{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:DUSTUGVJOPNL3ZCXGEPFNV7YQL","short_pith_number":"pith:DUSTUGVJ","canonical_record":{"source":{"id":"1809.00748","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2018-09-04T00:04:28Z","cross_cats_sorted":[],"title_canon_sha256":"efbe610193afa7c4bc80fff7771a7e55179d829729b108fcbd0c103aa464754d","abstract_canon_sha256":"8aa58b644c55544c67df1c38f4bd5b0d51e5750bb47d9f390064ff34b9647adf"},"schema_version":"1.0"},"canonical_sha256":"1d253a1aa973dabde457311e56d7f882d0a0898043bce6daf41dc47a89ab1bd7","source":{"kind":"arxiv","id":"1809.00748","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.00748","created_at":"2026-05-17T23:44:38Z"},{"alias_kind":"arxiv_version","alias_value":"1809.00748v2","created_at":"2026-05-17T23:44:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.00748","created_at":"2026-05-17T23:44:38Z"},{"alias_kind":"pith_short_12","alias_value":"DUSTUGVJOPNL","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"DUSTUGVJOPNL3ZCX","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"DUSTUGVJ","created_at":"2026-05-18T12:32:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:DUSTUGVJOPNL3ZCXGEPFNV7YQL","target":"record","payload":{"canonical_record":{"source":{"id":"1809.00748","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2018-09-04T00:04:28Z","cross_cats_sorted":[],"title_canon_sha256":"efbe610193afa7c4bc80fff7771a7e55179d829729b108fcbd0c103aa464754d","abstract_canon_sha256":"8aa58b644c55544c67df1c38f4bd5b0d51e5750bb47d9f390064ff34b9647adf"},"schema_version":"1.0"},"canonical_sha256":"1d253a1aa973dabde457311e56d7f882d0a0898043bce6daf41dc47a89ab1bd7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:38.091902Z","signature_b64":"GkypQNNUQqlwXiQR5ZD4+i2k65Dnjl22FgJYA8bPb8duoGUVDw//4S2AYlOPSSqXmun8fKfHAAyKCT45zIbpBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1d253a1aa973dabde457311e56d7f882d0a0898043bce6daf41dc47a89ab1bd7","last_reissued_at":"2026-05-17T23:44:38.091345Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:38.091345Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1809.00748","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:44:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5qpSSyz+ANHr1xdfXWEgy3wkDNp1X8z7DlJWmEK07NbOm+/JvsPfS/v0obCrKEGOQruSnwA9AJwvplyd9shLAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T12:53:39.208712Z"},"content_sha256":"728af2593253282d8db92a26efa0e3d064670bb92de87525bcafd8a639157b8b","schema_version":"1.0","event_id":"sha256:728af2593253282d8db92a26efa0e3d064670bb92de87525bcafd8a639157b8b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:DUSTUGVJOPNL3ZCXGEPFNV7YQL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Integrable systems on $Fl_n \\times Fl_n \\times Fl_n //SU(n)$ and $SU(n)$ tensor product multiplicities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SG","authors_text":"Jonathan Weitsman","submitted_at":"2018-09-04T00:04:28Z","abstract_excerpt":"We construct a densely defined torus action on the symplectic quotient of the product of three complete flag varieties. The closure of the image of the corresponding moment map is a convex polytope. The dimension of the geometric quantization of this space gives the structure constants of the representation ring of $SU(n)$, which we show is given by counting lattice points in the image of the moment map. Such lattice point formulas for the structure constants were given by Berenstein-Zelevinsky; our results show how such formulas arise geometrically as an example of `invariance of polarization"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.00748","kind":"arxiv","version":2},"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":"2026-05-17T23:44:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AxkIQys1S0LQM2HTp5S9XYEVhpiOFv2ysMcU4sIsw/ifq5xWkHNUAtQ+a4HO5hTNEo41wyLI1XsyOtTKBzvADg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T12:53:39.209065Z"},"content_sha256":"b4a5295ecf07d5606dee497bdb30ecfd53697db2e870b2c34e2b5675af5a9c77","schema_version":"1.0","event_id":"sha256:b4a5295ecf07d5606dee497bdb30ecfd53697db2e870b2c34e2b5675af5a9c77"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DUSTUGVJOPNL3ZCXGEPFNV7YQL/bundle.json","state_url":"https