{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:SCHMUSXHQAEMOXVSUNH2NIE6GY","short_pith_number":"pith:SCHMUSXH","canonical_record":{"source":{"id":"1209.3970","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-18T14:11:16Z","cross_cats_sorted":[],"title_canon_sha256":"a13c368f314f909e8901227d8176b10a03d4b44b737550cf415ae8118448fe11","abstract_canon_sha256":"21f75240275cf9f1ec30e6983c7fbf95d3c9c5a647d7fefa3d359e02b1bf65a4"},"schema_version":"1.0"},"canonical_sha256":"908eca4ae78008c75eb2a34fa6a09e36352ddab9dae5b6ac69aab42cb8c4bb8b","source":{"kind":"arxiv","id":"1209.3970","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.3970","created_at":"2026-05-18T03:38:55Z"},{"alias_kind":"arxiv_version","alias_value":"1209.3970v3","created_at":"2026-05-18T03:38:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3970","created_at":"2026-05-18T03:38:55Z"},{"alias_kind":"pith_short_12","alias_value":"SCHMUSXHQAEM","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"SCHMUSXHQAEMOXVS","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"SCHMUSXH","created_at":"2026-05-18T12:27:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:SCHMUSXHQAEMOXVSUNH2NIE6GY","target":"record","payload":{"canonical_record":{"source":{"id":"1209.3970","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-18T14:11:16Z","cross_cats_sorted":[],"title_canon_sha256":"a13c368f314f909e8901227d8176b10a03d4b44b737550cf415ae8118448fe11","abstract_canon_sha256":"21f75240275cf9f1ec30e6983c7fbf95d3c9c5a647d7fefa3d359e02b1bf65a4"},"schema_version":"1.0"},"canonical_sha256":"908eca4ae78008c75eb2a34fa6a09e36352ddab9dae5b6ac69aab42cb8c4bb8b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:38:55.646396Z","signature_b64":"dpfP9hhaZHqgDye2mAvGBpCO6wbROAL/mbxb43+6aQfk+r1sNRtOFb12eP0S+nU8iV1aA/fSdFtGJgFjdBsrAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"908eca4ae78008c75eb2a34fa6a09e36352ddab9dae5b6ac69aab42cb8c4bb8b","last_reissued_at":"2026-05-18T03:38:55.645783Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:38:55.645783Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.3970","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:38:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f8U/9ec2FT9Zvf5kHUmpnkw2pCR9qgPkw3PeLgnW1OyiNNyAeUBXZass5Er2VsZ91IPs2OyT2iR1x3pcuPJnCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T04:51:09.296063Z"},"content_sha256":"5c277bfb969890ed3e0273fe7426a45baf041c7898d839fd90234676d6a59ce8","schema_version":"1.0","event_id":"sha256:5c277bfb969890ed3e0273fe7426a45baf041c7898d839fd90234676d6a59ce8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:SCHMUSXHQAEMOXVSUNH2NIE6GY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The branching problem for generalized Verma modules, with application to the pair $(so(7),Lie G_2)$, extended version with tables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Petr Somberg, Todor Milev","submitted_at":"2012-09-18T14:11:16Z","abstract_excerpt":"We discuss the branching problem for generalized Verma modules $M_\\lambda(\\mathfrak g, \\mathfrak p)$ applied to couples of reductive Lie algebras $\\bar{\\mathfrak g}\\hookrightarrow \\mathfrak g$. The analysis is based on projecting character formulas to quantify the branching, and on the action of the center of $U(\\bar{\\mathfrak g})$ to explicitly construct singular vectors realizing part of the branching. We demonstrate the results on the pair $\\mathrm{Lie}G_2\\hookrightarrow{so(7)}$ for both strongly and weakly compatible with $i(\\mathrm {Lie} G_2)$ parabolic subalgebras and a large class of in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3970","kind":"arxiv","version":3},"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":"2026-05-18T03:38:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u4Hh732GOCoPsV1kHPk7iNz3hivii0YqXHvEVVjabPJv4OqfY84dX9zJfVaCj/RGGF+1n6ZtozToc8QHSB1wBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T04:51:09.296441Z"},"content_sha256":"e07d8b1a9304386912db30747c88ecf77f924714358632b4c165e3830f9fc3a6","schema_version":"1.0","event_id":"sha256:e07d8b1a9304386912db30747c88ecf77f924714358632b4c165e3830f9fc3a6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SCHMUSXHQAEMOXVSUNH2NIE6GY/bundle.json","state_url":"https://pith.science/