{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:XL6A5YTCOZLX6AIBTYOK3NLF7S","short_pith_number":"pith:XL6A5YTC","canonical_record":{"source":{"id":"1309.5520","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-09-21T19:20:43Z","cross_cats_sorted":["math.AT","math.CO","math.RT","nlin.SI"],"title_canon_sha256":"ba997e0b2f2cd02f70bf8d200cebfba733df2683d08f32cd50310071119f461a","abstract_canon_sha256":"8dca9b940fc373fd63250cc43fb8917d508cff9d029a16d476ee2103e0dee8bf"},"schema_version":"1.0"},"canonical_sha256":"bafc0ee26276577f01019e1cadb565fc85b45c3b321a9ea5c0a10b935fb63c6c","source":{"kind":"arxiv","id":"1309.5520","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.5520","created_at":"2026-05-18T03:12:36Z"},{"alias_kind":"arxiv_version","alias_value":"1309.5520v1","created_at":"2026-05-18T03:12:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.5520","created_at":"2026-05-18T03:12:36Z"},{"alias_kind":"pith_short_12","alias_value":"XL6A5YTCOZLX","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"XL6A5YTCOZLX6AIB","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"XL6A5YTC","created_at":"2026-05-18T12:28:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:XL6A5YTCOZLX6AIBTYOK3NLF7S","target":"record","payload":{"canonical_record":{"source":{"id":"1309.5520","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-09-21T19:20:43Z","cross_cats_sorted":["math.AT","math.CO","math.RT","nlin.SI"],"title_canon_sha256":"ba997e0b2f2cd02f70bf8d200cebfba733df2683d08f32cd50310071119f461a","abstract_canon_sha256":"8dca9b940fc373fd63250cc43fb8917d508cff9d029a16d476ee2103e0dee8bf"},"schema_version":"1.0"},"canonical_sha256":"bafc0ee26276577f01019e1cadb565fc85b45c3b321a9ea5c0a10b935fb63c6c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:36.075038Z","signature_b64":"0/VixEyGQol1vnnuyICQQ9LbH/kFYHQ40wgRmNswhxeL9aUWxzC/UZPCo7DUXASOXWTfvg0D7AS0uoJ20gbmDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bafc0ee26276577f01019e1cadb565fc85b45c3b321a9ea5c0a10b935fb63c6c","last_reissued_at":"2026-05-18T03:12:36.074337Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:36.074337Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.5520","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":"2026-05-18T03:12:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d02WO7rFc9SdyRNp4FwMh4yOH6rcJHYkpT5PCy60UhVeCDxHbArPUHLrg/AT7DPMxaSWC10QdZna2c/GHhRgAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T00:27:11.576967Z"},"content_sha256":"eeeaf9ae9a0df5fefc34883eb46bda4c8426ca566608bfd201724ba811a82c25","schema_version":"1.0","event_id":"sha256:eeeaf9ae9a0df5fefc34883eb46bda4c8426ca566608bfd201724ba811a82c25"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:XL6A5YTCOZLX6AIBTYOK3NLF7S","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the cohomology of real Grassmann manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.CO","math.RT","nlin.SI"],"primary_cat":"math.AG","authors_text":"Luis Casian, Yuji Kodama","submitted_at":"2013-09-21T19:20:43Z","abstract_excerpt":"We give an explicit and simple construction of the incidence graph for the integral cohomology of real Grassmann manifold Gr(k,n) in terms of the Young diagrams filled with the letter q in checkered pattern. It turns out that there are two types of graphs, one for the trivial coefficients and other for the twisted coefficients, and they compute the homology groups of the orientable and non-orientable cases of Gr(k,n) via the Poincar\\'e-Verdier duality. We also give an explicit formula of the Poincar\\'e polynomial for Gr(k,n) and show that the Poincar\\'e polynomial is also related to the number"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5520","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":"2026-05-18T03:12:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1asOfiSqOs83+DFFEHXDxz7iSQohsrNpBsPVpi+xvgmpgOWJ/8vmSdi6jwNH5UNsHhRmY5p8NxBqeKrD6v3hCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T00:27:11.577414Z"},"content_sha256":"82c3b0938c3c234abcfd475603ca1641d7095b9bd7fbdff09226e83a23470991","schema_version":"1.0","event_id":"sha256:82c3b0938c3c234abcfd475603ca1641d7095b9bd7fbdff09226e83a23470991"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XL6A5YTCOZLX6AIBTYOK3NLF7S/bundle.json","state_url":"https://pith.science/pith/XL6A5