{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:NU7AA7RCOXTKTFI3TFXLZZKO57","short_pith_number":"pith:NU7AA7RC","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"},"canonical_sha256":"6d3e007e2275e6a9951b996ebce54eefc8177b099fecdc3c0f8fe1c8a1eb6377","source":{"kind":"arxiv","id":"1101.2189","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.2189","created_at":"2026-05-18T03:10:37Z"},{"alias_kind":"arxiv_version","alias_value":"1101.2189v3","created_at":"2026-05-18T03:10:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.2189","created_at":"2026-05-18T03:10:37Z"},{"alias_kind":"pith_short_12","alias_value":"NU7AA7RCOXTK","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"NU7AA7RCOXTKTFI3","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"NU7AA7RC","created_at":"2026-05-18T12:26:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:NU7AA7RCOXTKTFI3TFXLZZKO57","target":"record","payload":{"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"},"canonical_sha256":"6d3e007e2275e6a9951b996ebce54eefc8177b099fecdc3c0f8fe1c8a1eb6377","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"},"source_kind":"arxiv","source_id":"1101.2189","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:10:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4uN2NQue2/s2EjCnzvJ/zUd0HZdvDPtPNFsnqGK4GQ8Q8RuLuId9z8tVOTeoAPTz0N6sflCUq3dkm3j/jtQBAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T08:15:29.425750Z"},"content_sha256":"788160349b3fecf080b728be9508aece30774363b14236af8696f6917f1a24bd","schema_version":"1.0","event_id":"sha256:788160349b3fecf080b728be9508aece30774363b14236af8696f6917f1a24bd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:NU7AA7RCOXTKTFI3TFXLZZKO57","target":"graph","payload":{"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. The answer is g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.2189","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:10:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"14rEyDXwxN4389GRbu29TdKgSJ8Es6tHOy1bxXY9UNEmQoK5x/2qlYHru6dda13fa7t8LX87OZ13tnRwUwAiDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T08:15:29.426091Z"},"content_sha256":"61b2f6faee340fa90315079fd889afe6b7ea953809e7162d98d8976a86c6b125","schema_version":"1.0","event_id":"sha256:61b2f6faee340fa90315079fd889afe6b7ea953809e7162d98d8976a86c6b125"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NU7AA7RCOXTKTFI3TFXLZZKO57/bundle.json","state_url":"https://pith.science/pith/NU7AA7RCOXTKTFI3TFXLZZKO57/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NU7AA7RCOXTKTFI3TFXLZZKO57/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-02T08:15:29Z","links":{"resolver":"https://pith.science/pith/NU7AA7RCOXTKTFI3TFXLZZKO57","bundle":"https://pith.science/pith/NU7AA7RCOXTKTFI3TFXLZZKO57/bundle.json","state":"https://pith.science/pith/NU7AA7RCOXTKTFI3TFXLZZKO57/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NU7AA7RCOXTKTFI3TFXLZZKO57/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:NU7AA7RCOXTKTFI3TFXLZZKO57","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e522a49f7aa700d1eb351282d3f73adfc41c77d27990c0270d76ec74f9525160","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-01-11T19:51:59Z","title_canon_sha256":"62a8413a027cce7591e51052ddc6cedcf078429d92b78185c5ded9d0cb930b6a"},"schema_version":"1.0","source":{"id":"1101.2189","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.2189","created_at":"2026-05-18T03:10:37Z"},{"alias_kind":"arxiv_version","alias_value":"1101.2189v3","created_at":"2026-05-18T03:10:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.2189","created_at":"2026-05-18T03:10:37Z"},{"alias_kind":"pith_short_12","alias_value":"NU7AA7RCOXTK","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"NU7AA7RCOXTKTFI3","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"NU7AA7RC","created_at":"2026-05-18T12:26:37Z"}],"graph_snapshots":[{"event_id":"sha256:61b2f6faee340fa90315079fd889afe6b7ea953809e7162d98d8976a86c6b125","target":"graph","created_at":"2026-05-18T03:10:37Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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. The answer is g","authors_text":"Mikhail V. Ignatyev","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-01-11T19:51:59Z","title":"Combinatorics of $B$-orbits and Bruhat--Chevalley order on involutions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.2189","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:788160349b3fecf080b728be9508aece30774363b14236af8696f6917f1a24bd","target":"record","created_at":"2026-05-18T03:10:37Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e522a49f7aa700d1eb351282d3f73adfc41c77d27990c0270d76ec74f9525160","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-01-11T19:51:59Z","title_canon_sha256":"62a8413a027cce7591e51052ddc6cedcf078429d92b78185c5ded9d0cb930b6a"},"schema_version":"1.0","source":{"id":"1101.2189","kind":"arxiv","version":3}},"canonical_sha256":"6d3e007e2275e6a9951b996ebce54eefc8177b099fecdc3c0f8fe1c8a1eb6377","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6d3e007e2275e6a9951b996ebce54eefc8177b099fecdc3c0f8fe1c8a1eb6377","first_computed_at":"2026-05-18T03:10:37.696324Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:10:37.696324Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BGuJODoydzgWjjBoGYphNha8yguHyDN7dxq8Ty+wosUFr6JZ/0vc+gQwbDqQnIFW5dpLJhnzMNW7Cordk2LGCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:10:37.696870Z","signed_message":"canonical_sha256_bytes"},"source_id":"1101.2189","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:788160349b3fecf080b728be9508aece30774363b14236af8696f6917f1a24bd","sha256:61b2f6faee340fa90315079fd889afe6b7ea953809e7162d98d8976a86c6b125"],"state_sha256":"af3ab319b2690c0c53702c2e6186397ac8e0a40ad1d8b66e68e1025c4e98d902"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RFNsG/2fdTC5zjoqAQCbEaXtE10psvISOAEa+5x0GLY6t8nAgcInGs5go9AuKCmwY3WbHp/0VXsyyaXGJy9jDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T08:15:29.427978Z","bundle_sha256":"9f0ac43367478973761e3923f10f37a24c59377db4769f412e468fa3b221e06d"}}