{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:ZH3QFQBMFF4J63ISR2YD5CIFG3","short_pith_number":"pith:ZH3QFQBM","canonical_record":{"source":{"id":"1304.4645","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-04-16T23:41:48Z","cross_cats_sorted":[],"title_canon_sha256":"ead4af8c6a23bd988358381f3a54a616c51f04c09cffa353e283041ba5997480","abstract_canon_sha256":"9ce8f8afa08c304db77ec7b63cbaea01eb0e0b9a72abe95ca238350a1f015406"},"schema_version":"1.0"},"canonical_sha256":"c9f702c02c29789f6d128eb03e890536c796d359e3d76abb6ab4f1fc2c9d8ecb","source":{"kind":"arxiv","id":"1304.4645","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.4645","created_at":"2026-05-18T03:19:01Z"},{"alias_kind":"arxiv_version","alias_value":"1304.4645v2","created_at":"2026-05-18T03:19:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.4645","created_at":"2026-05-18T03:19:01Z"},{"alias_kind":"pith_short_12","alias_value":"ZH3QFQBMFF4J","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"ZH3QFQBMFF4J63IS","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"ZH3QFQBM","created_at":"2026-05-18T12:28:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:ZH3QFQBMFF4J63ISR2YD5CIFG3","target":"record","payload":{"canonical_record":{"source":{"id":"1304.4645","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-04-16T23:41:48Z","cross_cats_sorted":[],"title_canon_sha256":"ead4af8c6a23bd988358381f3a54a616c51f04c09cffa353e283041ba5997480","abstract_canon_sha256":"9ce8f8afa08c304db77ec7b63cbaea01eb0e0b9a72abe95ca238350a1f015406"},"schema_version":"1.0"},"canonical_sha256":"c9f702c02c29789f6d128eb03e890536c796d359e3d76abb6ab4f1fc2c9d8ecb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:19:01.173569Z","signature_b64":"X13vaAgzuU+HzLR2iU17tjaIPZ5db8PmkOG3ZVBKa8oJ9cZnpbGw2ZVBU2TKbXfJRioERj9BDqU6kfMBVjjvAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9f702c02c29789f6d128eb03e890536c796d359e3d76abb6ab4f1fc2c9d8ecb","last_reissued_at":"2026-05-18T03:19:01.172840Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:19:01.172840Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1304.4645","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":"2026-05-18T03:19:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dy+8pU+Bn30Gb2rILTsUaJAGqtqKjVUYWBteohNn3Z7kUMCkq8YGWuJVCxL1BJzT8vLVOkDyPPiiLa5NSdORDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T22:57:43.940179Z"},"content_sha256":"b483d485e4c475a8b751983d70b7339e26b33ddbd026ccdeee418337574d36b3","schema_version":"1.0","event_id":"sha256:b483d485e4c475a8b751983d70b7339e26b33ddbd026ccdeee418337574d36b3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:ZH3QFQBMFF4J63ISR2YD5CIFG3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Action of the Symmetric Group on the Cohomology of Groups Related to (Virtual) Braids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Peter Lee","submitted_at":"2013-04-16T23:41:48Z","abstract_excerpt":"In this paper we consider the cohomology of four groups related to the virtual braids of [Kauffman] and [Goussarov-Polyak-Viro], namely the pure and non-pure virtual braid groups (PvB_n and vB_n, respectively), and the pure and non-pure flat braid groups (PfB_n and fB_n, respectively). The cohomologies of PvB_n and PfB_n admit an action of the symmetric group S_n. We give a description of the cohomology modules H^i(PvB_n,Q) and H^i(PfB_n,Q) as sums of S_n-modules induced from certain one-dimensional representations of specific subgroups of S_n. This in particular allows us to conclude that H^i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.4645","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":"2026-05-18T03:19:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XK/EhN8YqNoHzMN6t4L0VaYpgIx5heVpQdCUP28Tlh5yJT6U323M9PWadqxnVSy5T4yKrn0uLdIqGc/sV/DJDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T22:57:43.940888Z"},"content_sha256":"c90be00d625a00d18b45342156fb2e956fc1c976676b5e241076a5b9fc247927","schema_version":"1.0","event_id":"sha256:c90be00d625a00d18b45342156fb2e956fc1c976676b5e241076a5b9fc247927"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZH3QFQBMFF4J63ISR2YD5CIFG3/bundle.json","state_url":"https://pith.science