{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:SOVA3GXL5SFHL3L5INHTX6E7GZ","short_pith_number":"pith:SOVA3GXL","canonical_record":{"source":{"id":"1508.06826","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-08-27T12:29:26Z","cross_cats_sorted":[],"title_canon_sha256":"a7787e26391fd128c7eb1d9baa74ddae22ef647db8ceb6072f04cad0e5e646b2","abstract_canon_sha256":"da329663141b82ad7e5779ca5279651aea60692bd4035d335bd42c0291506db6"},"schema_version":"1.0"},"canonical_sha256":"93aa0d9aebec8a75ed7d434f3bf89f3646459c0b70c1c66e706fd1ca4a16a9d8","source":{"kind":"arxiv","id":"1508.06826","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.06826","created_at":"2026-05-18T01:34:40Z"},{"alias_kind":"arxiv_version","alias_value":"1508.06826v1","created_at":"2026-05-18T01:34:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.06826","created_at":"2026-05-18T01:34:40Z"},{"alias_kind":"pith_short_12","alias_value":"SOVA3GXL5SFH","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"SOVA3GXL5SFHL3L5","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"SOVA3GXL","created_at":"2026-05-18T12:29:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:SOVA3GXL5SFHL3L5INHTX6E7GZ","target":"record","payload":{"canonical_record":{"source":{"id":"1508.06826","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-08-27T12:29:26Z","cross_cats_sorted":[],"title_canon_sha256":"a7787e26391fd128c7eb1d9baa74ddae22ef647db8ceb6072f04cad0e5e646b2","abstract_canon_sha256":"da329663141b82ad7e5779ca5279651aea60692bd4035d335bd42c0291506db6"},"schema_version":"1.0"},"canonical_sha256":"93aa0d9aebec8a75ed7d434f3bf89f3646459c0b70c1c66e706fd1ca4a16a9d8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:40.899345Z","signature_b64":"vLUz8uccW8raWuL0dnX7ZVoSoY40dd4vAR6A4N/QNm4jcsNQxu/HpvSZTUGbkTcBPyruyXWFCuLan0VzNCs5Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"93aa0d9aebec8a75ed7d434f3bf89f3646459c0b70c1c66e706fd1ca4a16a9d8","last_reissued_at":"2026-05-18T01:34:40.898646Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:40.898646Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.06826","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":"2026-05-18T01:34:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M/3a3UEhRrLJ+Gn01Wu19hP/tWmLtmsrkNYFEylK+SGvD8RxVwqQ+tn5La9acvYiE3Ar6FLeaBVJwrh25zzXBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T11:06:26.137870Z"},"content_sha256":"330d90248bbf2adad986e49c43f6bd38d10b2e03deb1852863662b937e356772","schema_version":"1.0","event_id":"sha256:330d90248bbf2adad986e49c43f6bd38d10b2e03deb1852863662b937e356772"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:SOVA3GXL5SFHL3L5INHTX6E7GZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Representation ring of Levi subgroups versus cohomology ring of flag varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Shrawan Kumar","submitted_at":"2015-08-27T12:29:26Z","abstract_excerpt":"Recall the classical result that the cup product structure constants for the singular cohomology with integral coefficients of the Grassmannian of r-planes coincide with the Littlewood-Richardson tensor product structure constants for GL(r). Specifically, the result asserts that there is an explicit ring homomorphism \\phi: \\Rep _{\\poly}(GL(r)) to H^*(Gr(r, n)), where Gr(r, n) denotes the Grassmannian of r-planes in C^n and \\Rep_{\\poly} (GL(r)) denotes the polynomial representation ring of GL(r).\n  This work seeks to achieve one possible generalization of this classical result for GL(r) and the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.06826","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":"2026-05-18T01:34:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ro9a2DyUr7SHNjdeZJDl7uS/aI5VBNhc2FcLpkaoi4v8xUo73GHyRYUkBAV5AtRNFdi5DlK9m4GC1Qp7DesdCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T11:06:26.138283Z"},"content_sha256":"1d9bf63c352f1f4548156fcdc41f0d2a9fd1b57c22b33eb99f120190776537ed","schema_version":"1.0","event_id":"sha256:1d9bf63c352f1f4548156fcdc41f0d2a9fd1b57c22b33eb99f120190776537ed"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SOVA3GXL5SFHL3L5INHTX6E7GZ/bundle.json","state_url":"https://pith.science/