{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:VBHXP5KAPCKO5TYELFGX4K7OUD","short_pith_number":"pith:VBHXP5KA","schema_version":"1.0","canonical_sha256":"a84f77f5407894eecf04594d7e2beea0dabae1b82a88429343aa6f9f858ab1c7","source":{"kind":"arxiv","id":"1806.11233","version":2},"attestation_state":"computed","paper":{"title":"Back stable Schubert calculus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.CO","authors_text":"Mark Shimozono, Seung Jin Lee, Thomas Lam","submitted_at":"2018-06-29T00:14:31Z","abstract_excerpt":"We study the back stable Schubert calculus of the infinite flag variety. Our main results are: 1) a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part; 2) a novel definition of double and triple Stanley symmetric functions; 3) a proof of the positivity of double Edelman-Greene coefficients generalizing the results of Edelman-Greene and Lascoux-Schutzenberger; 4) the definition of a new class of bumpless pipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form o"},"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":"1806.11233","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-29T00:14:31Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"af49996cb409383a60a4b10c0c688ee731c3be1b0ed20108a30e169857db1e7c","abstract_canon_sha256":"6844f17a451921a1b352f15cb676d614a3d2065fecf069b79a401ad1615d25fc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:54:09.501390Z","signature_b64":"/01nAJyLHLDtKeh522QiliG2IsYQdDPzvPp9uEis1Q4at71KNZ//uEWTQHzblBVXSoZhwh7XhVJacL0nr/XfDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a84f77f5407894eecf04594d7e2beea0dabae1b82a88429343aa6f9f858ab1c7","last_reissued_at":"2026-07-05T02:54:09.500972Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:54:09.500972Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Back stable Schubert calculus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.CO","authors_text":"Mark Shimozono, Seung Jin Lee, Thomas Lam","submitted_at":"2018-06-29T00:14:31Z","abstract_excerpt":"We study the back stable Schubert calculus of the infinite flag variety. Our main results are: 1) a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part; 2) a novel definition of double and triple Stanley symmetric functions; 3) a proof of the positivity of double Edelman-Greene coefficients generalizing the results of Edelman-Greene and Lascoux-Schutzenberger; 4) the definition of a new class of bumpless pipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.11233","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1806.11233/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":"1806.11233","created_at":"2026-07-05T02:54:09.501027+00:00"},{"alias_kind":"arxiv_version","alias_value":"1806.11233v2","created_at":"2026-07-05T02:54:09.501027+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.11233","created_at":"2026-07-05T02:54:09.501027+00:00"},{"alias_kind":"pith_short_12","alias_value":"VBHXP5KAPCKO","created_at":"2026-07-05T02:54:09.501027+00:00"},{"alias_kind":"pith_short_16","alias_value":"VBHXP5KAPCKO5TYE","created_at":"2026-07-05T02:54:09.501027+00:00"},{"alias_kind":"pith_short_8","alias_value":"VBHXP5KA","created_at":"2026-07-05T02:54:09.501027+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2509.01857","citing_title":"Hybrid pipe dreams for the lower-upper scheme","ref_index":17,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VBHXP5KAPCKO5TYELFGX4K7OUD","json":"https://pith.science/pith/VBHXP5KAPCKO5TYELFGX4K7OUD.json","graph_json":"https://pith.science/api/pith-number/VBHXP5KAPCKO5TYELFGX4K7OUD/graph.json","events_json":"https://pith.science/api/pith-number/VBHXP5KAPCKO5TYELFGX4K7OUD/events.json","paper":"https://pith.science/paper/VBHXP5KA"},"agent_actions":{"view_html":"https://pith.science/pith/VBHXP5KAPCKO5TYELFGX4K7OUD","download_json":"https://pith.science/pith/VBHXP5KAPCKO5TYELFGX4K7OUD.json","view_paper":"https://pith.science/paper/VBHXP5KA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1806.11233&json=true","fetch_graph":"https://pith.science/api/pith-number/VBHXP5KAPCKO5TYELFGX4K7OUD/graph.json","fetch_events":"https://pith.science/api/pith-number/VBHXP5KAPCKO5TYELFGX4K7OUD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VBHXP5KAPCKO5TYELFGX4K7OUD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VBHXP5KAPCKO5TYELFGX4K7OUD/action/storage_attestation","attest_author":"https://pith.science/pith/VBHXP5KAPCKO5TYELFGX4K7OUD/action/author_attestation","sign_citation":"https://pith.science/pith/VBHXP5KAPCKO5TYELFGX4K7OUD/action/citation_signature","submit_replication":"https://pith.science/pith/VBHXP5KAPCKO5TYELFGX4K7OUD/action/replication_record"}},"created_at":"2026-07-05T02:54:09.501027+00:00","updated_at":"2026-07-05T02:54:09.501027+00:00"}