{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:B2YCGELEO6OJDJOPMYSQBGBUIJ","short_pith_number":"pith:B2YCGELE","canonical_record":{"source":{"id":"math/0607002","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.RT","submitted_at":"2006-06-30T22:27:19Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"627dbe3ebde672738eb2f82d1158a1ecb0a3c71bd389728eefb10c7a54c07c4a","abstract_canon_sha256":"16e1984013235dfdc31c11ec69a0b91dcec7cb8e022aad669fe896bfde6fd406"},"schema_version":"1.0"},"canonical_sha256":"0eb0231164779c91a5cf662500983442701604bee43562642e4136dc7e36f3ee","source":{"kind":"arxiv","id":"math/0607002","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0607002","created_at":"2026-05-18T04:19:32Z"},{"alias_kind":"arxiv_version","alias_value":"math/0607002v1","created_at":"2026-05-18T04:19:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0607002","created_at":"2026-05-18T04:19:32Z"},{"alias_kind":"pith_short_12","alias_value":"B2YCGELEO6OJ","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"B2YCGELEO6OJDJOP","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"B2YCGELE","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:B2YCGELEO6OJDJOPMYSQBGBUIJ","target":"record","payload":{"canonical_record":{"source":{"id":"math/0607002","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.RT","submitted_at":"2006-06-30T22:27:19Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"627dbe3ebde672738eb2f82d1158a1ecb0a3c71bd389728eefb10c7a54c07c4a","abstract_canon_sha256":"16e1984013235dfdc31c11ec69a0b91dcec7cb8e022aad669fe896bfde6fd406"},"schema_version":"1.0"},"canonical_sha256":"0eb0231164779c91a5cf662500983442701604bee43562642e4136dc7e36f3ee","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:19:32.789834Z","signature_b64":"8jShg/5vRTQM0bTDXyTqw1I0QgY9pEhCl7e2Ffgi/b/RrZ/JitYCYWcsBKAM8g27hE/nafahHfiRprHb9LwrDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0eb0231164779c91a5cf662500983442701604bee43562642e4136dc7e36f3ee","last_reissued_at":"2026-05-18T04:19:32.789464Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:19:32.789464Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0607002","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:19:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c6Uy9JQCjQDqFNr30zWq19mrbtwhGMUtZbTiTKH9KDFbSdeWEgRhqkbwd1sRpxhUFbK5Gp4nLFteyesbymyUAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T20:42:56.319589Z"},"content_sha256":"da770f80e14f58b30ca63625519245420db390b946225e0aa4d71bc291067074","schema_version":"1.0","event_id":"sha256:da770f80e14f58b30ca63625519245420db390b946225e0aa4d71bc291067074"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:B2YCGELEO6OJDJOPMYSQBGBUIJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs","license":"","headline":"","cross_cats":["math.DG"],"primary_cat":"math.RT","authors_text":"Toshiyuki Kobayashi","submitted_at":"2006-06-30T22:27:19Z","abstract_excerpt":"The complex analytic methods have found a wide range of applications in the study of multiplicity-free representations. This article discusses, in particular, its applications to the question of restricting highest weight modules with respect to reductive symmetric pairs. We present a number of multiplicity-free branching theorems that include the multiplicity-free property of some of known results such as the Clebsh--Gordan--Pieri formula for tensor products, the Plancherel theorem for Hermitian symmetric spaces (also for line bundle cases), the Hua--Kostant--Schmid $K$-type formula, and the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607002","kind":"arxiv","version":1},"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":"2026-05-18T04:19:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JMage5gGYu1KBPQ8VAhI+Ftc3/n6Vn0C9GVFLvZM0XwHU9ovRZywR3tRqJZYOvZKxlH6WO2ATOPornGbGgGmDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T20:42:56.320252Z"},"content_sha256":"d68dac62f1d26d1caf24ab915d33a492149b7fa698cb96ad1583b6886f7e072c","schema_version":"1.0","event_id":"sha256:d68dac62f1d26d1caf24ab915d33a492149b7fa698cb96ad1583b6886f7e072c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B2YCGELEO6OJDJOPMYSQBGBUIJ/bundle.json","state_url":"https://pith.science/pith/B2YCGELEO6OJDJOPMYSQBGBUIJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B2YCGELEO6OJDJOPMYSQBGBUIJ/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":"2026-05-26T20:42:56Z","links":{"resolver":"https://pith.science/pith/B2YCGELEO6OJDJOPMYSQBGBUIJ","bundle":"https://pith.science/pith/B2YCGELEO6OJDJOPMYSQBGBUIJ/bundle.json","state":"https://pith.science/pith/B2YCGELEO6OJDJOPMYSQBGBUIJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B2YCGELEO6OJDJOPMYSQBGBUIJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:B2YCGELEO6OJDJOPMYSQBGBUIJ","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":"16e1984013235dfdc31c11ec69a0b91dcec7cb8e022aad669fe896bfde6fd406","cross_cats_sorted":["math.DG"],"license":"","primary_cat":"math.RT","submitted_at":"2006-06-30T22:27:19Z","title_canon_sha256":"627dbe3ebde672738eb2f82d1158a1ecb0a3c71bd389728eefb10c7a54c07c4a"},"schema_version":"1.0","source":{"id":"math/0607002","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0607002","created_at":"2026-05-18T04:19:32Z"},{"alias_kind":"arxiv_version","alias_value":"math/0607002v1","created_at":"2026-05-18T04:19:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0607002","created_at":"2026-05-18T04:19:32Z"},{"alias_kind":"pith_short_12","alias_value":"B2YCGELEO6OJ","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"B2YCGELEO6OJDJOP","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"B2YCGELE","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:d68dac62f1d26d1caf24ab915d33a492149b7fa698cb96ad1583b6886f7e072c","target":"graph","created_at":"2026-05-18T04:19:32Z","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":"The complex analytic methods have found a wide range of applications in the study of multiplicity-free representations. This article discusses, in particular, its applications to the question of restricting highest weight modules with respect to reductive symmetric pairs. We present a number of multiplicity-free branching theorems that include the multiplicity-free property of some of known results such as the Clebsh--Gordan--Pieri formula for tensor products, the Plancherel theorem for Hermitian symmetric spaces (also for line bundle cases), the Hua--Kostant--Schmid $K$-type formula, and the ","authors_text":"Toshiyuki Kobayashi","cross_cats":["math.DG"],"headline":"","license":"","primary_cat":"math.RT","submitted_at":"2006-06-30T22:27:19Z","title":"Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607002","kind":"arxiv","version":1},"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:da770f80e14f58b30ca63625519245420db390b946225e0aa4d71bc291067074","target":"record","created_at":"2026-05-18T04:19:32Z","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":"16e1984013235dfdc31c11ec69a0b91dcec7cb8e022aad669fe896bfde6fd406","cross_cats_sorted":["math.DG"],"license":"","primary_cat":"math.RT","submitted_at":"2006-06-30T22:27:19Z","title_canon_sha256":"627dbe3ebde672738eb2f82d1158a1ecb0a3c71bd389728eefb10c7a54c07c4a"},"schema_version":"1.0","source":{"id":"math/0607002","kind":"arxiv","version":1}},"canonical_sha256":"0eb0231164779c91a5cf662500983442701604bee43562642e4136dc7e36f3ee","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0eb0231164779c91a5cf662500983442701604bee43562642e4136dc7e36f3ee","first_computed_at":"2026-05-18T04:19:32.789464Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:19:32.789464Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8jShg/5vRTQM0bTDXyTqw1I0QgY9pEhCl7e2Ffgi/b/RrZ/JitYCYWcsBKAM8g27hE/nafahHfiRprHb9LwrDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:19:32.789834Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0607002","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:da770f80e14f58b30ca63625519245420db390b946225e0aa4d71bc291067074","sha256:d68dac62f1d26d1caf24ab915d33a492149b7fa698cb96ad1583b6886f7e072c"],"state_sha256":"81090f7e601c2b6e491172fbe8d727d65db7adbf42102de218e3d790cf98c4ac"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4NV7Bcw4XcdGgM7jM7X73kKM91mwHak1rygsJ4/jAD9HrFQRqWRXW0ZT5hravxMa5wbao5IFwPoFRLpKcWfACA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T20:42:56.324294Z","bundle_sha256":"bf3e32bb17a68e7b0ce4717b146b3c640b7b0eb2beb0d973a9f70acdb22dcbaa"}}