{"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"}