{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:O3HLDQJ5AG6EAIFG6XKJ2DWXCF","short_pith_number":"pith:O3HLDQJ5","canonical_record":{"source":{"id":"1405.4887","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-05-19T20:38:12Z","cross_cats_sorted":["hep-th","math.MP","math.QA","math.RT"],"title_canon_sha256":"b47ab0f769ff1554df5a850559133c8827bcd58b8df68bde04a232b13996adff","abstract_canon_sha256":"22f5c3fa29c878fadd7a7859032d45fe219bac8c95592e2e245ffe5ca559905c"},"schema_version":"1.0"},"canonical_sha256":"76ceb1c13d01bc4020a6f5d49d0ed71172685951b23bbea7c28298153e6ed055","source":{"kind":"arxiv","id":"1405.4887","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.4887","created_at":"2026-05-18T02:38:09Z"},{"alias_kind":"arxiv_version","alias_value":"1405.4887v2","created_at":"2026-05-18T02:38:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.4887","created_at":"2026-05-18T02:38:09Z"},{"alias_kind":"pith_short_12","alias_value":"O3HLDQJ5AG6E","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"O3HLDQJ5AG6EAIFG","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"O3HLDQJ5","created_at":"2026-05-18T12:28:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:O3HLDQJ5AG6EAIFG6XKJ2DWXCF","target":"record","payload":{"canonical_record":{"source":{"id":"1405.4887","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-05-19T20:38:12Z","cross_cats_sorted":["hep-th","math.MP","math.QA","math.RT"],"title_canon_sha256":"b47ab0f769ff1554df5a850559133c8827bcd58b8df68bde04a232b13996adff","abstract_canon_sha256":"22f5c3fa29c878fadd7a7859032d45fe219bac8c95592e2e245ffe5ca559905c"},"schema_version":"1.0"},"canonical_sha256":"76ceb1c13d01bc4020a6f5d49d0ed71172685951b23bbea7c28298153e6ed055","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:09.870532Z","signature_b64":"1IIkFQoPq29CfdzchQFkZo0ef42lrqVRPqKPxkpczU8QSgrh62IJFG0YHNiUhAF+HMq40VF5459N/3LCpKWxDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"76ceb1c13d01bc4020a6f5d49d0ed71172685951b23bbea7c28298153e6ed055","last_reissued_at":"2026-05-18T02:38:09.869887Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:09.869887Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1405.4887","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":"2026-05-18T02:38:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2yfst/DMhAJ+bJPRW0hm2I+KOL3Z6fZD5J2tXxI1K5T33/4zqTAoRPoF1hdo6YDIwy7n6U7EnD8rJsh82gNtBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T15:42:40.391478Z"},"content_sha256":"17140621967d94db07ff614fb3cc56be73a64c749d756c298044026b57a81e63","schema_version":"1.0","event_id":"sha256:17140621967d94db07ff614fb3cc56be73a64c749d756c298044026b57a81e63"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:O3HLDQJ5AG6EAIFG6XKJ2DWXCF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Conjugation properties of tensor product multiplicities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math.MP","math.QA","math.RT"],"primary_cat":"math-ph","authors_text":"Jean-Bernard Zuber, Robert Coquereaux","submitted_at":"2014-05-19T20:38:12Z","abstract_excerpt":"It was recently proven that the total multiplicity in the decomposition into irreducibles of the tensor product lambda x mu of two irreducible representations of a simple Lie algebra is invariant under conjugation of one of them; at a given level, this also applies to the fusion multiplicities of affine algebras. Here, we show that, in the case of SU(3), the lists of multiplicities, in the tensor products lambda x mu and lambda x bar{mu}, are identical up to permutations. This latter property does not hold in general for other Lie algebras. We conjecture that the same property should hold for "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.4887","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":"2026-05-18T02:38:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0cyAAz1bt9iVvifrc4GtWbvGAR+Z0jdHdqlR4BmVAQ8ckiq3/CUTWZuCN2Ni1FHpegrrvUbrz3XaopKHHxS7Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T15:42:40.391871Z"},"content_sha256":"5a98c631069015c0bad34bad6b70fea756bf98e2926edc15bad334be15e92949","schema_version":"1.0","event_id":"sha256:5a98c631069015c0bad34bad6b70fea756bf98e2926edc15bad334be15e92949"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/O3HLDQJ5AG6EAIFG6XKJ2DWXCF/bundle.json","state_url":"https://pith.science/