{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:4KHAZY7JQK6KQLT5LEKE3XIY4E","short_pith_number":"pith:4KHAZY7J","canonical_record":{"source":{"id":"0904.1679","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-04-10T11:32:33Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"198df9d01588c2c4199f06f7eee068ff36f75560bcdc109cd769e5e4d4da41b0","abstract_canon_sha256":"8b8bec3c5d1c9c26bd9170a0868448f75f6d7d6e7459e874d95c90237eafc124"},"schema_version":"1.0"},"canonical_sha256":"e28e0ce3e982bca82e7d59144ddd18e1067bcd057baf33e7332c71d91d3da03c","source":{"kind":"arxiv","id":"0904.1679","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0904.1679","created_at":"2026-05-17T23:54:27Z"},{"alias_kind":"arxiv_version","alias_value":"0904.1679v2","created_at":"2026-05-17T23:54:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0904.1679","created_at":"2026-05-17T23:54:27Z"},{"alias_kind":"pith_short_12","alias_value":"4KHAZY7JQK6K","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"4KHAZY7JQK6KQLT5","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"4KHAZY7J","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:4KHAZY7JQK6KQLT5LEKE3XIY4E","target":"record","payload":{"canonical_record":{"source":{"id":"0904.1679","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-04-10T11:32:33Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"198df9d01588c2c4199f06f7eee068ff36f75560bcdc109cd769e5e4d4da41b0","abstract_canon_sha256":"8b8bec3c5d1c9c26bd9170a0868448f75f6d7d6e7459e874d95c90237eafc124"},"schema_version":"1.0"},"canonical_sha256":"e28e0ce3e982bca82e7d59144ddd18e1067bcd057baf33e7332c71d91d3da03c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:27.518336Z","signature_b64":"6AJ6N3GeYR8dbolS5gLWBnJ9WrlwfGAWiNs+/Az0fJWk0F47Z1Yv7XIZwgBKwNrargcsKfHV019J2ZBCPd9nBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e28e0ce3e982bca82e7d59144ddd18e1067bcd057baf33e7332c71d91d3da03c","last_reissued_at":"2026-05-17T23:54:27.517638Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:27.517638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0904.1679","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:54:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wV/NUpyzdYT4pWViF3CCnDd7CMeF2eK01zI2YCUnhXK8Bw7wy+VnDWHhycJH4jvApvR+BjdNI0a5hgMGO/kmDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T11:13:29.886648Z"},"content_sha256":"573a58958db308cf63c879126ca1c2a2047b2523517cc53022b17f4d3d2b3973","schema_version":"1.0","event_id":"sha256:573a58958db308cf63c879126ca1c2a2047b2523517cc53022b17f4d3d2b3973"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:4KHAZY7JQK6KQLT5LEKE3XIY4E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Heisenberg action in the equivariant K-theory of Hilbert schemes via Shuffle Algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.RT","authors_text":"Alexander Tsymbaliuk, Boris Feigin","submitted_at":"2009-04-10T11:32:33Z","abstract_excerpt":"In this paper we construct the action of Ding-Iohara and shuffle algebras in the sum of localized equivariant K-groups of Hilbert schemes of points on C^2. We show that commutative elements K_i of shuffle algebra act through vertex operators over positive part {h_i}_{i>0} of the Heisenberg algebra in these K-groups. Hence we get the action of Heisenberg algebra itself. Finally, we normalize the basis of the structure sheaves of fixed points in such a way that it corresponds to the basis of Macdonald polynomials in the Fock space k[h_1,h_2,...]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.1679","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:54:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V46HrHm3Dz+0dOc80Gr/LV7nr7LC7Pecy1m0vWacdDporDckU1YtKxyuAPw7aSAQMZ85a6h42nPoHB/x5KYKBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T11:13:29.887297Z"},"content_sha256":"213f3b753b07bab5e800190f039ab6d2c966c531630e31437c2ba4123a57b5f6","schema_version":"1.0","event_id":"sha256:213f3b753b07bab5e800190f039ab6d2c966c531630e31437c2ba4123a57b5f6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4KHAZY7JQK6KQLT5LEKE3XIY4E/bundle.json","state_url":"https://pith.science/pith/4KHAZY7JQK6KQLT5LEKE3XIY4E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4KHAZY7JQK6KQLT5LEKE3XIY4E/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-22T11:13:29Z","links":{"resolver":"https://pith.science/pith/4KHAZY7JQK6KQLT5LEKE3XIY4E","bundle":"https://pith.science/pith/4KHAZY7JQK6KQLT5LEKE3XIY4E/bundle.json","state":"https://pith.science/pith/4KHAZY7JQK6KQLT5LEKE3XIY4E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4KHAZY7JQK6KQLT5LEKE3XIY4E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:4KHAZY7JQK6KQLT5LEKE3XIY4E","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":"8b8bec3c5d1c9c26bd9170a0868448f75f6d7d6e7459e874d95c90237eafc124","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-04-10T11:32:33Z","title_canon_sha256":"198df9d01588c2c4199f06f7eee068ff36f75560bcdc109cd769e5e4d4da41b0"},"schema_version":"1.0","source":{"id":"0904.1679","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0904.1679","created_at":"2026-05-17T23:54:27Z"},{"alias_kind":"arxiv_version","alias_value":"0904.1679v2","created_at":"2026-05-17T23:54:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0904.1679","created_at":"2026-05-17T23:54:27Z"},{"alias_kind":"pith_short_12","alias_value":"4KHAZY7JQK6K","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"4KHAZY7JQK6KQLT5","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"4KHAZY7J","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:213f3b753b07bab5e800190f039ab6d2c966c531630e31437c2ba4123a57b5f6","target":"graph","created_at":"2026-05-17T23:54:27Z","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":"In this paper we construct the action of Ding-Iohara and shuffle algebras in the sum of localized equivariant K-groups of Hilbert schemes of points on C^2. We show that commutative elements K_i of shuffle algebra act through vertex operators over positive part {h_i}_{i>0} of the Heisenberg algebra in these K-groups. Hence we get the action of Heisenberg algebra itself. Finally, we normalize the basis of the structure sheaves of fixed points in such a way that it corresponds to the basis of Macdonald polynomials in the Fock space k[h_1,h_2,...].","authors_text":"Alexander Tsymbaliuk, Boris Feigin","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-04-10T11:32:33Z","title":"Heisenberg action in the equivariant K-theory of Hilbert schemes via Shuffle Algebra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.1679","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":"sha256:573a58958db308cf63c879126ca1c2a2047b2523517cc53022b17f4d3d2b3973","target":"record","created_at":"2026-05-17T23:54:27Z","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":"8b8bec3c5d1c9c26bd9170a0868448f75f6d7d6e7459e874d95c90237eafc124","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-04-10T11:32:33Z","title_canon_sha256":"198df9d01588c2c4199f06f7eee068ff36f75560bcdc109cd769e5e4d4da41b0"},"schema_version":"1.0","source":{"id":"0904.1679","kind":"arxiv","version":2}},"canonical_sha256":"e28e0ce3e982bca82e7d59144ddd18e1067bcd057baf33e7332c71d91d3da03c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e28e0ce3e982bca82e7d59144ddd18e1067bcd057baf33e7332c71d91d3da03c","first_computed_at":"2026-05-17T23:54:27.517638Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:54:27.517638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6AJ6N3GeYR8dbolS5gLWBnJ9WrlwfGAWiNs+/Az0fJWk0F47Z1Yv7XIZwgBKwNrargcsKfHV019J2ZBCPd9nBA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:54:27.518336Z","signed_message":"canonical_sha256_bytes"},"source_id":"0904.1679","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:573a58958db308cf63c879126ca1c2a2047b2523517cc53022b17f4d3d2b3973","sha256:213f3b753b07bab5e800190f039ab6d2c966c531630e31437c2ba4123a57b5f6"],"state_sha256":"617f59bcdf9383f104fc1380cda86a7a005db30360ea2d2c08abe3edefaed746"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w0TPLR2O7ECHs+IHG9juIeM5LXdMtDc/tWvdnr/Jm1FgFYkgXRv1ShMsxsn80ssRXjZ4RPu5sS635svGF8uMAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-22T11:13:29.890918Z","bundle_sha256":"653dec55e53ab7cbd145395bbc0030fa8cd7e33a1d3fa014d9f83e65a4bd5f88"}}