{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:TACQIIRSOMJ7JIEHJOKQPDLCFE","short_pith_number":"pith:TACQIIRS","schema_version":"1.0","canonical_sha256":"98050422327313f4a0874b95078d6229023469b81bab036b39696df5fcebaf0f","source":{"kind":"arxiv","id":"2506.21052","version":1},"attestation_state":"computed","paper":{"title":"Cauchy identities for Grothendieck polynomials and a dual RSK correspondence through pipe dreams","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hugh Dennin","submitted_at":"2025-06-26T06:58:47Z","abstract_excerpt":"The Cauchy identity gives a recipe for decomposing a double Grothendieck polynomial $\\mathfrak{G}^{(\\beta)}_w(x;y)$ as a sum of products $\\mathfrak{G}^{(\\beta)}_v(x)\\mathfrak{G}^{(\\beta)}_u(y)$ of single Grothendieck polynomials. Combinatorially, this identity suggests the existence of a weight-preserving bijection between certain families of diagrams called pipe dreams. In this paper, we provide such a bijection using an algorithm called pipe dream rectification. In turn, rectification is built from a new class of flow operators which themselves exhibit a surprising symmetry. Finally, we exam"},"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":"2506.21052","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-06-26T06:58:47Z","cross_cats_sorted":[],"title_canon_sha256":"094bbadbf07f376aee081a8db908257b0bd300bcc0eb84aa52d8bd9043a422ed","abstract_canon_sha256":"c2c8fe1762d41551f5d3669377b34588a33fe2a383b5dcef69da2bf8204b1f8d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:27:38.801688Z","signature_b64":"jZoNIDKgJm+YiXFx/EEq599dJdYgF6HzAUQ1+p8XTqJtuhH2OsiT5kzS6KVtlrVkqV+LI1JtPpmChVJ5BQDQDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98050422327313f4a0874b95078d6229023469b81bab036b39696df5fcebaf0f","last_reissued_at":"2026-07-05T11:27:38.801228Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:27:38.801228Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cauchy identities for Grothendieck polynomials and a dual RSK correspondence through pipe dreams","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hugh Dennin","submitted_at":"2025-06-26T06:58:47Z","abstract_excerpt":"The Cauchy identity gives a recipe for decomposing a double Grothendieck polynomial $\\mathfrak{G}^{(\\beta)}_w(x;y)$ as a sum of products $\\mathfrak{G}^{(\\beta)}_v(x)\\mathfrak{G}^{(\\beta)}_u(y)$ of single Grothendieck polynomials. Combinatorially, this identity suggests the existence of a weight-preserving bijection between certain families of diagrams called pipe dreams. In this paper, we provide such a bijection using an algorithm called pipe dream rectification. In turn, rectification is built from a new class of flow operators which themselves exhibit a surprising symmetry. Finally, we exam"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.21052","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/2506.21052/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":"2506.21052","created_at":"2026-07-05T11:27:38.801279+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.21052v1","created_at":"2026-07-05T11:27:38.801279+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.21052","created_at":"2026-07-05T11:27:38.801279+00:00"},{"alias_kind":"pith_short_12","alias_value":"TACQIIRSOMJ7","created_at":"2026-07-05T11:27:38.801279+00:00"},{"alias_kind":"pith_short_16","alias_value":"TACQIIRSOMJ7JIEH","created_at":"2026-07-05T11:27:38.801279+00:00"},{"alias_kind":"pith_short_8","alias_value":"TACQIIRS","created_at":"2026-07-05T11:27:38.801279+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TACQIIRSOMJ7JIEHJOKQPDLCFE","json":"https://pith.science/pith/TACQIIRSOMJ7JIEHJOKQPDLCFE.json","graph_json":"https://pith.science/api/pith-number/TACQIIRSOMJ7JIEHJOKQPDLCFE/graph.json","events_json":"https://pith.science/api/pith-number/TACQIIRSOMJ7JIEHJOKQPDLCFE/events.json","paper":"https://pith.science/paper/TACQIIRS"},"agent_actions":{"view_html":"https://pith.science/pith/TACQIIRSOMJ7JIEHJOKQPDLCFE","download_json":"https://pith.science/pith/TACQIIRSOMJ7JIEHJOKQPDLCFE.json","view_paper":"https://pith.science/paper/TACQIIRS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.21052&json=true","fetch_graph":"https://pith.science/api/pith-number/TACQIIRSOMJ7JIEHJOKQPDLCFE/graph.json","fetch_events":"https://pith.science/api/pith-number/TACQIIRSOMJ7JIEHJOKQPDLCFE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TACQIIRSOMJ7JIEHJOKQPDLCFE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TACQIIRSOMJ7JIEHJOKQPDLCFE/action/storage_attestation","attest_author":"https://pith.science/pith/TACQIIRSOMJ7JIEHJOKQPDLCFE/action/author_attestation","sign_citation":"https://pith.science/pith/TACQIIRSOMJ7JIEHJOKQPDLCFE/action/citation_signature","submit_replication":"https://pith.science/pith/TACQIIRSOMJ7JIEHJOKQPDLCFE/action/replication_record"}},"created_at":"2026-07-05T11:27:38.801279+00:00","updated_at":"2026-07-05T11:27:38.801279+00:00"}