{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:ZCOZKFUT65ZYCYXKSOAAHVR4O7","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":"b657484782ccb2de4630df9b81798b9a42d0c8996768dcd7c5009786a82d3e09","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2014-08-17T00:50:36Z","title_canon_sha256":"d5bdb2dec39e8d25f4a0caa0d88f5d668c89b2fddb50a0e5c92c4bb0998ba0db"},"schema_version":"1.0","source":{"id":"1408.3782","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.3782","created_at":"2026-07-05T09:28:26Z"},{"alias_kind":"arxiv_version","alias_value":"1408.3782v6","created_at":"2026-07-05T09:28:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.3782","created_at":"2026-07-05T09:28:26Z"},{"alias_kind":"pith_short_12","alias_value":"ZCOZKFUT65ZY","created_at":"2026-07-05T09:28:26Z"},{"alias_kind":"pith_short_16","alias_value":"ZCOZKFUT65ZYCYXK","created_at":"2026-07-05T09:28:26Z"},{"alias_kind":"pith_short_8","alias_value":"ZCOZKFUT","created_at":"2026-07-05T09:28:26Z"}],"graph_snapshots":[{"event_id":"sha256:2c46f006580e2f535dbc29daa0d1a5e030e340092ee73cd075684dfc91c9d2df","target":"graph","created_at":"2026-07-05T09:28:26Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1408.3782/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The integral formulae pertaining to the unitary group $\\mathsf{U}(d)$ have been comprehensively reviewed, yielding fresh results and innovative proofs. Central to the derivation of these formulae lies the employment of Schur-Weyl duality, a classical and powerful theorem from the representation theory of groups. This duality serves as a bridge, establishing a profound connection between the representation theory of finite groups (or permutation groups) and that of classical Lie groups, specifically the unitary groups. From the perspective of Schur-Weyl duality, it becomes evident that the comp","authors_text":"Lin Zhang","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2014-08-17T00:50:36Z","title":"Matrix integrals over unitary groups: An application of Schur-Weyl duality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.3782","kind":"arxiv","version":6},"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":"sha256:0cfeec96ebdb3f6c03b2e715debfb718ad39f5a17736db3a4aabb7c1e7fdd1a4","target":"record","created_at":"2026-07-05T09:28:26Z","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":"b657484782ccb2de4630df9b81798b9a42d0c8996768dcd7c5009786a82d3e09","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2014-08-17T00:50:36Z","title_canon_sha256":"d5bdb2dec39e8d25f4a0caa0d88f5d668c89b2fddb50a0e5c92c4bb0998ba0db"},"schema_version":"1.0","source":{"id":"1408.3782","kind":"arxiv","version":6}},"canonical_sha256":"c89d951693f7738162ea938003d63c77d613017f5929fad3ba8ec75bc7821649","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c89d951693f7738162ea938003d63c77d613017f5929fad3ba8ec75bc7821649","first_computed_at":"2026-07-05T09:28:26.360298Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:28:26.360298Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/f+zR6ODt5zKZB8iAek49HB88y92FRLIIuPxaC0vIRnyoevFCk8z6CGYw+D9kHjXD2HF+dFbh9kYO8y7tjq5BA==","signature_status":"signed_v1","signed_at":"2026-07-05T09:28:26.360887Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.3782","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0cfeec96ebdb3f6c03b2e715debfb718ad39f5a17736db3a4aabb7c1e7fdd1a4","sha256:2c46f006580e2f535dbc29daa0d1a5e030e340092ee73cd075684dfc91c9d2df"],"state_sha256":"782e64267bf821e988c4f466104676a1120e8f96e478491ba215e117cf4e786c"}