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The group $B$ acts on $\\mathfrak{u}^*$ by $(g.f)(x)=f(gxg^{-1})$, $g\\in B$, $f\\in\\mathfrak{u}^*$, $x\\in\\mathfrak{u}$.\n  To each involution $\\sigma$ in $S_n$, the symmetric group on $n$ letters, one can assign the $B$-orbit $\\Omega_{\\sigma}\\in\\mathfrak{u}^*$. We present a combinatorial description of the partial order on the set of involutions induced by the orbit closures. The answer is 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":"1101.2189","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-01-11T19:51:59Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"62a8413a027cce7591e51052ddc6cedcf078429d92b78185c5ded9d0cb930b6a","abstract_canon_sha256":"e522a49f7aa700d1eb351282d3f73adfc41c77d27990c0270d76ec74f9525160"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:10:37.696870Z","signature_b64":"BGuJODoydzgWjjBoGYphNha8yguHyDN7dxq8Ty+wosUFr6JZ/0vc+gQwbDqQnIFW5dpLJhnzMNW7Cordk2LGCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6d3e007e2275e6a9951b996ebce54eefc8177b099fecdc3c0f8fe1c8a1eb6377","last_reissued_at":"2026-05-18T03:10:37.696324Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:10:37.696324Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Combinatorics of $B$-orbits and Bruhat--Chevalley order on involutions","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":"2011-01-11T19:51:59Z","abstract_excerpt":"Let $B$ be the group of invertible upper-triangular complex $n\\times n$ matrices, $\\mathfrak{u}$ the space of upper-triangular complex matrices with zeroes on the diagonal and $\\mathfrak{u}^*$ its dual space. The group $B$ acts on $\\mathfrak{u}^*$ by $(g.f)(x)=f(gxg^{-1})$, $g\\in B$, $f\\in\\mathfrak{u}^*$, $x\\in\\mathfrak{u}$.\n  To each involution $\\sigma$ in $S_n$, the symmetric group on $n$ letters, one can assign the $B$-orbit $\\Omega_{\\sigma}\\in\\mathfrak{u}^*$. We present a combinatorial description of the partial order on the set of involutions induced by the orbit closures. 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