{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:PFSJL3ST6VEFUFNCFUM45IETYK","short_pith_number":"pith:PFSJL3ST","canonical_record":{"source":{"id":"1201.3776","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-01-18T13:02:32Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"e50ffc17f9cba30be425cd113c2f1ab4ede829e7d8e1a00356ca14ca3fd96bab","abstract_canon_sha256":"7d95345802984b6aabf615b19cdbd332e900f2461d3c54c66421635dd771acbd"},"schema_version":"1.0"},"canonical_sha256":"796495ee53f5485a15a22d19cea093c2b768e5a9cbe6669374550bf3fa6922fb","source":{"kind":"arxiv","id":"1201.3776","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.3776","created_at":"2026-05-18T04:04:18Z"},{"alias_kind":"arxiv_version","alias_value":"1201.3776v1","created_at":"2026-05-18T04:04:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.3776","created_at":"2026-05-18T04:04:18Z"},{"alias_kind":"pith_short_12","alias_value":"PFSJL3ST6VEF","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"PFSJL3ST6VEFUFNC","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"PFSJL3ST","created_at":"2026-05-18T12:27:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:PFSJL3ST6VEFUFNCFUM45IETYK","target":"record","payload":{"canonical_record":{"source":{"id":"1201.3776","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-01-18T13:02:32Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"e50ffc17f9cba30be425cd113c2f1ab4ede829e7d8e1a00356ca14ca3fd96bab","abstract_canon_sha256":"7d95345802984b6aabf615b19cdbd332e900f2461d3c54c66421635dd771acbd"},"schema_version":"1.0"},"canonical_sha256":"796495ee53f5485a15a22d19cea093c2b768e5a9cbe6669374550bf3fa6922fb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:04:18.055804Z","signature_b64":"4TtbhKkA8BKYhqdnhURUKscV5MjkEmEMh8ouu7a5j6n/CiHdE9iYTcte57VvcrdMKGpFsKDBDKTyguw63czYBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"796495ee53f5485a15a22d19cea093c2b768e5a9cbe6669374550bf3fa6922fb","last_reissued_at":"2026-05-18T04:04:18.054993Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:04:18.054993Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1201.3776","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":"2026-05-18T04:04:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"75aZm3djkRRTwrdaJe004CVonxAn5HXHBFthrMOqp9zLYWiJzIMndE7sOsYVkS0bAzOEjIA9HV4M4NU/2KovDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T15:00:36.736182Z"},"content_sha256":"84f13314e6ec12bfdd0ddd64a24c4ab0e5a512f84f8a66bba604c298cc251d2b","schema_version":"1.0","event_id":"sha256:84f13314e6ec12bfdd0ddd64a24c4ab0e5a512f84f8a66bba604c298cc251d2b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:PFSJL3ST6VEFUFNCFUM45IETYK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CA","authors_text":"Michael Voit","submitted_at":"2012-01-18T13:02:32Z","abstract_excerpt":"Let $p,q$ positive integers. The groups $U_p(\\b C)$ and $U_p(\\b C)\\times U_q(\\b C) $ act on the Heisenberg group $H_{p,q}:=M_{p,q}(\\b C)\\times \\b R$ canonically as groups of automorphisms where $M_{p,q}(\\b C)$ is the vector space of all complex $p\\times q$-matrices. The associated orbit spaces may be identified with $\\Pi_q\\times \\b R$ and $\\Xi_q\\times \\b R$ respectively with the cone $\\Pi_q$ of positive semidefinite matrices and the Weyl chamber $\\Xi_q={x\\in\\b R^q: x_1\\ge...\\ge x_q\\ge 0}$.\n  In this paper we compute the associated convolutions on $\\Pi_q\\times \\b R$ and $\\Xi_q\\times \\b R$ expli"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.3776","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":"2026-05-18T04:04:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Jt91tkBM4QZdC8xuJLHrth1cSLwaFTpYaPK6sxPZlP+56+0yapX4XO5sQMJTNg5Bob1I7N1IMcHBgc1qYd+gDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T15:00:36.736539Z"},"content_sha256":"a93f3c3770d22dd902054281934345f02434727fc35d31a3a4279e0399699f03","schema_version":"1.0","event_id":"sha256:a93f3c3770d22dd902054281934345f02434727fc35d31a3a4279e0399699f03"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PFSJL3ST6VEFUFNCFUM45IETYK/bundle.json","state_url":"https://pith.science