{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:HB7A76CUKUKSXZC2JLGMOAYJRC","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":"4a61e9ab5887ac2f1d9a59311c5c858b5efaa5b2ae3f3b291ace785c74f99bea","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-09-15T21:07:29Z","title_canon_sha256":"05ad5f489f8f8699263901db2dc62dff1445f7779ea05241f3a61afe2a8f780f"},"schema_version":"1.0","source":{"id":"1009.3040","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.3040","created_at":"2026-05-18T04:07:52Z"},{"alias_kind":"arxiv_version","alias_value":"1009.3040v2","created_at":"2026-05-18T04:07:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.3040","created_at":"2026-05-18T04:07:52Z"},{"alias_kind":"pith_short_12","alias_value":"HB7A76CUKUKS","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"HB7A76CUKUKSXZC2","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"HB7A76CU","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:d7af39f87e44abeb018677c11a6cbe14381f6b64d7e26f74e6e423e4265e67f8","target":"graph","created_at":"2026-05-18T04:07:52Z","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 denote either a special orthogonal group or a symplectic group defined over the complex numbers. We prove the following saturation result for G: given dominant weights \\lambda^1, ..., \\lambda^r such that the tensor product V_{N\\lambda^1} \\otimes ... \\otimes V_{N\\lambda^r} contains nonzero G-invariants for some N \\ge 1, we show that the tensor product V_{2\\lambda^1} \\otimes ... \\otimes V_{2\\lambda^r} also contains nonzero G-invariants. This extends results of Kapovich-Millson and Belkale-Kumar and complements similar results for the general linear group due to Knutson-Tao and Derksen-Weym","authors_text":"Steven V Sam","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-09-15T21:07:29Z","title":"Symmetric quivers, invariant theory, and saturation theorems for the classical groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.3040","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:e8b334ae6cd1a309c600b74f25ba63124f0b6ef01d4e26eb39e9d0c3c1e3b25d","target":"record","created_at":"2026-05-18T04:07:52Z","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":"4a61e9ab5887ac2f1d9a59311c5c858b5efaa5b2ae3f3b291ace785c74f99bea","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-09-15T21:07:29Z","title_canon_sha256":"05ad5f489f8f8699263901db2dc62dff1445f7779ea05241f3a61afe2a8f780f"},"schema_version":"1.0","source":{"id":"1009.3040","kind":"arxiv","version":2}},"canonical_sha256":"387e0ff85455152be45a4accc7030988839d06fac2328a8c5937e9de48dbf0f2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"387e0ff85455152be45a4accc7030988839d06fac2328a8c5937e9de48dbf0f2","first_computed_at":"2026-05-18T04:07:52.713770Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:07:52.713770Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/YEZp5wxYC1V1oToh4jSs/AWWpvAS6gQT8YZ2ZpQQOEWSBgnxs0GaTbV+2gxjRmE/o9oZW4ipI98BPWF27RWDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:07:52.714237Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.3040","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e8b334ae6cd1a309c600b74f25ba63124f0b6ef01d4e26eb39e9d0c3c1e3b25d","sha256:d7af39f87e44abeb018677c11a6cbe14381f6b64d7e26f74e6e423e4265e67f8"],"state_sha256":"3edbfdbc53cdfedd0e9a81454a5e3e59f830275b1a3905d97e6539a9f8020a47"}