YTCOZLX6AIBTYOK3NLF7S/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XL6A5YTCOZLX6AIBTYOK3NLF7S/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":"2026-05-27T00:27:11Z","links":{"resolver":"https://pith.science/pith/XL6A5YTCOZLX6AIBTYOK3NLF7S","bundle":"https://pith.science/pith/XL6A5YTCOZLX6AIBTYOK3NLF7S/bundle.json","state":"https://pith.science/pith/XL6A5YTCOZLX6AIBTYOK3NLF7S/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XL6A5YTCOZLX6AIBTYOK3NLF7S/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:XL6A5YTCOZLX6AIBTYOK3NLF7S","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":"8dca9b940fc373fd63250cc43fb8917d508cff9d029a16d476ee2103e0dee8bf","cross_cats_sorted":["math.AT","math.CO","math.RT","nlin.SI"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-09-21T19:20:43Z","title_canon_sha256":"ba997e0b2f2cd02f70bf8d200cebfba733df2683d08f32cd50310071119f461a"},"schema_version":"1.0","source":{"id":"1309.5520","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.5520","created_at":"2026-05-18T03:12:36Z"},{"alias_kind":"arxiv_version","alias_value":"1309.5520v1","created_at":"2026-05-18T03:12:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.5520","created_at":"2026-05-18T03:12:36Z"},{"alias_kind":"pith_short_12","alias_value":"XL6A5YTCOZLX","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"XL6A5YTCOZLX6AIB","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"XL6A5YTC","created_at":"2026-05-18T12:28:06Z"}],"graph_snapshots":[{"event_id":"sha256:82c3b0938c3c234abcfd475603ca1641d7095b9bd7fbdff09226e83a23470991","target":"graph","created_at":"2026-05-18T03:12:36Z","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":"We give an explicit and simple construction of the incidence graph for the integral cohomology of real Grassmann manifold Gr(k,n) in terms of the Young diagrams filled with the letter q in checkered pattern. It turns out that there are two types of graphs, one for the trivial coefficients and other for the twisted coefficients, and they compute the homology groups of the orientable and non-orientable cases of Gr(k,n) via the Poincar\\'e-Verdier duality. We also give an explicit formula of the Poincar\\'e polynomial for Gr(k,n) and show that the Poincar\\'e polynomial is also related to the number","authors_text":"Luis Casian, Yuji Kodama","cross_cats":["math.AT","math.CO","math.RT","nlin.SI"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-09-21T19:20:43Z","title":"On the cohomology of real Grassmann manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5520","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:eeeaf9ae9a0df5fefc34883eb46bda4c8426ca566608bfd201724ba811a82c25","target":"record","created_at":"2026-05-18T03:12:36Z","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":"8dca9b940fc373fd63250cc43fb8917d508cff9d029a16d476ee2103e0dee8bf","cross_cats_sorted":["math.AT","math.CO","math.RT","nlin.SI"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-09-21T19:20:43Z","title_canon_sha256":"ba997e0b2f2cd02f70bf8d200cebfba733df2683d08f32cd50310071119f461a"},"schema_version":"1.0","source":{"id":"1309.5520","kind":"arxiv","version":1}},"canonical_sha256":"bafc0ee26276577f01019e1cadb565fc85b45c3b321a9ea5c0a10b935fb63c6c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bafc0ee26276577f01019e1cadb565fc85b45c3b321a9ea5c0a10b935fb63c6c","first_computed_at":"2026-05-18T03:12:36.074337Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:12:36.074337Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0/VixEyGQol1vnnuyICQQ9LbH/kFYHQ40wgRmNswhxeL9aUWxzC/UZPCo7DUXASOXWTfvg0D7AS0uoJ20gbmDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:12:36.075038Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.5520","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:eeeaf9ae9a0df5fefc34883eb46bda4c8426ca566608bfd201724ba811a82c25","sha256:82c3b0938c3c234abcfd475603ca1641d7095b9bd7fbdff09226e83a23470991"],"state_sha256":"b8f6095958ad1be95987cebe9a761d71f1f34ddf43d1ad8b5b18010b9c2d563e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1u9jxxN5hiJHn5Z5+vUDpxKjosuLLQJs6NGDBnCmGQ1nTyD/UOWj11oWwznx/nOrFe83Zx00l6OZaoR528luAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T00:27:11.580703Z","bundle_sha256":"9dcc3fd0852406158fe318fb415f96e7f6e4463c5a45916c53c3891775c0a3c9"}}