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Let $D$ be an orthogonal subset of $\\Phi$ and $\\Omega$ be a coadjoint orbit of $U$ associated with $D$. We construct a polarization of $\\mathfrak{u}$ at the canonical form on $\\Omega$. We also find the dimension of $\\Omega$ in terms of the Weyl group of $\\Phi$. As a corollary, we determine all possible dimensions of irreducible complex represenations"},"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":"0904.2841","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-04-18T16:46:47Z","cross_cats_sorted":[],"title_canon_sha256":"df213fe24df59b6b130d487d38a3bee31af6c8acf784c0917279b8b3912c6248","abstract_canon_sha256":"408e7de36d79ad1235307f66ae86f50b2c46ef99310bea2ff31899c77c4360c4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:10:37.853982Z","signature_b64":"sO1wAD+uypKxqu9UgV2qO1LzvOIuGjx4LhVmUNAL9ZsiW6PyJl2BQzaUWQMFlGtfb/E3QHZeSRBzVL2wvGdlCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"887ab8acb822a14c9e5304ffefd30cdcded8eef6b5dc046903ad0c82e10e6194","last_reissued_at":"2026-05-18T03:10:37.853465Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:10:37.853465Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Orthogonal subsets of classical root systems and coadjoint orbits of unipotent groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Mikhail V. Ignatyev","submitted_at":"2009-04-18T16:46:47Z","abstract_excerpt":"Let $\\Phi$ be a classical root system and $k$ be a field of sufficiently large characteristic. Let $G$ be the classical group over $k$ with the root system $\\Phi$, $U$ be its maximal unipotent subgroup and $\\mathfrak{u}$ be the Lie algebra of $U$. Let $D$ be an orthogonal subset of $\\Phi$ and $\\Omega$ be a coadjoint orbit of $U$ associated with $D$. We construct a polarization of $\\mathfrak{u}$ at the canonical form on $\\Omega$. We also find the dimension of $\\Omega$ in terms of the Weyl group of $\\Phi$. 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