pith/O3HLDQJ5AG6EAIFG6XKJ2DWXCF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/O3HLDQJ5AG6EAIFG6XKJ2DWXCF/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":"2026-06-23T15:42:40Z","links":{"resolver":"https://pith.science/pith/O3HLDQJ5AG6EAIFG6XKJ2DWXCF","bundle":"https://pith.science/pith/O3HLDQJ5AG6EAIFG6XKJ2DWXCF/bundle.json","state":"https://pith.science/pith/O3HLDQJ5AG6EAIFG6XKJ2DWXCF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/O3HLDQJ5AG6EAIFG6XKJ2DWXCF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:O3HLDQJ5AG6EAIFG6XKJ2DWXCF","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":"22f5c3fa29c878fadd7a7859032d45fe219bac8c95592e2e245ffe5ca559905c","cross_cats_sorted":["hep-th","math.MP","math.QA","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-05-19T20:38:12Z","title_canon_sha256":"b47ab0f769ff1554df5a850559133c8827bcd58b8df68bde04a232b13996adff"},"schema_version":"1.0","source":{"id":"1405.4887","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.4887","created_at":"2026-05-18T02:38:09Z"},{"alias_kind":"arxiv_version","alias_value":"1405.4887v2","created_at":"2026-05-18T02:38:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.4887","created_at":"2026-05-18T02:38:09Z"},{"alias_kind":"pith_short_12","alias_value":"O3HLDQJ5AG6E","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"O3HLDQJ5AG6EAIFG","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"O3HLDQJ5","created_at":"2026-05-18T12:28:41Z"}],"graph_snapshots":[{"event_id":"sha256:5a98c631069015c0bad34bad6b70fea756bf98e2926edc15bad334be15e92949","target":"graph","created_at":"2026-05-18T02:38:09Z","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":"It was recently proven that the total multiplicity in the decomposition into irreducibles of the tensor product lambda x mu of two irreducible representations of a simple Lie algebra is invariant under conjugation of one of them; at a given level, this also applies to the fusion multiplicities of affine algebras. Here, we show that, in the case of SU(3), the lists of multiplicities, in the tensor products lambda x mu and lambda x bar{mu}, are identical up to permutations. This latter property does not hold in general for other Lie algebras. We conjecture that the same property should hold for ","authors_text":"Jean-Bernard Zuber, Robert Coquereaux","cross_cats":["hep-th","math.MP","math.QA","math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-05-19T20:38:12Z","title":"Conjugation properties of tensor product multiplicities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.4887","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":"sha256:17140621967d94db07ff614fb3cc56be73a64c749d756c298044026b57a81e63","target":"record","created_at":"2026-05-18T02:38:09Z","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":"22f5c3fa29c878fadd7a7859032d45fe219bac8c95592e2e245ffe5ca559905c","cross_cats_sorted":["hep-th","math.MP","math.QA","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-05-19T20:38:12Z","title_canon_sha256":"b47ab0f769ff1554df5a850559133c8827bcd58b8df68bde04a232b13996adff"},"schema_version":"1.0","source":{"id":"1405.4887","kind":"arxiv","version":2}},"canonical_sha256":"76ceb1c13d01bc4020a6f5d49d0ed71172685951b23bbea7c28298153e6ed055","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"76ceb1c13d01bc4020a6f5d49d0ed71172685951b23bbea7c28298153e6ed055","first_computed_at":"2026-05-18T02:38:09.869887Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:38:09.869887Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1IIkFQoPq29CfdzchQFkZo0ef42lrqVRPqKPxkpczU8QSgrh62IJFG0YHNiUhAF+HMq40VF5459N/3LCpKWxDw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:38:09.870532Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.4887","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:17140621967d94db07ff614fb3cc56be73a64c749d756c298044026b57a81e63","sha256:5a98c631069015c0bad34bad6b70fea756bf98e2926edc15bad334be15e92949"],"state_sha256":"442f03c3154f95a06987bce120c04c265bd2620ad6179112ef803f4758d3a714"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xg/vQgv4w7/rcrFVLZt3RaK4kzK5a+wojzHZdKLjYhgymAb+vw33k+l/JIxOwD99PZRdni84VBWY0HyH8iMHBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T15:42:40.393726Z","bundle_sha256":"e08b2fc4f7cbcbe9c0a40d862b6cc87866c88a7c6dccae20aea8c4382c3d4e18"}}