{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:OPW5JEMDTDFCLGKNB7BVOMSU7Z","short_pith_number":"pith:OPW5JEMD","canonical_record":{"source":{"id":"0903.4501","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2009-03-26T09:14:48Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"33cfded00797eb38f4c03127b84f4a75f1f768c62431b3535b63f245b37c459d","abstract_canon_sha256":"19b469eae02079b4ac70749f633ddaf73b7d8f4225989bf5632a159fc2058a21"},"schema_version":"1.0"},"canonical_sha256":"73edd4918398ca25994d0fc3573254fe52e95c8f6564e91fc06ecb29b3ed8d5b","source":{"kind":"arxiv","id":"0903.4501","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.4501","created_at":"2026-05-18T03:02:44Z"},{"alias_kind":"arxiv_version","alias_value":"0903.4501v4","created_at":"2026-05-18T03:02:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.4501","created_at":"2026-05-18T03:02:44Z"},{"alias_kind":"pith_short_12","alias_value":"OPW5JEMDTDFC","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"OPW5JEMDTDFCLGKN","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"OPW5JEMD","created_at":"2026-05-18T12:26:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:OPW5JEMDTDFCLGKNB7BVOMSU7Z","target":"record","payload":{"canonical_record":{"source":{"id":"0903.4501","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2009-03-26T09:14:48Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"33cfded00797eb38f4c03127b84f4a75f1f768c62431b3535b63f245b37c459d","abstract_canon_sha256":"19b469eae02079b4ac70749f633ddaf73b7d8f4225989bf5632a159fc2058a21"},"schema_version":"1.0"},"canonical_sha256":"73edd4918398ca25994d0fc3573254fe52e95c8f6564e91fc06ecb29b3ed8d5b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:02:44.693050Z","signature_b64":"z+UCidUrOA6uRgII9DkIVY3EH016MwDdGkNrpwB6R7lO2LYH4Z2PSizjQDxslPonpOv2Uyjblr344JcwJy77DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"73edd4918398ca25994d0fc3573254fe52e95c8f6564e91fc06ecb29b3ed8d5b","last_reissued_at":"2026-05-18T03:02:44.692268Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:02:44.692268Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0903.4501","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:02:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BfKWejpTmp7d1sasDYo+tXrFSwJYiP1yPiM1/DbvrSNDKuJwT8T2Nr06wvNXkSfb3lFUEVsGWD2BTCzhNqXeCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T15:04:17.848563Z"},"content_sha256":"da1545f4cb37b895ea4b0bfca7a059e4c418d5a796e85ffb871af174f1bd26fd","schema_version":"1.0","event_id":"sha256:da1545f4cb37b895ea4b0bfca7a059e4c418d5a796e85ffb871af174f1bd26fd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:OPW5JEMDTDFCLGKNB7BVOMSU7Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Schubert calculus and the Hopf algebra structures of exceptional Lie groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AT","authors_text":"Haibao Duan, Xuezhi Zhao","submitted_at":"2009-03-26T09:14:48Z","abstract_excerpt":"Let G be an exceptional Lie group with a maximal torus T. Based on common properties in the Schubert presentation of the cohomology ring H*(G/T;F_{p}) DZ1, and concrete expressions of generalized Weyl invariants for G over F_{p}, we obtain a unified approach to the structure of H*(G;F_{p}) as a Hopf algebra over the Steenrod algebra A_{p}. The results has been applied in Du2 to determine the near--Hopf ring structure on the integral cohomology of all exceptional Lie groups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.4501","kind":"arxiv","version":4},"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":"2026-05-18T03:02:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1KF9aePaUOZoobJJJe+I/0II5zpzX0fFh0wKVay0hBUKZFNu0cFMBEaL0mXKMYiM9NBXg1iDBUOItOSwjZRIAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T15:04:17.849259Z"},"content_sha256":"9cc3cd345782e6fa3837e3b423e00e6d728d2e32f6cb6997082f39fcb0355ef7","schema_version":"1.0","event_id":"sha256:9cc3cd345782e6fa3837e3b423e00e6d728d2e32f6cb6997082f39fcb0355ef7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OPW5JEMDTDFCLGKNB7BVOMSU7Z/bundle.json","state_url":