{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:NZSVVSQ5CMABAI2B7FR6LXA3NX","short_pith_number":"pith:NZSVVSQ5","canonical_record":{"source":{"id":"0907.3247","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-07-18T23:23:43Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"6c9ef7f8812a83b92a429cfb1f4919d75206beaeedde381adbfb35c4544e73d0","abstract_canon_sha256":"4a6e64adc3ad0be1f202d1fcf31098721989604e0db29cc1110518f0d9df61cf"},"schema_version":"1.0"},"canonical_sha256":"6e655aca1d1300102341f963e5dc1b6df2dda966e457ef146fe1119646b29a8a","source":{"kind":"arxiv","id":"0907.3247","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0907.3247","created_at":"2026-05-18T03:46:32Z"},{"alias_kind":"arxiv_version","alias_value":"0907.3247v2","created_at":"2026-05-18T03:46:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.3247","created_at":"2026-05-18T03:46:32Z"},{"alias_kind":"pith_short_12","alias_value":"NZSVVSQ5CMAB","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"NZSVVSQ5CMABAI2B","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"NZSVVSQ5","created_at":"2026-05-18T12:26:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:NZSVVSQ5CMABAI2B7FR6LXA3NX","target":"record","payload":{"canonical_record":{"source":{"id":"0907.3247","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-07-18T23:23:43Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"6c9ef7f8812a83b92a429cfb1f4919d75206beaeedde381adbfb35c4544e73d0","abstract_canon_sha256":"4a6e64adc3ad0be1f202d1fcf31098721989604e0db29cc1110518f0d9df61cf"},"schema_version":"1.0"},"canonical_sha256":"6e655aca1d1300102341f963e5dc1b6df2dda966e457ef146fe1119646b29a8a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:46:32.483667Z","signature_b64":"kWnjSRYynU/1PFBF9zDkUQoDf8SmPzvU9LB+UYlSms1MId0sopJQXV9d2QVqtD99aGi8t/+Q/I6sxKmy9V4xDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6e655aca1d1300102341f963e5dc1b6df2dda966e457ef146fe1119646b29a8a","last_reissued_at":"2026-05-18T03:46:32.482876Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:46:32.482876Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0907.3247","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":"2026-05-18T03:46:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xhwlIsT9zWUYayMc3lRkO7Hy+ZpLWxs4dWDNUOMOnw+1/dWY0ojHxDK+3hJC8E14hfDc1Zyo4UrA17SkxErAAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T16:25:39.751825Z"},"content_sha256":"3257456567b42756f1e8818dd8095d046e9192acacdc2a400470a6c841832039","schema_version":"1.0","event_id":"sha256:3257456567b42756f1e8818dd8095d046e9192acacdc2a400470a6c841832039"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:NZSVVSQ5CMABAI2B7FR6LXA3NX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An Analysis of the Multiplicity Spaces in Branching of Symplectic Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.RT","authors_text":"Oded Yacobi","submitted_at":"2009-07-18T23:23:43Z","abstract_excerpt":"Branching of symplectic groups is not multiplicity-free.  We describe a new approach to resolving these multiplicities that is based on studying the associated branching algebra $B$. The algebra $B$ is a graded algebra whose components encode the multiplicities of irreducible representations of $Sp_{2n-2}$ in irreducible representations of $Sp_{2n}$. Our first theorem states that the map taking an element of $Sp_{2n}$ to its principal $n \\times (n+1)$ submatrix induces an isomorphism of $\\B$ to a different branching algebra $\\B'$.  The algebra $\\B'$ encodes multiplicities of irreducible repres"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.3247","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":"2026-05-18T03:46:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sY7FF8O95UW7Vw7Enteq6sHEGsqYdjmVTKPySeBCF4XDOCnRBcWg57N9w/5iZ8g89uhekAS4yXUKDbMZaILSBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T16:25:39.752476Z"},"content_sha256":"dab04412329e6de09299c8cabc8e77fa390baa8badf8ea1968254f46841eb5d3","schema_version":"1.0","event_id":"sha256:dab04412329e6de09299c8cabc8e77fa390baa8badf8ea1968254f46841eb5d3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NZSVVSQ5CMABAI2B7FR6LXA3NX/bundle.json","state_url":"https