{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2003:F4QTSTJ4KEGACHRIKBF46THJMA","short_pith_number":"pith:F4QTSTJ4","schema_version":"1.0","canonical_sha256":"2f21394d3c510c011e28504bcf4ce96029f680f098f4ebbcc9ceefa3c93de980","source":{"kind":"arxiv","id":"math/0306304","version":1},"attestation_state":"computed","paper":{"title":"A Schubert calculus recurrence from the noncomplex W-action on G/B","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Allen Knutson","submitted_at":"2003-06-20T00:19:09Z","abstract_excerpt":"In this paper, as in our previous \"Descent-cycling in Schubert calculus\" math.CO/0009112, we study the structure constants in equivariant cohomology of flag manifolds G/B. In this one we give a recurrence (which is frequently, but alas not always, positive) to compute these one by one, using the non-complex action of the Weyl group on G/B.\n  Probably the most noteworthy feature of this recurrence is that to compute a particular structure constant c_{lambda,mu}^nu, one does not have to compute the whole product S_lambda * S_mu."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0306304","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2003-06-20T00:19:09Z","cross_cats_sorted":[],"title_canon_sha256":"52a6115992af5f0e089c77b9170a0ce217e715618c4231df1d81e127b90bde61","abstract_canon_sha256":"b0dbe487302491527ba3326ce0fed63391bc2d6e2afd17f4848ad4dea1405e95"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:37:32.435191Z","signature_b64":"ARSS9JdIFE9o5L9+dVZJo89YOgFN9RzQkduD4fPu/Em6YjzIe18mbHioOLDH91iuYu9RzTzYDwrlOC6TzGszAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f21394d3c510c011e28504bcf4ce96029f680f098f4ebbcc9ceefa3c93de980","last_reissued_at":"2026-07-04T14:37:32.434784Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:37:32.434784Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Schubert calculus recurrence from the noncomplex W-action on G/B","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Allen Knutson","submitted_at":"2003-06-20T00:19:09Z","abstract_excerpt":"In this paper, as in our previous \"Descent-cycling in Schubert calculus\" math.CO/0009112, we study the structure constants in equivariant cohomology of flag manifolds G/B. In this one we give a recurrence (which is frequently, but alas not always, positive) to compute these one by one, using the non-complex action of the Weyl group on G/B.\n  Probably the most noteworthy feature of this recurrence is that to compute a particular structure constant c_{lambda,mu}^nu, one does not have to compute the whole product S_lambda * S_mu."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0306304","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0306304/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0306304","created_at":"2026-07-04T14:37:32.434848+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0306304v1","created_at":"2026-07-04T14:37:32.434848+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0306304","created_at":"2026-07-04T14:37:32.434848+00:00"},{"alias_kind":"pith_short_12","alias_value":"F4QTSTJ4KEGA","created_at":"2026-07-04T14:37:32.434848+00:00"},{"alias_kind":"pith_short_16","alias_value":"F4QTSTJ4KEGACHRI","created_at":"2026-07-04T14:37:32.434848+00:00"},{"alias_kind":"pith_short_8","alias_value":"F4QTSTJ4","created_at":"2026-07-04T14:37:32.434848+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2504.17734","citing_title":"Signed puzzles for Schubert coefficients","ref_index":12,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/F4QTSTJ4KEGACHRIKBF46THJMA","json":"https://pith.science/pith/F4QTSTJ4KEGACHRIKBF46THJMA.json","graph_json":"https://pith.science/api/pith-number/F4QTSTJ4KEGACHRIKBF46THJMA/graph.json","events_json":"https://pith.science/api/pith-number/F4QTSTJ4KEGACHRIKBF46THJMA/events.json","paper":"https://pith.science/paper/F4QTSTJ4"},"agent_actions":{"view_html":"https://pith.science/pith/F4QTSTJ4KEGACHRIKBF46THJMA","download_json":"https://pith.science/pith/F4QTSTJ4KEGACHRIKBF46THJMA.json","view_paper":"https://pith.science/paper/F4QTSTJ4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0306304&json=true","fetch_graph":"https://pith.science/api/pith-number/F4QTSTJ4KEGACHRIKBF46THJMA/graph.json","fetch_events":"https://pith.science/api/pith-number/F4QTSTJ4KEGACHRIKBF46THJMA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/F4QTSTJ4KEGACHRIKBF46THJMA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/F4QTSTJ4KEGACHRIKBF46THJMA/action/storage_attestation","attest_author":"https://pith.science/pith/F4QTSTJ4KEGACHRIKBF46THJMA/action/author_attestation","sign_citation":"https://pith.science/pith/F4QTSTJ4KEGACHRIKBF46THJMA/action/citation_signature","submit_replication":"https://pith.science/pith/F4QTSTJ4KEGACHRIKBF46THJMA/action/replication_record"}},"created_at":"2026-07-04T14:37:32.434848+00:00","updated_at":"2026-07-04T14:37:32.434848+00:00"}