{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:GOUPAQEP3J3PYLQFEC35QEBPQK","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":"10d913c60784a8459e5c3b9a7f5a153126f05ae4a5a0eeec07d405e676463d7e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-08-21T01:30:49Z","title_canon_sha256":"db22ef7970a8f0a9a4a0aff10ba36c5c4a22927b8b13d20221d489c69c561b7a"},"schema_version":"1.0","source":{"id":"1308.4464","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.4464","created_at":"2026-05-18T03:05:18Z"},{"alias_kind":"arxiv_version","alias_value":"1308.4464v3","created_at":"2026-05-18T03:05:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.4464","created_at":"2026-05-18T03:05:18Z"},{"alias_kind":"pith_short_12","alias_value":"GOUPAQEP3J3P","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"GOUPAQEP3J3PYLQF","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"GOUPAQEP","created_at":"2026-05-18T12:27:45Z"}],"graph_snapshots":[{"event_id":"sha256:9b6e359b6926dfaf5d541a6e68bc77ea6c8520c5ce8edcfe2c365021fa1adae8","target":"graph","created_at":"2026-05-18T03:05:18Z","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 $\\frb$ be a fixed Borel subalgebra of a finite-dimensional complex simple Lie algebra $\\frg$. The Shi bijection associates to every ad-nilpotent ideal $\\fri$ of $\\frb$ a region $V_{\\fri}$. In this paper, we show that $\\fri$ is abelian if and only if $V_{\\fri}\\cap 2A$ is empty, if and only if the volume of $V_{\\fri}\\cap 2A$ equals to that of $A$, where $A$ is the fundamental alcove of the affine Weyl group. For certain flag of abelian ideals, we record an ascending property of their associated regions. We also determine the maximal eigenvalue $m_{r-1}$ of the Casimir operator on $\\wedge^{r-","authors_text":"Chao-Ping Dong","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-08-21T01:30:49Z","title":"Remarks on the abelian ideals of a Borel subalgebra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.4464","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:beb6de921a256fb7b402e52bf7aa79b90d665da86de20ee9cfaa749373190804","target":"record","created_at":"2026-05-18T03:05:18Z","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":"10d913c60784a8459e5c3b9a7f5a153126f05ae4a5a0eeec07d405e676463d7e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-08-21T01:30:49Z","title_canon_sha256":"db22ef7970a8f0a9a4a0aff10ba36c5c4a22927b8b13d20221d489c69c561b7a"},"schema_version":"1.0","source":{"id":"1308.4464","kind":"arxiv","version":3}},"canonical_sha256":"33a8f0408fda76fc2e0520b7d8102f8281d235ab8c35a8151e5309b5e1a37c5e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"33a8f0408fda76fc2e0520b7d8102f8281d235ab8c35a8151e5309b5e1a37c5e","first_computed_at":"2026-05-18T03:05:18.161838Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:05:18.161838Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"K4202A5fuPKyID+nBeMlOJgO2ctFBK+U5dWd92faYXM+2Ir7ipGgu07WvrDF6F65QT/EZLJeNhUWT27FaJ7FDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:05:18.162379Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.4464","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:beb6de921a256fb7b402e52bf7aa79b90d665da86de20ee9cfaa749373190804","sha256:9b6e359b6926dfaf5d541a6e68bc77ea6c8520c5ce8edcfe2c365021fa1adae8"],"state_sha256":"4a77ec07f372ea42f46ef6e438819ad58098e3cfbcf5c8cc24c859321a4e56bf"}