pith/SCHMUSXHQAEMOXVSUNH2NIE6GY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SCHMUSXHQAEMOXVSUNH2NIE6GY/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":"2026-06-03T04:51:09Z","links":{"resolver":"https://pith.science/pith/SCHMUSXHQAEMOXVSUNH2NIE6GY","bundle":"https://pith.science/pith/SCHMUSXHQAEMOXVSUNH2NIE6GY/bundle.json","state":"https://pith.science/pith/SCHMUSXHQAEMOXVSUNH2NIE6GY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SCHMUSXHQAEMOXVSUNH2NIE6GY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:SCHMUSXHQAEMOXVSUNH2NIE6GY","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":"21f75240275cf9f1ec30e6983c7fbf95d3c9c5a647d7fefa3d359e02b1bf65a4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-18T14:11:16Z","title_canon_sha256":"a13c368f314f909e8901227d8176b10a03d4b44b737550cf415ae8118448fe11"},"schema_version":"1.0","source":{"id":"1209.3970","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.3970","created_at":"2026-05-18T03:38:55Z"},{"alias_kind":"arxiv_version","alias_value":"1209.3970v3","created_at":"2026-05-18T03:38:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3970","created_at":"2026-05-18T03:38:55Z"},{"alias_kind":"pith_short_12","alias_value":"SCHMUSXHQAEM","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"SCHMUSXHQAEMOXVS","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"SCHMUSXH","created_at":"2026-05-18T12:27:20Z"}],"graph_snapshots":[{"event_id":"sha256:e07d8b1a9304386912db30747c88ecf77f924714358632b4c165e3830f9fc3a6","target":"graph","created_at":"2026-05-18T03:38:55Z","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 discuss the branching problem for generalized Verma modules $M_\\lambda(\\mathfrak g, \\mathfrak p)$ applied to couples of reductive Lie algebras $\\bar{\\mathfrak g}\\hookrightarrow \\mathfrak g$. The analysis is based on projecting character formulas to quantify the branching, and on the action of the center of $U(\\bar{\\mathfrak g})$ to explicitly construct singular vectors realizing part of the branching. We demonstrate the results on the pair $\\mathrm{Lie}G_2\\hookrightarrow{so(7)}$ for both strongly and weakly compatible with $i(\\mathrm {Lie} G_2)$ parabolic subalgebras and a large class of in","authors_text":"Petr Somberg, Todor Milev","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-18T14:11:16Z","title":"The branching problem for generalized Verma modules, with application to the pair $(so(7),Lie G_2)$, extended version with tables"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3970","kind":"arxiv","version":3},"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":"sha256:5c277bfb969890ed3e0273fe7426a45baf041c7898d839fd90234676d6a59ce8","target":"record","created_at":"2026-05-18T03:38:55Z","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":"21f75240275cf9f1ec30e6983c7fbf95d3c9c5a647d7fefa3d359e02b1bf65a4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-18T14:11:16Z","title_canon_sha256":"a13c368f314f909e8901227d8176b10a03d4b44b737550cf415ae8118448fe11"},"schema_version":"1.0","source":{"id":"1209.3970","kind":"arxiv","version":3}},"canonical_sha256":"908eca4ae78008c75eb2a34fa6a09e36352ddab9dae5b6ac69aab42cb8c4bb8b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"908eca4ae78008c75eb2a34fa6a09e36352ddab9dae5b6ac69aab42cb8c4bb8b","first_computed_at":"2026-05-18T03:38:55.645783Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:38:55.645783Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dpfP9hhaZHqgDye2mAvGBpCO6wbROAL/mbxb43+6aQfk+r1sNRtOFb12eP0S+nU8iV1aA/fSdFtGJgFjdBsrAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:38:55.646396Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.3970","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5c277bfb969890ed3e0273fe7426a45baf041c7898d839fd90234676d6a59ce8","sha256:e07d8b1a9304386912db30747c88ecf77f924714358632b4c165e3830f9fc3a6"],"state_sha256":"6554ce6a5fab9dad947212f75819ea2efe35bf8ee26b18693224f63971b75e65"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u1t+iEGezCN044H0CdvqMWd5cukZLmgwkkVyZPc85HNiFczPr4bZah60G0+s1h8ZwX2GdsAqpE8QfGzFzYSiBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T04:51:09.298418Z","bundle_sha256":"f36bab32ece8c9c8f035907cedf7db26c102d1ad4d9f6e74bbd7d6379454c725"}}