://pith.science/pith/DUSTUGVJOPNL3ZCXGEPFNV7YQL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DUSTUGVJOPNL3ZCXGEPFNV7YQL/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":"2026-05-31T12:53:39Z","links":{"resolver":"https://pith.science/pith/DUSTUGVJOPNL3ZCXGEPFNV7YQL","bundle":"https://pith.science/pith/DUSTUGVJOPNL3ZCXGEPFNV7YQL/bundle.json","state":"https://pith.science/pith/DUSTUGVJOPNL3ZCXGEPFNV7YQL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DUSTUGVJOPNL3ZCXGEPFNV7YQL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:DUSTUGVJOPNL3ZCXGEPFNV7YQL","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":"8aa58b644c55544c67df1c38f4bd5b0d51e5750bb47d9f390064ff34b9647adf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2018-09-04T00:04:28Z","title_canon_sha256":"efbe610193afa7c4bc80fff7771a7e55179d829729b108fcbd0c103aa464754d"},"schema_version":"1.0","source":{"id":"1809.00748","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.00748","created_at":"2026-05-17T23:44:38Z"},{"alias_kind":"arxiv_version","alias_value":"1809.00748v2","created_at":"2026-05-17T23:44:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.00748","created_at":"2026-05-17T23:44:38Z"},{"alias_kind":"pith_short_12","alias_value":"DUSTUGVJOPNL","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"DUSTUGVJOPNL3ZCX","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"DUSTUGVJ","created_at":"2026-05-18T12:32:19Z"}],"graph_snapshots":[{"event_id":"sha256:b4a5295ecf07d5606dee497bdb30ecfd53697db2e870b2c34e2b5675af5a9c77","target":"graph","created_at":"2026-05-17T23:44:38Z","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":"We construct a densely defined torus action on the symplectic quotient of the product of three complete flag varieties. The closure of the image of the corresponding moment map is a convex polytope. The dimension of the geometric quantization of this space gives the structure constants of the representation ring of $SU(n)$, which we show is given by counting lattice points in the image of the moment map. Such lattice point formulas for the structure constants were given by Berenstein-Zelevinsky; our results show how such formulas arise geometrically as an example of `invariance of polarization","authors_text":"Jonathan Weitsman","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2018-09-04T00:04:28Z","title":"Integrable systems on $Fl_n \\times Fl_n \\times Fl_n //SU(n)$ and $SU(n)$ tensor product multiplicities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.00748","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":"sha256:728af2593253282d8db92a26efa0e3d064670bb92de87525bcafd8a639157b8b","target":"record","created_at":"2026-05-17T23:44:38Z","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":"8aa58b644c55544c67df1c38f4bd5b0d51e5750bb47d9f390064ff34b9647adf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2018-09-04T00:04:28Z","title_canon_sha256":"efbe610193afa7c4bc80fff7771a7e55179d829729b108fcbd0c103aa464754d"},"schema_version":"1.0","source":{"id":"1809.00748","kind":"arxiv","version":2}},"canonical_sha256":"1d253a1aa973dabde457311e56d7f882d0a0898043bce6daf41dc47a89ab1bd7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1d253a1aa973dabde457311e56d7f882d0a0898043bce6daf41dc47a89ab1bd7","first_computed_at":"2026-05-17T23:44:38.091345Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:44:38.091345Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GkypQNNUQqlwXiQR5ZD4+i2k65Dnjl22FgJYA8bPb8duoGUVDw//4S2AYlOPSSqXmun8fKfHAAyKCT45zIbpBA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:44:38.091902Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.00748","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:728af2593253282d8db92a26efa0e3d064670bb92de87525bcafd8a639157b8b","sha256:b4a5295ecf07d5606dee497bdb30ecfd53697db2e870b2c34e2b5675af5a9c77"],"state_sha256":"7a81b036d8f305ce19b8d9fa56b6d371455f1198b54d683bf36cebac6dd3fe75"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+Lxey2h6JDtz3YH6uMKFt264mfzJ+uqfchQ5qZVOl6HaROSBaAyzvIdzb/Jx3teCaYHn2pk+3HUfRGWxukRDAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T12:53:39.211557Z","bundle_sha256":"8e0757cd87dce37592019a79c52288b15e864ba70ec7c395768a884e1110f1fa"}}