://pith.science/pith/NZSVVSQ5CMABAI2B7FR6LXA3NX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NZSVVSQ5CMABAI2B7FR6LXA3NX/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":"2026-05-28T16:25:39Z","links":{"resolver":"https://pith.science/pith/NZSVVSQ5CMABAI2B7FR6LXA3NX","bundle":"https://pith.science/pith/NZSVVSQ5CMABAI2B7FR6LXA3NX/bundle.json","state":"https://pith.science/pith/NZSVVSQ5CMABAI2B7FR6LXA3NX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NZSVVSQ5CMABAI2B7FR6LXA3NX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:NZSVVSQ5CMABAI2B7FR6LXA3NX","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":"4a6e64adc3ad0be1f202d1fcf31098721989604e0db29cc1110518f0d9df61cf","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-07-18T23:23:43Z","title_canon_sha256":"6c9ef7f8812a83b92a429cfb1f4919d75206beaeedde381adbfb35c4544e73d0"},"schema_version":"1.0","source":{"id":"0907.3247","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0907.3247","created_at":"2026-05-18T03:46:32Z"},{"alias_kind":"arxiv_version","alias_value":"0907.3247v2","created_at":"2026-05-18T03:46:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.3247","created_at":"2026-05-18T03:46:32Z"},{"alias_kind":"pith_short_12","alias_value":"NZSVVSQ5CMAB","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"NZSVVSQ5CMABAI2B","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"NZSVVSQ5","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:dab04412329e6de09299c8cabc8e77fa390baa8badf8ea1968254f46841eb5d3","target":"graph","created_at":"2026-05-18T03:46:32Z","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":"Branching of symplectic groups is not multiplicity-free.  We describe a new approach to resolving these multiplicities that is based on studying the associated branching algebra $B$. The algebra $B$ is a graded algebra whose components encode the multiplicities of irreducible representations of $Sp_{2n-2}$ in irreducible representations of $Sp_{2n}$. Our first theorem states that the map taking an element of $Sp_{2n}$ to its principal $n \\times (n+1)$ submatrix induces an isomorphism of $\\B$ to a different branching algebra $\\B'$.  The algebra $\\B'$ encodes multiplicities of irreducible repres","authors_text":"Oded Yacobi","cross_cats":["math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-07-18T23:23:43Z","title":"An Analysis of the Multiplicity Spaces in Branching of Symplectic Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.3247","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":"sha256:3257456567b42756f1e8818dd8095d046e9192acacdc2a400470a6c841832039","target":"record","created_at":"2026-05-18T03:46:32Z","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":"4a6e64adc3ad0be1f202d1fcf31098721989604e0db29cc1110518f0d9df61cf","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-07-18T23:23:43Z","title_canon_sha256":"6c9ef7f8812a83b92a429cfb1f4919d75206beaeedde381adbfb35c4544e73d0"},"schema_version":"1.0","source":{"id":"0907.3247","kind":"arxiv","version":2}},"canonical_sha256":"6e655aca1d1300102341f963e5dc1b6df2dda966e457ef146fe1119646b29a8a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6e655aca1d1300102341f963e5dc1b6df2dda966e457ef146fe1119646b29a8a","first_computed_at":"2026-05-18T03:46:32.482876Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:46:32.482876Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kWnjSRYynU/1PFBF9zDkUQoDf8SmPzvU9LB+UYlSms1MId0sopJQXV9d2QVqtD99aGi8t/+Q/I6sxKmy9V4xDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:46:32.483667Z","signed_message":"canonical_sha256_bytes"},"source_id":"0907.3247","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3257456567b42756f1e8818dd8095d046e9192acacdc2a400470a6c841832039","sha256:dab04412329e6de09299c8cabc8e77fa390baa8badf8ea1968254f46841eb5d3"],"state_sha256":"366c28ec8e32193faf31cc1e4a9206c7a90a196d57aea915122d9cdc3c08faf5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ocIstobUnpqy0FlhyF+TuenzCUn4eFpn/Es4ac6njUl+mihKD2+Y7NvSnSZjLsQ7e+xPQ3PKTFkTkhr5ylZCCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T16:25:39.755858Z","bundle_sha256":"c001403f614adc419b4180bf4539026156622f5b3853e2c303fbdae43ec43b0f"}}