{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:YQLHOH62L5OHSU5JHQYHP6TPEC","short_pith_number":"pith:YQLHOH62","schema_version":"1.0","canonical_sha256":"c416771fda5f5c7953a93c3077fa6f20b4cad21ad1b8d65b9be73a4d3bfc5a83","source":{"kind":"arxiv","id":"1808.02251","version":1},"attestation_state":"computed","paper":{"title":"Automorphisms on the ring of symmetric functions and stable and dual stable Grothendieck polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Motoki Takigiku","submitted_at":"2018-08-07T08:09:35Z","abstract_excerpt":"The dual stable Grothendieck polynomials $g_\\lambda$ and their sums $\\sum_{\\mu\\subset\\lambda} g_\\mu$ (which represent $K$-homology classes of boundary ideal sheaves and structure sheaves of Schubert varieties in the Grassmannians) have the same product structure constants. In this paper we first explain that the ring automorphism $g_\\lambda\\mapsto\\sum_{\\mu\\subset\\lambda} g_\\mu$ on the ring of symmetric functions is described as the operator $F^\\perp$, the adjoint of the multiplication $(F\\cdot)$, by a \"group-like\" element $F=\\sum_{i} h_i$ where $h_i$ is the complete symmetric function. Next we"},"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":"1808.02251","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-07T08:09:35Z","cross_cats_sorted":[],"title_canon_sha256":"a3f10ad3860e568fff9a0e4c8aabdc5c99714b4a8d987ad7b0608e01ba5caf81","abstract_canon_sha256":"8fbf1f2b79ef5937d854b54c97f43f1617d314fa67785b5fc25c61a9d7df606a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:47.122721Z","signature_b64":"aUpNJvLbGODVdEVzTbhtW4ltmtAaXtYRXwDLfDK0oBYcNE1uudFOKOIrWIwclxYFLCzmHDBX9DNMr/ijUTppDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c416771fda5f5c7953a93c3077fa6f20b4cad21ad1b8d65b9be73a4d3bfc5a83","last_reissued_at":"2026-05-18T00:08:47.122012Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:47.122012Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Automorphisms on the ring of symmetric functions and stable and dual stable Grothendieck polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Motoki Takigiku","submitted_at":"2018-08-07T08:09:35Z","abstract_excerpt":"The dual stable Grothendieck polynomials $g_\\lambda$ and their sums $\\sum_{\\mu\\subset\\lambda} g_\\mu$ (which represent $K$-homology classes of boundary ideal sheaves and structure sheaves of Schubert varieties in the Grassmannians) have the same product structure constants. In this paper we first explain that the ring automorphism $g_\\lambda\\mapsto\\sum_{\\mu\\subset\\lambda} g_\\mu$ on the ring of symmetric functions is described as the operator $F^\\perp$, the adjoint of the multiplication $(F\\cdot)$, by a \"group-like\" element $F=\\sum_{i} h_i$ where $h_i$ is the complete symmetric function. Next we"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.02251","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1808.02251","created_at":"2026-05-18T00:08:47.122116+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.02251v1","created_at":"2026-05-18T00:08:47.122116+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.02251","created_at":"2026-05-18T00:08:47.122116+00:00"},{"alias_kind":"pith_short_12","alias_value":"YQLHOH62L5OH","created_at":"2026-05-18T12:33:04.347982+00:00"},{"alias_kind":"pith_short_16","alias_value":"YQLHOH62L5OHSU5J","created_at":"2026-05-18T12:33:04.347982+00:00"},{"alias_kind":"pith_short_8","alias_value":"YQLHOH62","created_at":"2026-05-18T12:33:04.347982+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2502.05618","citing_title":"Lowering operators on $K$-$k$-Schur functions and a lowering operator formula for closed $K$-$k$-Schur functions","ref_index":20,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/YQLHOH62L5OHSU5JHQYHP6TPEC","json":"https://pith.science/pith/YQLHOH62L5OHSU5JHQYHP6TPEC.json","graph_json":"https://pith.science/api/pith-number/YQLHOH62L5OHSU5JHQYHP6TPEC/graph.json","events_json":"https://pith.science/api/pith-number/YQLHOH62L5OHSU5JHQYHP6TPEC/events.json","paper":"https://pith.science/paper/YQLHOH62"},"agent_actions":{"view_html":"https://pith.science/pith/YQLHOH62L5OHSU5JHQYHP6TPEC","download_json":"https://pith.science/pith/YQLHOH62L5OHSU5JHQYHP6TPEC.json","view_paper":"https://pith.science/paper/YQLHOH62","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.02251&json=true","fetch_graph":"https://pith.science/api/pith-number/YQLHOH62L5OHSU5JHQYHP6TPEC/graph.json","fetch_events":"https://pith.science/api/pith-number/YQLHOH62L5OHSU5JHQYHP6TPEC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YQLHOH62L5OHSU5JHQYHP6TPEC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YQLHOH62L5OHSU5JHQYHP6TPEC/action/storage_attestation","attest_author":"https://pith.science/pith/YQLHOH62L5OHSU5JHQYHP6TPEC/action/author_attestation","sign_citation":"https://pith.science/pith/YQLHOH62L5OHSU5JHQYHP6TPEC/action/citation_signature","submit_replication":"https://pith.science/pith/YQLHOH62L5OHSU5JHQYHP6TPEC/action/replication_record"}},"created_at":"2026-05-18T00:08:47.122116+00:00","updated_at":"2026-05-18T00:08:47.122116+00:00"}