{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:MPXW2HSME7PW7467QZZCNMDC4R","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":"6e1ea71e2270686adb6075f19f849f87e9cba6ed64ca23ef11acc70100e5b2fb","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.QA","submitted_at":"2014-09-30T16:36:02Z","title_canon_sha256":"442a4ceb47fb2cf378d52594ff55570a694df854ffb2aede89bb715d94d2c234"},"schema_version":"1.0","source":{"id":"1409.8622","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.8622","created_at":"2026-05-18T02:18:04Z"},{"alias_kind":"arxiv_version","alias_value":"1409.8622v2","created_at":"2026-05-18T02:18:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.8622","created_at":"2026-05-18T02:18:04Z"},{"alias_kind":"pith_short_12","alias_value":"MPXW2HSME7PW","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"MPXW2HSME7PW7467","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"MPXW2HSM","created_at":"2026-05-18T12:28:38Z"}],"graph_snapshots":[{"event_id":"sha256:6138b5a3cce27cf256fae4cb726d36fd0105937ad5356ecf45306094c1b846c4","target":"graph","created_at":"2026-05-18T02:18:04Z","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 simply connected simple algebraic group over $\\mathbb{C}$, $B$ and $B_-$ be two opposite Borel subgroups in $G$ and $W$ be the Weyl group. For $u$, $v\\in W$, it is known that the coordinate ring ${\\mathbb C}[G^{u,v}]$ of the double Bruhat cell $G^{u,v}=BuB\\cap B_-vB_-$ is isomorphic to an upper cluster algebra $\\bar{{\\mathcal A}}({\\bf i})_{{\\mathbb C}}$ and the generalized minors $\\{\\Delta(k;{\\bf i})\\}$ are the cluster variables belonging to a given initial seed in ${\\mathbb C}[G^{u,v}]$ [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52, math.RT/0305434]. In ","authors_text":"Toshiki Nakashima, Yuki Kanakubo","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.QA","submitted_at":"2014-09-30T16:36:02Z","title":"Cluster Variables on Certain Double Bruhat Cells of Type $(u,e)$ and Monomial Realizations of Crystal Bases of Type A"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.8622","kind":"arxiv","version":2},"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":"sha256:6cde9c8aa51029b962a71d01a7c0168a6fc5a91b2eb4b953eb9f441cf4825fd3","target":"record","created_at":"2026-05-18T02:18:04Z","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":"6e1ea71e2270686adb6075f19f849f87e9cba6ed64ca23ef11acc70100e5b2fb","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.QA","submitted_at":"2014-09-30T16:36:02Z","title_canon_sha256":"442a4ceb47fb2cf378d52594ff55570a694df854ffb2aede89bb715d94d2c234"},"schema_version":"1.0","source":{"id":"1409.8622","kind":"arxiv","version":2}},"canonical_sha256":"63ef6d1e4c27df6ff3df867226b062e4727a5f3f5ca215424ff5492b781c15ac","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"63ef6d1e4c27df6ff3df867226b062e4727a5f3f5ca215424ff5492b781c15ac","first_computed_at":"2026-05-18T02:18:04.054877Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:18:04.054877Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xernXjCQdVJnbz5YRdEdY3xvtUv/fj+/MRDF8nkPPeL5nGHGO+gsOaURq/qSVi6HlMTtceEZ3F/c+8F8fY7bDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:18:04.055709Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.8622","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6cde9c8aa51029b962a71d01a7c0168a6fc5a91b2eb4b953eb9f441cf4825fd3","sha256:6138b5a3cce27cf256fae4cb726d36fd0105937ad5356ecf45306094c1b846c4"],"state_sha256":"834259f64d5b87d9cf5c3aa752fa9b022a191a5ef7e269bc3d51d5ad54ed1cf4"}