{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:6VPFU4BUSS5OU7VBAC22VG4T5Z","short_pith_number":"pith:6VPFU4BU","canonical_record":{"source":{"id":"1209.4074","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-18T19:53:08Z","cross_cats_sorted":[],"title_canon_sha256":"e7d7ae9c486332e57043770ffb1cbbdacecc5571722995b1bf483e612d126e3c","abstract_canon_sha256":"d627f5d875e254f16261ef29110b30b34de0f28381a36470f89ee05e8e8137d4"},"schema_version":"1.0"},"canonical_sha256":"f55e5a703494baea7ea100b5aa9b93ee62bebd6cf3240ae78a78141625f21523","source":{"kind":"arxiv","id":"1209.4074","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.4074","created_at":"2026-05-18T03:45:21Z"},{"alias_kind":"arxiv_version","alias_value":"1209.4074v1","created_at":"2026-05-18T03:45:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.4074","created_at":"2026-05-18T03:45:21Z"},{"alias_kind":"pith_short_12","alias_value":"6VPFU4BUSS5O","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"6VPFU4BUSS5OU7VB","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"6VPFU4BU","created_at":"2026-05-18T12:26:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:6VPFU4BUSS5OU7VBAC22VG4T5Z","target":"record","payload":{"canonical_record":{"source":{"id":"1209.4074","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-18T19:53:08Z","cross_cats_sorted":[],"title_canon_sha256":"e7d7ae9c486332e57043770ffb1cbbdacecc5571722995b1bf483e612d126e3c","abstract_canon_sha256":"d627f5d875e254f16261ef29110b30b34de0f28381a36470f89ee05e8e8137d4"},"schema_version":"1.0"},"canonical_sha256":"f55e5a703494baea7ea100b5aa9b93ee62bebd6cf3240ae78a78141625f21523","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:45:21.841047Z","signature_b64":"gmoNyhyYH0aSiFFEtw0uxNtr6HsJIIMZKoJFu5A9HGHEDT/+cETaG/hzwQq900+v65BCJwB/EDDOyuZrrzUPAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f55e5a703494baea7ea100b5aa9b93ee62bebd6cf3240ae78a78141625f21523","last_reissued_at":"2026-05-18T03:45:21.840491Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:45:21.840491Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.4074","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":"2026-05-18T03:45:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J9Fe1JC49irXpV0O52P8G9bBIzBPA6PcnZy60w1R4A7cEC7quTAKV6GTZhiAsvKbjbg0tKCkw/xiLF6cDNocAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T15:12:19.207645Z"},"content_sha256":"dfd444a44abee9cfe859291f4dc56330f83d93d9af657a42771749eb63ec781f","schema_version":"1.0","event_id":"sha256:dfd444a44abee9cfe859291f4dc56330f83d93d9af657a42771749eb63ec781f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:6VPFU4BUSS5OU7VBAC22VG4T5Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Representations of The miraculous Klein group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Jan Minac, Sunil K. Chebolu","submitted_at":"2012-09-18T19:53:08Z","abstract_excerpt":"The Klein group contains only four elements. Nevertheless this little group contains a number of remarkable entry points to current highways of modern representation theory of groups. In this paper, we shall describe all possible ways in which the Klein group can act on vector spaces over a field of two elements. These are called representations of the Klein group. This description involves some powerful visual methods of representation theory which builds on the work of generations of mathematicians starting roughly with the work of K. Weiestrass. We also discuss some applications to properti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.4074","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":"2026-05-18T03:45:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yRmEBavRC0Xcs1W3BBdYxLiPgVwDSHAoHyCr7E9OXrOxswq8J0AWqvNh8JtXt5uYioqIb0epCsFS2uPCxbv0CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T15:12:19.207983Z"},"content_sha256":"8d4ff4a42aa339e20b970c3bc7c88bae670fb87d0b562d64fd16d8ebed2ea9d0","schema_version":"1.0","event_id":"sha256:8d4ff4a42aa339e20b970c3bc7c88bae670fb87d0b562d64fd16d8ebed2ea9d0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6VPFU4BUSS5OU7VBAC22VG4T5Z/bundle.json","state_url":"https://