pith/SOVA3GXL5SFHL3L5INHTX6E7GZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SOVA3GXL5SFHL3L5INHTX6E7GZ/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":"2026-06-06T11:06:26Z","links":{"resolver":"https://pith.science/pith/SOVA3GXL5SFHL3L5INHTX6E7GZ","bundle":"https://pith.science/pith/SOVA3GXL5SFHL3L5INHTX6E7GZ/bundle.json","state":"https://pith.science/pith/SOVA3GXL5SFHL3L5INHTX6E7GZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SOVA3GXL5SFHL3L5INHTX6E7GZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:SOVA3GXL5SFHL3L5INHTX6E7GZ","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":"da329663141b82ad7e5779ca5279651aea60692bd4035d335bd42c0291506db6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-08-27T12:29:26Z","title_canon_sha256":"a7787e26391fd128c7eb1d9baa74ddae22ef647db8ceb6072f04cad0e5e646b2"},"schema_version":"1.0","source":{"id":"1508.06826","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.06826","created_at":"2026-05-18T01:34:40Z"},{"alias_kind":"arxiv_version","alias_value":"1508.06826v1","created_at":"2026-05-18T01:34:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.06826","created_at":"2026-05-18T01:34:40Z"},{"alias_kind":"pith_short_12","alias_value":"SOVA3GXL5SFH","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"SOVA3GXL5SFHL3L5","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"SOVA3GXL","created_at":"2026-05-18T12:29:42Z"}],"graph_snapshots":[{"event_id":"sha256:1d9bf63c352f1f4548156fcdc41f0d2a9fd1b57c22b33eb99f120190776537ed","target":"graph","created_at":"2026-05-18T01:34:40Z","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":"Recall the classical result that the cup product structure constants for the singular cohomology with integral coefficients of the Grassmannian of r-planes coincide with the Littlewood-Richardson tensor product structure constants for GL(r). Specifically, the result asserts that there is an explicit ring homomorphism \\phi: \\Rep _{\\poly}(GL(r)) to H^*(Gr(r, n)), where Gr(r, n) denotes the Grassmannian of r-planes in C^n and \\Rep_{\\poly} (GL(r)) denotes the polynomial representation ring of GL(r).\n  This work seeks to achieve one possible generalization of this classical result for GL(r) and the","authors_text":"Shrawan Kumar","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-08-27T12:29:26Z","title":"Representation ring of Levi subgroups versus cohomology ring of flag varieties"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.06826","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":"sha256:330d90248bbf2adad986e49c43f6bd38d10b2e03deb1852863662b937e356772","target":"record","created_at":"2026-05-18T01:34:40Z","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":"da329663141b82ad7e5779ca5279651aea60692bd4035d335bd42c0291506db6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-08-27T12:29:26Z","title_canon_sha256":"a7787e26391fd128c7eb1d9baa74ddae22ef647db8ceb6072f04cad0e5e646b2"},"schema_version":"1.0","source":{"id":"1508.06826","kind":"arxiv","version":1}},"canonical_sha256":"93aa0d9aebec8a75ed7d434f3bf89f3646459c0b70c1c66e706fd1ca4a16a9d8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"93aa0d9aebec8a75ed7d434f3bf89f3646459c0b70c1c66e706fd1ca4a16a9d8","first_computed_at":"2026-05-18T01:34:40.898646Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:34:40.898646Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vLUz8uccW8raWuL0dnX7ZVoSoY40dd4vAR6A4N/QNm4jcsNQxu/HpvSZTUGbkTcBPyruyXWFCuLan0VzNCs5Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:34:40.899345Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.06826","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:330d90248bbf2adad986e49c43f6bd38d10b2e03deb1852863662b937e356772","sha256:1d9bf63c352f1f4548156fcdc41f0d2a9fd1b57c22b33eb99f120190776537ed"],"state_sha256":"3fdef1910a609f04268754b99deaeb7ea3e0a2d514f22da604338668afc95fc9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vPSyFqtpjJNPYVuzQC1+J6C/4lchXYS9xr6dWxUpTzjFJWX0EhkYnoQ88Skajq4QcRZrqV8j+NZOpnFO+9jsCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T11:06:26.142198Z","bundle_sha256":"5421debb4c31e0f5e3f96a94a7af006d7afefcc6471afb60839a1b29dd5de2c5"}}