/pith/ZH3QFQBMFF4J63ISR2YD5CIFG3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZH3QFQBMFF4J63ISR2YD5CIFG3/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":"2026-06-11T22:57:43Z","links":{"resolver":"https://pith.science/pith/ZH3QFQBMFF4J63ISR2YD5CIFG3","bundle":"https://pith.science/pith/ZH3QFQBMFF4J63ISR2YD5CIFG3/bundle.json","state":"https://pith.science/pith/ZH3QFQBMFF4J63ISR2YD5CIFG3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZH3QFQBMFF4J63ISR2YD5CIFG3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:ZH3QFQBMFF4J63ISR2YD5CIFG3","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":"9ce8f8afa08c304db77ec7b63cbaea01eb0e0b9a72abe95ca238350a1f015406","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-04-16T23:41:48Z","title_canon_sha256":"ead4af8c6a23bd988358381f3a54a616c51f04c09cffa353e283041ba5997480"},"schema_version":"1.0","source":{"id":"1304.4645","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.4645","created_at":"2026-05-18T03:19:01Z"},{"alias_kind":"arxiv_version","alias_value":"1304.4645v2","created_at":"2026-05-18T03:19:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.4645","created_at":"2026-05-18T03:19:01Z"},{"alias_kind":"pith_short_12","alias_value":"ZH3QFQBMFF4J","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"ZH3QFQBMFF4J63IS","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"ZH3QFQBM","created_at":"2026-05-18T12:28:09Z"}],"graph_snapshots":[{"event_id":"sha256:c90be00d625a00d18b45342156fb2e956fc1c976676b5e241076a5b9fc247927","target":"graph","created_at":"2026-05-18T03:19:01Z","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":"In this paper we consider the cohomology of four groups related to the virtual braids of [Kauffman] and [Goussarov-Polyak-Viro], namely the pure and non-pure virtual braid groups (PvB_n and vB_n, respectively), and the pure and non-pure flat braid groups (PfB_n and fB_n, respectively). The cohomologies of PvB_n and PfB_n admit an action of the symmetric group S_n. We give a description of the cohomology modules H^i(PvB_n,Q) and H^i(PfB_n,Q) as sums of S_n-modules induced from certain one-dimensional representations of specific subgroups of S_n. This in particular allows us to conclude that H^i","authors_text":"Peter Lee","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-04-16T23:41:48Z","title":"On the Action of the Symmetric Group on the Cohomology of Groups Related to (Virtual) Braids"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.4645","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:b483d485e4c475a8b751983d70b7339e26b33ddbd026ccdeee418337574d36b3","target":"record","created_at":"2026-05-18T03:19:01Z","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":"9ce8f8afa08c304db77ec7b63cbaea01eb0e0b9a72abe95ca238350a1f015406","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-04-16T23:41:48Z","title_canon_sha256":"ead4af8c6a23bd988358381f3a54a616c51f04c09cffa353e283041ba5997480"},"schema_version":"1.0","source":{"id":"1304.4645","kind":"arxiv","version":2}},"canonical_sha256":"c9f702c02c29789f6d128eb03e890536c796d359e3d76abb6ab4f1fc2c9d8ecb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c9f702c02c29789f6d128eb03e890536c796d359e3d76abb6ab4f1fc2c9d8ecb","first_computed_at":"2026-05-18T03:19:01.172840Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:19:01.172840Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"X13vaAgzuU+HzLR2iU17tjaIPZ5db8PmkOG3ZVBKa8oJ9cZnpbGw2ZVBU2TKbXfJRioERj9BDqU6kfMBVjjvAA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:19:01.173569Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.4645","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b483d485e4c475a8b751983d70b7339e26b33ddbd026ccdeee418337574d36b3","sha256:c90be00d625a00d18b45342156fb2e956fc1c976676b5e241076a5b9fc247927"],"state_sha256":"6ab28316ec647cbbe5889d9759156b9823e6303ce66849276aea00214b6e5665"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Wd11Th6TdNXeJ74cD/M5jD7Yfa/RV5TRmg9EjwfckIDR/LtpefnVWz/JmRgFDNystK/Kf0HBhzRvVxmY8GVnDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T22:57:43.945025Z","bundle_sha256":"27f82b170b892f3bff5b1fbcc96d6e41d17bb3349fb2f6aeb7685517fd1cdd4f"}}