{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:64NUFBHAYL6BK4FTN5RGA2DVRX","short_pith_number":"pith:64NUFBHA","canonical_record":{"source":{"id":"1806.06067","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-06-14T21:24:49Z","cross_cats_sorted":[],"title_canon_sha256":"59b1bd47c74cc7b46dc9f048f546d26d32f67821ed783ebebd3e316962ab5d27","abstract_canon_sha256":"5dbd2e419f467d8b6bc5bdc7ae0e3c76cd9923fa9564195ac61d3d5befacbe7a"},"schema_version":"1.0"},"canonical_sha256":"f71b4284e0c2fc1570b36f626068758dd0a69d91d5f4ed9723cbd2f1a5348a91","source":{"kind":"arxiv","id":"1806.06067","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.06067","created_at":"2026-05-18T00:13:03Z"},{"alias_kind":"arxiv_version","alias_value":"1806.06067v1","created_at":"2026-05-18T00:13:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.06067","created_at":"2026-05-18T00:13:03Z"},{"alias_kind":"pith_short_12","alias_value":"64NUFBHAYL6B","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_16","alias_value":"64NUFBHAYL6BK4FT","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_8","alias_value":"64NUFBHA","created_at":"2026-05-18T12:32:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:64NUFBHAYL6BK4FTN5RGA2DVRX","target":"record","payload":{"canonical_record":{"source":{"id":"1806.06067","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-06-14T21:24:49Z","cross_cats_sorted":[],"title_canon_sha256":"59b1bd47c74cc7b46dc9f048f546d26d32f67821ed783ebebd3e316962ab5d27","abstract_canon_sha256":"5dbd2e419f467d8b6bc5bdc7ae0e3c76cd9923fa9564195ac61d3d5befacbe7a"},"schema_version":"1.0"},"canonical_sha256":"f71b4284e0c2fc1570b36f626068758dd0a69d91d5f4ed9723cbd2f1a5348a91","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:03.742884Z","signature_b64":"mT5UExYrMBf51IChqv75mDe/nGmrLZo3gn1p/yDOeXIC1sxEMCWfjOdxxkxYA/luaFJu6fD0C8ABKgacHFmvBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f71b4284e0c2fc1570b36f626068758dd0a69d91d5f4ed9723cbd2f1a5348a91","last_reissued_at":"2026-05-18T00:13:03.742262Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:03.742262Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1806.06067","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":"2026-05-18T00:13:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wNnrU4p7EQqAunTfd3wK/wJ407WbXlbX4O70Z2/Fl0T3Wd42OchacOPMyNJb9OdI2YZ5KAeh0oNbmCv7Beb3DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T01:08:27.671360Z"},"content_sha256":"243ff14cf029bedd85c0b31d6c6d1cb66936aabadf007cec5df0acae0f47b3bb","schema_version":"1.0","event_id":"sha256:243ff14cf029bedd85c0b31d6c6d1cb66936aabadf007cec5df0acae0f47b3bb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:64NUFBHAYL6BK4FTN5RGA2DVRX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Fourier transform of a projective group frame","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Shayne Waldron","submitted_at":"2018-06-14T21:24:49Z","abstract_excerpt":"Many tight frames of interest are constructed via their Gramian matrix (which determines the frame up to unitary equivalence). Given such a Gramian, it can be determined whether or not the tight frame is projective group frame, i.e., is the projective orbit of some group $G$ (which may not be unique). On the other hand, there is complete description of the projective group frames in terms of the irreducible projective representations of $G$. Here we consider the inverse problem of taking the Gramian of a projective group frame for a group $G$, and identifying the cocycle and constructing the f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.06067","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":"2026-05-18T00:13:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4HfYhR9i+jZu5ucMrQQ34rS9jlQ+doPbKEm6eoakyVsgfHvAUWR7Z9mrrJGWdNyv3tmW0RS8TCyYNChT6BwPAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T01:08:27.671711Z"},"content_sha256":"243581b2c028888708ef8bcc181e6784341f21ba2e00d1034c33922476023352","schema_version":"1.0","event_id":"sha256:243581b2c028888708ef8bcc181e6784341f21ba2e00d1034c33922476023352"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/64NUFBHAYL6BK4FTN5RGA2DVRX/bundle.json","state_url":"https://pith.science