{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:72TE5Z5LIOOU544IZTJGQTVT6D","short_pith_number":"pith:72TE5Z5L","schema_version":"1.0","canonical_sha256":"fea64ee7ab439d4ef388ccd2684eb3f0d1350ee2ed5dd880d41288d44c4618c6","source":{"kind":"arxiv","id":"1805.01718","version":9},"attestation_state":"computed","paper":{"title":"Loop structure on equivariant $K$-theory of semi-infinite flag manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA","math.RT"],"primary_cat":"math.AG","authors_text":"Syu Kato","submitted_at":"2018-05-04T11:27:53Z","abstract_excerpt":"We explain that the Pontryagin product structure on the equivariant $K$-group of an affine Grassmannian considered in [Lam-Schilling-Shimozono, Compos. Math. {\\bf 146} (2010)] coincides with the tensor structure on the equivariant $K$-group of a semi-infinite flag manifold considered in [K-Naito-Sagaki, Duke Math. {\\bf 169} (2020)]. Then, we construct an explicit isomorphism between the equivariant $K$-group of a semi-infinite flag manifold with a suitably localized equivariant quantum $K$-group of the corresponding flag manifold. These exhibit a new framework to understand the ring structure "},"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":"1805.01718","kind":"arxiv","version":9},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2018-05-04T11:27:53Z","cross_cats_sorted":["math.QA","math.RT"],"title_canon_sha256":"ef829c131de9df8a96261df44e92b8d95446e0cfb16f42d7e814418e594fca59","abstract_canon_sha256":"ca6be8693611161c3c8a9dce09d5c056db01e3bde715366bc5c8d641a4c36abf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:53:58.452787Z","signature_b64":"KruuKynDG3s8RI7tGj0GGGumRJVBzS5fALs1LnCjTwSsnn8YNxEGoWfqhEVErIsShNLM1xvmLRgHt9CTzTqTAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fea64ee7ab439d4ef388ccd2684eb3f0d1350ee2ed5dd880d41288d44c4618c6","last_reissued_at":"2026-07-05T09:53:58.452434Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:53:58.452434Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Loop structure on equivariant $K$-theory of semi-infinite flag manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA","math.RT"],"primary_cat":"math.AG","authors_text":"Syu Kato","submitted_at":"2018-05-04T11:27:53Z","abstract_excerpt":"We explain that the Pontryagin product structure on the equivariant $K$-group of an affine Grassmannian considered in [Lam-Schilling-Shimozono, Compos. Math. {\\bf 146} (2010)] coincides with the tensor structure on the equivariant $K$-group of a semi-infinite flag manifold considered in [K-Naito-Sagaki, Duke Math. {\\bf 169} (2020)]. Then, we construct an explicit isomorphism between the equivariant $K$-group of a semi-infinite flag manifold with a suitably localized equivariant quantum $K$-group of the corresponding flag manifold. These exhibit a new framework to understand the ring structure "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.01718","kind":"arxiv","version":9},"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/1805.01718/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":"1805.01718","created_at":"2026-07-05T09:53:58.452488+00:00"},{"alias_kind":"arxiv_version","alias_value":"1805.01718v9","created_at":"2026-07-05T09:53:58.452488+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.01718","created_at":"2026-07-05T09:53:58.452488+00:00"},{"alias_kind":"pith_short_12","alias_value":"72TE5Z5LIOOU","created_at":"2026-07-05T09:53:58.452488+00:00"},{"alias_kind":"pith_short_16","alias_value":"72TE5Z5LIOOU544I","created_at":"2026-07-05T09:53:58.452488+00:00"},{"alias_kind":"pith_short_8","alias_value":"72TE5Z5L","created_at":"2026-07-05T09:53:58.452488+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"1906.09343","citing_title":"On quantum $K$-groups of partial flag manifolds","ref_index":21,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/72TE5Z5LIOOU544IZTJGQTVT6D","json":"https://pith.science/pith/72TE5Z5LIOOU544IZTJGQTVT6D.json","graph_json":"https://pith.science/api/pith-number/72TE5Z5LIOOU544IZTJGQTVT6D/graph.json","events_json":"https://pith.science/api/pith-number/72TE5Z5LIOOU544IZTJGQTVT6D/events.json","paper":"https://pith.science/paper/72TE5Z5L"},"agent_actions":{"view_html":"https://pith.science/pith/72TE5Z5LIOOU544IZTJGQTVT6D","download_json":"https://pith.science/pith/72TE5Z5LIOOU544IZTJGQTVT6D.json","view_paper":"https://pith.science/paper/72TE5Z5L","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1805.01718&json=true","fetch_graph":"https://pith.science/api/pith-number/72TE5Z5LIOOU544IZTJGQTVT6D/graph.json","fetch_events":"https://pith.science/api/pith-number/72TE5Z5LIOOU544IZTJGQTVT6D/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/72TE5Z5LIOOU544IZTJGQTVT6D/action/timestamp_anchor","attest_storage":"https://pith.science/pith/72TE5Z5LIOOU544IZTJGQTVT6D/action/storage_attestation","attest_author":"https://pith.science/pith/72TE5Z5LIOOU544IZTJGQTVT6D/action/author_attestation","sign_citation":"https://pith.science/pith/72TE5Z5LIOOU544IZTJGQTVT6D/action/citation_signature","submit_replication":"https://pith.science/pith/72TE5Z5LIOOU544IZTJGQTVT6D/action/replication_record"}},"created_at":"2026-07-05T09:53:58.452488+00:00","updated_at":"2026-07-05T09:53:58.452488+00:00"}