{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:DA44CVW7ZF67YFWGWGM3PRIZQJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"7602b2687ba8c49378baff610fa9301b3566f636f71d207db25adaf6c6624eb6","cross_cats_sorted":["math.AG","math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-01T23:41:22Z","title_canon_sha256":"85aa64c96755b4a69abae220dad7d3fd9a3244b043bdbbd3987093d5a85cf082"},"schema_version":"1.0","source":{"id":"1408.0320","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.0320","created_at":"2026-05-18T01:13:09Z"},{"alias_kind":"arxiv_version","alias_value":"1408.0320v2","created_at":"2026-05-18T01:13:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.0320","created_at":"2026-05-18T01:13:09Z"},{"alias_kind":"pith_short_12","alias_value":"DA44CVW7ZF67","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_16","alias_value":"DA44CVW7ZF67YFWG","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_8","alias_value":"DA44CVW7","created_at":"2026-05-18T12:28:25Z"}],"graph_snapshots":[{"event_id":"sha256:36fd28284603e2c97981441ec614ba4b845618f93423e325302f5e97c12988bf","target":"graph","created_at":"2026-05-18T01:13:09Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We apply crystal theory to affine Schubert calculus, Gromov-Witten invariants for the complete flag manifold, and the positroid stratification of the positive Grassmannian. We introduce operators on decompositions of elements in the type-$A$ affine Weyl group and produce a crystal reflecting the internal structure of the generalized Young modules whose Frobenius image is represented by stable Schubert polynomials. We apply the crystal framework to products of a Schur function with a $k$-Schur function, consequently proving that a subclass of 3-point Gromov-Witten invariants of complete flag va","authors_text":"Anne Schilling, Jennifer Morse","cross_cats":["math.AG","math.QA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-01T23:41:22Z","title":"Crystal approach to affine Schubert calculus"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.0320","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:3a5efa8ce8dbe0039d9bf9dd312baa61ec435c2a9e22896872592a03585d20b9","target":"record","created_at":"2026-05-18T01:13:09Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"7602b2687ba8c49378baff610fa9301b3566f636f71d207db25adaf6c6624eb6","cross_cats_sorted":["math.AG","math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-01T23:41:22Z","title_canon_sha256":"85aa64c96755b4a69abae220dad7d3fd9a3244b043bdbbd3987093d5a85cf082"},"schema_version":"1.0","source":{"id":"1408.0320","kind":"arxiv","version":2}},"canonical_sha256":"1839c156dfc97dfc16c6b199b7c519824a4dffc1b98086ad3b56d063768eaca5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1839c156dfc97dfc16c6b199b7c519824a4dffc1b98086ad3b56d063768eaca5","first_computed_at":"2026-05-18T01:13:09.870094Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:13:09.870094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Wn3dXGgSBq+ZGDHOefto87sfvHUtCbABx+Gzrkh5XcUgWIuoaqBQVikUMwgKtFRL/PPQnenYR/1Vg/O0Jh+QDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:13:09.870475Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.0320","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3a5efa8ce8dbe0039d9bf9dd312baa61ec435c2a9e22896872592a03585d20b9","sha256:36fd28284603e2c97981441ec614ba4b845618f93423e325302f5e97c12988bf"],"state_sha256":"52a6c35ccc53a841dff5ade73a33c2772240f1f4ec723c73f08133ab2cf189c6"}