/pith/PFSJL3ST6VEFUFNCFUM45IETYK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PFSJL3ST6VEFUFNCFUM45IETYK/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":"2026-06-08T15:00:36Z","links":{"resolver":"https://pith.science/pith/PFSJL3ST6VEFUFNCFUM45IETYK","bundle":"https://pith.science/pith/PFSJL3ST6VEFUFNCFUM45IETYK/bundle.json","state":"https://pith.science/pith/PFSJL3ST6VEFUFNCFUM45IETYK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PFSJL3ST6VEFUFNCFUM45IETYK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:PFSJL3ST6VEFUFNCFUM45IETYK","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":"7d95345802984b6aabf615b19cdbd332e900f2461d3c54c66421635dd771acbd","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-01-18T13:02:32Z","title_canon_sha256":"e50ffc17f9cba30be425cd113c2f1ab4ede829e7d8e1a00356ca14ca3fd96bab"},"schema_version":"1.0","source":{"id":"1201.3776","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.3776","created_at":"2026-05-18T04:04:18Z"},{"alias_kind":"arxiv_version","alias_value":"1201.3776v1","created_at":"2026-05-18T04:04:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.3776","created_at":"2026-05-18T04:04:18Z"},{"alias_kind":"pith_short_12","alias_value":"PFSJL3ST6VEF","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"PFSJL3ST6VEFUFNC","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"PFSJL3ST","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:a93f3c3770d22dd902054281934345f02434727fc35d31a3a4279e0399699f03","target":"graph","created_at":"2026-05-18T04:04:18Z","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":"Let $p,q$ positive integers. The groups $U_p(\\b C)$ and $U_p(\\b C)\\times U_q(\\b C) $ act on the Heisenberg group $H_{p,q}:=M_{p,q}(\\b C)\\times \\b R$ canonically as groups of automorphisms where $M_{p,q}(\\b C)$ is the vector space of all complex $p\\times q$-matrices. The associated orbit spaces may be identified with $\\Pi_q\\times \\b R$ and $\\Xi_q\\times \\b R$ respectively with the cone $\\Pi_q$ of positive semidefinite matrices and the Weyl chamber $\\Xi_q={x\\in\\b R^q: x_1\\ge...\\ge x_q\\ge 0}$.\n  In this paper we compute the associated convolutions on $\\Pi_q\\times \\b R$ and $\\Xi_q\\times \\b R$ expli","authors_text":"Michael Voit","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-01-18T13:02:32Z","title":"Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.3776","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":"sha256:84f13314e6ec12bfdd0ddd64a24c4ab0e5a512f84f8a66bba604c298cc251d2b","target":"record","created_at":"2026-05-18T04:04:18Z","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":"7d95345802984b6aabf615b19cdbd332e900f2461d3c54c66421635dd771acbd","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-01-18T13:02:32Z","title_canon_sha256":"e50ffc17f9cba30be425cd113c2f1ab4ede829e7d8e1a00356ca14ca3fd96bab"},"schema_version":"1.0","source":{"id":"1201.3776","kind":"arxiv","version":1}},"canonical_sha256":"796495ee53f5485a15a22d19cea093c2b768e5a9cbe6669374550bf3fa6922fb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"796495ee53f5485a15a22d19cea093c2b768e5a9cbe6669374550bf3fa6922fb","first_computed_at":"2026-05-18T04:04:18.054993Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:04:18.054993Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4TtbhKkA8BKYhqdnhURUKscV5MjkEmEMh8ouu7a5j6n/CiHdE9iYTcte57VvcrdMKGpFsKDBDKTyguw63czYBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:04:18.055804Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.3776","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:84f13314e6ec12bfdd0ddd64a24c4ab0e5a512f84f8a66bba604c298cc251d2b","sha256:a93f3c3770d22dd902054281934345f02434727fc35d31a3a4279e0399699f03"],"state_sha256":"ff73efc36a19613a68795eef945947895c3ef26501220a65e3b4193a69c85240"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tQQv6S8YymqzHh4OYnzvUNVpYIXGLcC2VXiXNXGkycNDdLnrRL8ok2l1HK+p9De57mO7LEoeNuKctwOmUWg5DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T15:00:36.738431Z","bundle_sha256":"366a26f6df5848f1633c605a9957f89d817153c41bd4e843fd68c2cf37c29a0f"}}