/pith/64NUFBHAYL6BK4FTN5RGA2DVRX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/64NUFBHAYL6BK4FTN5RGA2DVRX/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":"2026-07-03T01:08:27Z","links":{"resolver":"https://pith.science/pith/64NUFBHAYL6BK4FTN5RGA2DVRX","bundle":"https://pith.science/pith/64NUFBHAYL6BK4FTN5RGA2DVRX/bundle.json","state":"https://pith.science/pith/64NUFBHAYL6BK4FTN5RGA2DVRX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/64NUFBHAYL6BK4FTN5RGA2DVRX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:64NUFBHAYL6BK4FTN5RGA2DVRX","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":"5dbd2e419f467d8b6bc5bdc7ae0e3c76cd9923fa9564195ac61d3d5befacbe7a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-06-14T21:24:49Z","title_canon_sha256":"59b1bd47c74cc7b46dc9f048f546d26d32f67821ed783ebebd3e316962ab5d27"},"schema_version":"1.0","source":{"id":"1806.06067","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.06067","created_at":"2026-05-18T00:13:03Z"},{"alias_kind":"arxiv_version","alias_value":"1806.06067v1","created_at":"2026-05-18T00:13:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.06067","created_at":"2026-05-18T00:13:03Z"},{"alias_kind":"pith_short_12","alias_value":"64NUFBHAYL6B","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_16","alias_value":"64NUFBHAYL6BK4FT","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_8","alias_value":"64NUFBHA","created_at":"2026-05-18T12:32:08Z"}],"graph_snapshots":[{"event_id":"sha256:243581b2c028888708ef8bcc181e6784341f21ba2e00d1034c33922476023352","target":"graph","created_at":"2026-05-18T00:13:03Z","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":"Many tight frames of interest are constructed via their Gramian matrix (which determines the frame up to unitary equivalence). Given such a Gramian, it can be determined whether or not the tight frame is projective group frame, i.e., is the projective orbit of some group $G$ (which may not be unique). On the other hand, there is complete description of the projective group frames in terms of the irreducible projective representations of $G$. Here we consider the inverse problem of taking the Gramian of a projective group frame for a group $G$, and identifying the cocycle and constructing the f","authors_text":"Shayne Waldron","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-06-14T21:24:49Z","title":"The Fourier transform of a projective group frame"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.06067","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":"sha256:243ff14cf029bedd85c0b31d6c6d1cb66936aabadf007cec5df0acae0f47b3bb","target":"record","created_at":"2026-05-18T00:13:03Z","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":"5dbd2e419f467d8b6bc5bdc7ae0e3c76cd9923fa9564195ac61d3d5befacbe7a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-06-14T21:24:49Z","title_canon_sha256":"59b1bd47c74cc7b46dc9f048f546d26d32f67821ed783ebebd3e316962ab5d27"},"schema_version":"1.0","source":{"id":"1806.06067","kind":"arxiv","version":1}},"canonical_sha256":"f71b4284e0c2fc1570b36f626068758dd0a69d91d5f4ed9723cbd2f1a5348a91","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f71b4284e0c2fc1570b36f626068758dd0a69d91d5f4ed9723cbd2f1a5348a91","first_computed_at":"2026-05-18T00:13:03.742262Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:13:03.742262Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mT5UExYrMBf51IChqv75mDe/nGmrLZo3gn1p/yDOeXIC1sxEMCWfjOdxxkxYA/luaFJu6fD0C8ABKgacHFmvBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:13:03.742884Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.06067","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:243ff14cf029bedd85c0b31d6c6d1cb66936aabadf007cec5df0acae0f47b3bb","sha256:243581b2c028888708ef8bcc181e6784341f21ba2e00d1034c33922476023352"],"state_sha256":"79d7de3853dc3898b6f340caee362096f3ace88502e324db126003c66776daa6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6qInYoGHw7SkxifLwUm0c4kPw/OfxXOMFnXaZzr3IofZ4hdUL3a4dKPIyTr6imQVpBSFvtXCZvFV+l653ck8Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T01:08:27.673710Z","bundle_sha256":"40e00ab74f70a0ef917679d1242365500a1952ddd8f4f660c8c374a17e39a3d6"}}