pith.science/pith/6VPFU4BUSS5OU7VBAC22VG4T5Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6VPFU4BUSS5OU7VBAC22VG4T5Z/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":"2026-06-27T15:12:19Z","links":{"resolver":"https://pith.science/pith/6VPFU4BUSS5OU7VBAC22VG4T5Z","bundle":"https://pith.science/pith/6VPFU4BUSS5OU7VBAC22VG4T5Z/bundle.json","state":"https://pith.science/pith/6VPFU4BUSS5OU7VBAC22VG4T5Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6VPFU4BUSS5OU7VBAC22VG4T5Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:6VPFU4BUSS5OU7VBAC22VG4T5Z","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":"d627f5d875e254f16261ef29110b30b34de0f28381a36470f89ee05e8e8137d4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-18T19:53:08Z","title_canon_sha256":"e7d7ae9c486332e57043770ffb1cbbdacecc5571722995b1bf483e612d126e3c"},"schema_version":"1.0","source":{"id":"1209.4074","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.4074","created_at":"2026-05-18T03:45:21Z"},{"alias_kind":"arxiv_version","alias_value":"1209.4074v1","created_at":"2026-05-18T03:45:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.4074","created_at":"2026-05-18T03:45:21Z"},{"alias_kind":"pith_short_12","alias_value":"6VPFU4BUSS5O","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"6VPFU4BUSS5OU7VB","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"6VPFU4BU","created_at":"2026-05-18T12:26:56Z"}],"graph_snapshots":[{"event_id":"sha256:8d4ff4a42aa339e20b970c3bc7c88bae670fb87d0b562d64fd16d8ebed2ea9d0","target":"graph","created_at":"2026-05-18T03:45:21Z","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 Klein group contains only four elements. Nevertheless this little group contains a number of remarkable entry points to current highways of modern representation theory of groups. In this paper, we shall describe all possible ways in which the Klein group can act on vector spaces over a field of two elements. These are called representations of the Klein group. This description involves some powerful visual methods of representation theory which builds on the work of generations of mathematicians starting roughly with the work of K. Weiestrass. We also discuss some applications to properti","authors_text":"Jan Minac, Sunil K. Chebolu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-18T19:53:08Z","title":"Representations of The miraculous Klein group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.4074","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:dfd444a44abee9cfe859291f4dc56330f83d93d9af657a42771749eb63ec781f","target":"record","created_at":"2026-05-18T03:45:21Z","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":"d627f5d875e254f16261ef29110b30b34de0f28381a36470f89ee05e8e8137d4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-18T19:53:08Z","title_canon_sha256":"e7d7ae9c486332e57043770ffb1cbbdacecc5571722995b1bf483e612d126e3c"},"schema_version":"1.0","source":{"id":"1209.4074","kind":"arxiv","version":1}},"canonical_sha256":"f55e5a703494baea7ea100b5aa9b93ee62bebd6cf3240ae78a78141625f21523","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f55e5a703494baea7ea100b5aa9b93ee62bebd6cf3240ae78a78141625f21523","first_computed_at":"2026-05-18T03:45:21.840491Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:45:21.840491Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gmoNyhyYH0aSiFFEtw0uxNtr6HsJIIMZKoJFu5A9HGHEDT/+cETaG/hzwQq900+v65BCJwB/EDDOyuZrrzUPAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:45:21.841047Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.4074","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dfd444a44abee9cfe859291f4dc56330f83d93d9af657a42771749eb63ec781f","sha256:8d4ff4a42aa339e20b970c3bc7c88bae670fb87d0b562d64fd16d8ebed2ea9d0"],"state_sha256":"085defde2023a4a2fbc3d773032fe11c49c78d380cd6d5d42a041ad947346eae"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"45VNqUld54RsWv95RON1+0fveWDz1qzwMkJY4GsLK0PiylrhSeG48zJYaJbC7yhqkUnkTVHIBjQGXKH/K1mqAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T15:12:19.209858Z","bundle_sha256":"f6a81f388a916e0beea3e4df5a96c12310a6f8883640d802665f48ce92ba8fd1"}}