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Suppose that $k$ is an integer not greater than $n-2$ and consider the relation ${\\mathfrak R}_{i,j}$, $0\\le i\\le j\\le k+1$ formed by all pairs $(X,Y)\\in {\\mathcal G}_{k}(\\Pi)\\times {\\mathcal G}_{k}(\\Pi)$ such that $\\dim_{p}(X^{\\perp}\\cap Y)=k-i$ and $\\dim_{p} (X\\cap Y)=k-j$ ($X^{\\perp}$ consists of all points of $\\Pi$ collinear to every point of $X$). We show that every bijective transformation of ${\\mathcal G}_"},"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":"1307.2316","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-07-09T01:28:04Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d3c279a15a53ca77912973f707fe9c0e59420a7bfb6c4043e8ba38663b648fd2","abstract_canon_sha256":"f73f1ef4641f0a0431691f484f11b2a593c27b35f2ab0887e4f86d13d98d4db7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:52.877219Z","signature_b64":"X4oLpgeNrjyJYT1fy6BVoRqegP2abTehYYTZ0axleCttrgPyqE4IMJPM3ATaUb8LeoAVRzUZ2jrlQ7RnTrXVCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"abc053b2310e8607dfc50bcc676b636ef985ce06d5aeb5546b49d3eee2b45d3c","last_reissued_at":"2026-05-18T03:18:52.876539Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:52.876539Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Transformations of polar Grassmannians preserving certain intersecting relations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AG","authors_text":"Kaishun Wang, Mark Pankov, Wen Liu","submitted_at":"2013-07-09T01:28:04Z","abstract_excerpt":"Let $\\Pi$ be a polar space of rank $n\\ge 3$. Denote by ${\\mathcal G}_{k}(\\Pi)$ the polar Grassmannian formed by singular subspaces of $\\Pi$ whose projective dimension is equal to $k$. Suppose that $k$ is an integer not greater than $n-2$ and consider the relation ${\\mathfrak R}_{i,j}$, $0\\le i\\le j\\le k+1$ formed by all pairs $(X,Y)\\in {\\mathcal G}_{k}(\\Pi)\\times {\\mathcal G}_{k}(\\Pi)$ such that $\\dim_{p}(X^{\\perp}\\cap Y)=k-i$ and $\\dim_{p} (X\\cap Y)=k-j$ ($X^{\\perp}$ consists of all points of $\\Pi$ collinear to every point of $X$). 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