{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:FBPQS5TX36NMEW5SREXWCGC7SL","short_pith_number":"pith:FBPQS5TX","canonical_record":{"source":{"id":"1209.1289","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-06T14:16:42Z","cross_cats_sorted":[],"title_canon_sha256":"4c7c42741661b0fe31c26c984f7a036ad509b3e16d653c86948c146a7aa61362","abstract_canon_sha256":"59408f0af02fb7b480e8142040beb0e4863630f712dc8043b82c01c339908961"},"schema_version":"1.0"},"canonical_sha256":"285f097677df9ac25bb2892f61185f92d74a8a763b37f8f3144e71b3f91d6f66","source":{"kind":"arxiv","id":"1209.1289","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.1289","created_at":"2026-05-18T03:46:06Z"},{"alias_kind":"arxiv_version","alias_value":"1209.1289v1","created_at":"2026-05-18T03:46:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.1289","created_at":"2026-05-18T03:46:06Z"},{"alias_kind":"pith_short_12","alias_value":"FBPQS5TX36NM","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"FBPQS5TX36NMEW5S","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"FBPQS5TX","created_at":"2026-05-18T12:27:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:FBPQS5TX36NMEW5SREXWCGC7SL","target":"record","payload":{"canonical_record":{"source":{"id":"1209.1289","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-06T14:16:42Z","cross_cats_sorted":[],"title_canon_sha256":"4c7c42741661b0fe31c26c984f7a036ad509b3e16d653c86948c146a7aa61362","abstract_canon_sha256":"59408f0af02fb7b480e8142040beb0e4863630f712dc8043b82c01c339908961"},"schema_version":"1.0"},"canonical_sha256":"285f097677df9ac25bb2892f61185f92d74a8a763b37f8f3144e71b3f91d6f66","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:46:06.014428Z","signature_b64":"YwCgxJOgFqA62CUcKD1lkpGvBDrE9Q3JP98Iu1dr3SKnR/yMJhm66DWkiQf//31kkpXZfHSZnV46Eg6cXCghAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"285f097677df9ac25bb2892f61185f92d74a8a763b37f8f3144e71b3f91d6f66","last_reissued_at":"2026-05-18T03:46:06.013925Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:46:06.013925Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.1289","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:46:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ea9uYgMo62GdKoe5/2b6pHMQXNLkQcIt01/9b4HIDw8fJo922xbPIcIwhbZ1OL/9xlWzaJnMV40No0fyXmqDCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T23:57:25.613195Z"},"content_sha256":"b619c8c7aaa86f59a1c68a2a516452c49e7db50c9029773a16ccb141192efdc8","schema_version":"1.0","event_id":"sha256:b619c8c7aaa86f59a1c68a2a516452c49e7db50c9029773a16ccb141192efdc8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:FBPQS5TX36NMEW5SREXWCGC7SL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On commuting varieties of nilradicals of Borel subalgebras of reductive Lie algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Gerhard Roehrle, Simon Goodwin","submitted_at":"2012-09-06T14:16:42Z","abstract_excerpt":"Let $G$ be a connected reductive algebraic group defined over an algebraically closed field $\\mathbbm k$ of characteristic zero. We consider the commuting variety $\\mathcal C(\\mathfrak u)$ of the nilradical $\\mathfrak u$ of the Lie algebra $\\mathfrak b$ of a Borel subgroup $B$ of $G$. In case $B$ acts on $\\mathfrak u$ with only a finite number of orbits, we verify that $\\mathcal C(\\mathfrak u)$ is equidimensional and that the irreducible components are in correspondence with the {\\em distinguished} $B$-orbits in $\\mathfrak u$. We observe that in general $\\mathcal C(\\mathfrak u)$ is not equidim"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.1289","kind":"arxiv","version":1},"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":"2026-05-18T03:46:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kiIiWKFLTraxpFlEcWyL5s7TWs2P4K5jpfr5dnMDhshz0v04QylxpfdILQOX5PrYHhl+Snp3jKIeo9KSvM0ABA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T23:57:25.613574Z"},"content_sha256":"249f3b3e57587d51c7df737b57e4561db9a2f762d2bc866e2cada835e8d2430d","schema_version":"1.0","event_id":"sha256:249f3b3e57587d51c7df737b57e4561db9a2f762d2bc866e2cada835e8d2430d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FBPQS5TX36NMEW5SREXWCGC7SL/bundle.json","state_url":"https://pith.science