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We will calculate the cohomology modules of line bundles over this flag scheme. We will prove that the only non-trivial ones are isomorphic to the kernel or the cokernel of certain matrices with multinomial coefficients."},"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":"1908.08438","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2019-08-22T15:14:45Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e6637b4eda161e48d5710492eeb75fd40b733fd282fe2af94f272fc0f77f93d5","abstract_canon_sha256":"8cec5f5fffeddd246a4d325c757ef738515e68c0d19c2b71fa7fb8e406946267"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:41:28.472408Z","signature_b64":"0Ehw1HG+iRNWfZR2twUgoZbSsJky5VMPH44gBA6qIVPoCM6n8XN0U8CHuV+bTGDcGLmf134RmWdgrUo9wnw4BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bff7f19eed10023c7bfadcd5c512558ebadf714cea9fb8b6b288016c215ef6d1","last_reissued_at":"2026-07-05T01:41:28.471979Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:41:28.471979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the cohomology of line bundles over certain flag schemes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RT","authors_text":"Linyuan Liu","submitted_at":"2019-08-22T15:14:45Z","abstract_excerpt":"Let $G$ be the group scheme $\\operatorname{SL}_{d+1}$ over $\\mathbb{Z}$ and let $Q$ be the parabolic subgroup scheme corresponding to the simple roots $\\alpha_{2},\\cdots,\\alpha_{d-1}$. Then $G/Q$ is the $\\mathbb{Z} $-scheme of partial flags $\\{D_{1}\\subset H_{d}\\subset V\\}$. We will calculate the cohomology modules of line bundles over this flag scheme. 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