"https://pith.science/pith/OPW5JEMDTDFCLGKNB7BVOMSU7Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OPW5JEMDTDFCLGKNB7BVOMSU7Z/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":"2026-06-10T15:04:17Z","links":{"resolver":"https://pith.science/pith/OPW5JEMDTDFCLGKNB7BVOMSU7Z","bundle":"https://pith.science/pith/OPW5JEMDTDFCLGKNB7BVOMSU7Z/bundle.json","state":"https://pith.science/pith/OPW5JEMDTDFCLGKNB7BVOMSU7Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OPW5JEMDTDFCLGKNB7BVOMSU7Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:OPW5JEMDTDFCLGKNB7BVOMSU7Z","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":"19b469eae02079b4ac70749f633ddaf73b7d8f4225989bf5632a159fc2058a21","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2009-03-26T09:14:48Z","title_canon_sha256":"33cfded00797eb38f4c03127b84f4a75f1f768c62431b3535b63f245b37c459d"},"schema_version":"1.0","source":{"id":"0903.4501","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.4501","created_at":"2026-05-18T03:02:44Z"},{"alias_kind":"arxiv_version","alias_value":"0903.4501v4","created_at":"2026-05-18T03:02:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.4501","created_at":"2026-05-18T03:02:44Z"},{"alias_kind":"pith_short_12","alias_value":"OPW5JEMDTDFC","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"OPW5JEMDTDFCLGKN","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"OPW5JEMD","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:9cc3cd345782e6fa3837e3b423e00e6d728d2e32f6cb6997082f39fcb0355ef7","target":"graph","created_at":"2026-05-18T03:02:44Z","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 G be an exceptional Lie group with a maximal torus T. Based on common properties in the Schubert presentation of the cohomology ring H*(G/T;F_{p}) DZ1, and concrete expressions of generalized Weyl invariants for G over F_{p}, we obtain a unified approach to the structure of H*(G;F_{p}) as a Hopf algebra over the Steenrod algebra A_{p}. The results has been applied in Du2 to determine the near--Hopf ring structure on the integral cohomology of all exceptional Lie groups.","authors_text":"Haibao Duan, Xuezhi Zhao","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2009-03-26T09:14:48Z","title":"Schubert calculus and the Hopf algebra structures of exceptional Lie groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.4501","kind":"arxiv","version":4},"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:da1545f4cb37b895ea4b0bfca7a059e4c418d5a796e85ffb871af174f1bd26fd","target":"record","created_at":"2026-05-18T03:02:44Z","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":"19b469eae02079b4ac70749f633ddaf73b7d8f4225989bf5632a159fc2058a21","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2009-03-26T09:14:48Z","title_canon_sha256":"33cfded00797eb38f4c03127b84f4a75f1f768c62431b3535b63f245b37c459d"},"schema_version":"1.0","source":{"id":"0903.4501","kind":"arxiv","version":4}},"canonical_sha256":"73edd4918398ca25994d0fc3573254fe52e95c8f6564e91fc06ecb29b3ed8d5b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"73edd4918398ca25994d0fc3573254fe52e95c8f6564e91fc06ecb29b3ed8d5b","first_computed_at":"2026-05-18T03:02:44.692268Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:02:44.692268Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"z+UCidUrOA6uRgII9DkIVY3EH016MwDdGkNrpwB6R7lO2LYH4Z2PSizjQDxslPonpOv2Uyjblr344JcwJy77DA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:02:44.693050Z","signed_message":"canonical_sha256_bytes"},"source_id":"0903.4501","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:da1545f4cb37b895ea4b0bfca7a059e4c418d5a796e85ffb871af174f1bd26fd","sha256:9cc3cd345782e6fa3837e3b423e00e6d728d2e32f6cb6997082f39fcb0355ef7"],"state_sha256":"f27e37677696b03798949ea6b1dafabd580e5aa5305def173d23144131bc04c5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QVV6Nr46WClU3PPFNMlSQy/UrR32sZCY+7405/RGGnttwe6IC1581f+ZOcBHtLKJqva/U5RhRZzjYJaITMTeBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T15:04:17.853096Z","bundle_sha256":"8d580dc8f9950e7bd8ccdac1929087d71a4542fd0023db3b156eb4d70d02de04"}}