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To each basis involution $w$ in the Weyl group $W$ of $G$ one can assign the associated $B$-orbit $\\Omega_w$. We prove that, given basis involutions $\\sigma$, $\\tau$ in $W$, if the orbit $\\Omega_{\\sigma}$ is contained in the closure of the orbit $\\Omega_{\\tau}$ then $\\sigma$ is less than or equal to $\\tau$ with respect to the Bruhat order on $W$. For a basis involution $w$, we also compute the dimension of $\\O"},"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":"1810.02703","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-10-04T09:20:47Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"2b801b39c1238ccf5bbb7a2c490ea2ab41480765b0001e26af1482eddf22a477","abstract_canon_sha256":"329604f7e26422f9975f8026a4b1c41def90c0700ecc9e22aecf49159bb17d3e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:00.906789Z","signature_b64":"dVA+X2H1jzXmCmUklfhCgE49+BBjwwRL+JlPK9sGgi6F6jY/vnXNQkpU2BixWD0wWW8km1oV0675aRUvUev9Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"80dfe38d61132f8c486f6ceaa32c0b836b253c1956a1aee54d14865b86fd4cd9","last_reissued_at":"2026-05-18T00:04:00.906246Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:00.906246Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On involutions in the Weyl group and $B$-orbit closures in the orthogonal case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RT","authors_text":"Mikhail V. Ignatyev","submitted_at":"2018-10-04T09:20:47Z","abstract_excerpt":"We study coadjoint $B$-orbits on $\\mathfrak{n}^*$, where $B$ is a Borel subgroup of a complex orthogonal group $G$, and $\\mathfrak{n}$ is the Lie algebra of the unipotent radical of $B$. To each basis involution $w$ in the Weyl group $W$ of $G$ one can assign the associated $B$-orbit $\\Omega_w$. We prove that, given basis involutions $\\sigma$, $\\tau$ in $W$, if the orbit $\\Omega_{\\sigma}$ is contained in the closure of the orbit $\\Omega_{\\tau}$ then $\\sigma$ is less than or equal to $\\tau$ with respect to the Bruhat order on $W$. 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