/pith/FBPQS5TX36NMEW5SREXWCGC7SL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FBPQS5TX36NMEW5SREXWCGC7SL/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":"2026-05-27T23:57:25Z","links":{"resolver":"https://pith.science/pith/FBPQS5TX36NMEW5SREXWCGC7SL","bundle":"https://pith.science/pith/FBPQS5TX36NMEW5SREXWCGC7SL/bundle.json","state":"https://pith.science/pith/FBPQS5TX36NMEW5SREXWCGC7SL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FBPQS5TX36NMEW5SREXWCGC7SL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:FBPQS5TX36NMEW5SREXWCGC7SL","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":"59408f0af02fb7b480e8142040beb0e4863630f712dc8043b82c01c339908961","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-06T14:16:42Z","title_canon_sha256":"4c7c42741661b0fe31c26c984f7a036ad509b3e16d653c86948c146a7aa61362"},"schema_version":"1.0","source":{"id":"1209.1289","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.1289","created_at":"2026-05-18T03:46:06Z"},{"alias_kind":"arxiv_version","alias_value":"1209.1289v1","created_at":"2026-05-18T03:46:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.1289","created_at":"2026-05-18T03:46:06Z"},{"alias_kind":"pith_short_12","alias_value":"FBPQS5TX36NM","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"FBPQS5TX36NMEW5S","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"FBPQS5TX","created_at":"2026-05-18T12:27:04Z"}],"graph_snapshots":[{"event_id":"sha256:249f3b3e57587d51c7df737b57e4561db9a2f762d2bc866e2cada835e8d2430d","target":"graph","created_at":"2026-05-18T03:46:06Z","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 $G$ be a connected reductive algebraic group defined over an algebraically closed field $\\mathbbm k$ of characteristic zero. We consider the commuting variety $\\mathcal C(\\mathfrak u)$ of the nilradical $\\mathfrak u$ of the Lie algebra $\\mathfrak b$ of a Borel subgroup $B$ of $G$. In case $B$ acts on $\\mathfrak u$ with only a finite number of orbits, we verify that $\\mathcal C(\\mathfrak u)$ is equidimensional and that the irreducible components are in correspondence with the {\\em distinguished} $B$-orbits in $\\mathfrak u$. We observe that in general $\\mathcal C(\\mathfrak u)$ is not equidim","authors_text":"Gerhard Roehrle, Simon Goodwin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-06T14:16:42Z","title":"On commuting varieties of nilradicals of Borel subalgebras of reductive Lie algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.1289","kind":"arxiv","version":1},"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:b619c8c7aaa86f59a1c68a2a516452c49e7db50c9029773a16ccb141192efdc8","target":"record","created_at":"2026-05-18T03:46:06Z","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":"59408f0af02fb7b480e8142040beb0e4863630f712dc8043b82c01c339908961","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-06T14:16:42Z","title_canon_sha256":"4c7c42741661b0fe31c26c984f7a036ad509b3e16d653c86948c146a7aa61362"},"schema_version":"1.0","source":{"id":"1209.1289","kind":"arxiv","version":1}},"canonical_sha256":"285f097677df9ac25bb2892f61185f92d74a8a763b37f8f3144e71b3f91d6f66","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"285f097677df9ac25bb2892f61185f92d74a8a763b37f8f3144e71b3f91d6f66","first_computed_at":"2026-05-18T03:46:06.013925Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:46:06.013925Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YwCgxJOgFqA62CUcKD1lkpGvBDrE9Q3JP98Iu1dr3SKnR/yMJhm66DWkiQf//31kkpXZfHSZnV46Eg6cXCghAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:46:06.014428Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.1289","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b619c8c7aaa86f59a1c68a2a516452c49e7db50c9029773a16ccb141192efdc8","sha256:249f3b3e57587d51c7df737b57e4561db9a2f762d2bc866e2cada835e8d2430d"],"state_sha256":"19a8bf8f0d65ce15b12d22cf4e7b2e07858406d93fccab0c4c3de9f26d7cf61d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X6yTVHMdlPs6lY0GY38jFeRHkFBD2e+vW+BUojnAj+/Ovr+s1h3LMW5J/VqroU5dx61GG7RijOwaQDr6Juk1AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T23:57:25.615821Z","bundle_sha256":"4bd39aca3d7796a045f25f848a60aa0be0191e550565bd9b2aae15d3afe5c6e2"}}