLZEFU/action/storage_attestation","attest_author":"https://pith.science/pith/4ZEUKYLJ7IN7WQS5F3PRXLZEFU/action/author_attestation","sign_citation":"https://pith.science/pith/4ZEUKYLJ7IN7WQS5F3PRXLZEFU/action/citation_signature","submit_replication":"https://pith.science/pith/4ZEUKYLJ7IN7WQS5F3PRXLZEFU/action/replication_record"}},"created_at":"2026-07-05T01:41:28.456051+00:00","updated_at":"2026-07-05T01:41:28.456051+00:00"}