{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:4GZRJL24BZTUQMCMS5UIAWL5C3","short_pith_number":"pith:4GZRJL24","canonical_record":{"source":{"id":"1605.04817","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-16T15:58:59Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"3b07199169fce84a84664a597786e8da975b24bdac173898a72ee0f649fe9d39","abstract_canon_sha256":"c9e40b6f7a50ccde76b4e74355b4f96ab138a412cb0237b9fd7196d10e752fdf"},"schema_version":"1.0"},"canonical_sha256":"e1b314af5c0e6748304c976880597d16cca931eb9a2edf76b23b070b84092e34","source":{"kind":"arxiv","id":"1605.04817","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.04817","created_at":"2026-05-18T01:14:46Z"},{"alias_kind":"arxiv_version","alias_value":"1605.04817v1","created_at":"2026-05-18T01:14:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.04817","created_at":"2026-05-18T01:14:46Z"},{"alias_kind":"pith_short_12","alias_value":"4GZRJL24BZTU","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4GZRJL24BZTUQMCM","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4GZRJL24","created_at":"2026-05-18T12:29:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:4GZRJL24BZTUQMCMS5UIAWL5C3","target":"record","payload":{"canonical_record":{"source":{"id":"1605.04817","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-16T15:58:59Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"3b07199169fce84a84664a597786e8da975b24bdac173898a72ee0f649fe9d39","abstract_canon_sha256":"c9e40b6f7a50ccde76b4e74355b4f96ab138a412cb0237b9fd7196d10e752fdf"},"schema_version":"1.0"},"canonical_sha256":"e1b314af5c0e6748304c976880597d16cca931eb9a2edf76b23b070b84092e34","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:14:46.201215Z","signature_b64":"SXUA82uf1p3V5GKTrBZE6R0U5IJCUH1btBKBr+sCiCzlrmhfbpMW62JCNbXLxXuxcz1U2vlYmIrFSvNnMgTaDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e1b314af5c0e6748304c976880597d16cca931eb9a2edf76b23b070b84092e34","last_reissued_at":"2026-05-18T01:14:46.200380Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:14:46.200380Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1605.04817","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:14:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RGZ2muWptqHZWzanXpbIGjnX40GHHiZePZhDw4t/kwE8UU/ocP3LPiixC6+EfzDArAD+L/Q/iaVngmropsitBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T09:30:20.912717Z"},"content_sha256":"0953d7d55f45e9bf501b8038c73ca9cad15d289e4d1ba3df555ea5fd6aae944f","schema_version":"1.0","event_id":"sha256:0953d7d55f45e9bf501b8038c73ca9cad15d289e4d1ba3df555ea5fd6aae944f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:4GZRJL24BZTUQMCMS5UIAWL5C3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A t-generalization for Schubert Representatives of the Affine Grassmannian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.CO","authors_text":"Avinash J. Dalal, Jennifer Morse","submitted_at":"2016-05-16T15:58:59Z","abstract_excerpt":"We introduce two families of symmetric functions with an extra parameter t that specialize to Schubert representatives for cohomology and homology of the affine Grassmannian when t = 1. The families are defined by a statistic on combinatorial objects associated to the type-A affine Weyl group and their transition matrix with Hall-Littlewood polynomials is t-positive. We conjecture that one family is the set of k-atoms."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.04817","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:14:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/PhgHVIMySk714zFh1NmQCh41JAem0FjJZ2R/AjCpV8I1BcOa6FQjY9e7/48vcNS1SQGeeg6mYsBKTPczu37AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T09:30:20.913399Z"},"content_sha256":"188107f264b4d628a0b3bae4b5639c4fb527ce205e8264eeafbbf8358355c614","schema_version":"1.0","event_id":"sha256:188107f264b4d628a0b3bae4b5639c4fb527ce205e8264eeafbbf8358355c614"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4GZRJL24BZTUQMCMS5UIAWL5C3/bundle.json","state_url":"https://pith.science/pith/4GZRJL24BZTUQMCMS5UIAWL5C3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4GZRJL24BZTUQMCMS5UIAWL5C3/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":"2026-05-27T09:30:20Z","links":{"resolver":"https://pith.science/pith/4GZRJL24BZTUQMCMS5UIAWL5C3","bundle":"https://pith.science/pith/4GZRJL24BZTUQMCMS5UIAWL5C3/bundle.json","state":"https://pith.science/pith/4GZRJL24BZTUQMCMS5UIAWL5C3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4GZRJL24BZTUQMCMS5UIAWL5C3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:4GZRJL24BZTUQMCMS5UIAWL5C3","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":"c9e40b6f7a50ccde76b4e74355b4f96ab138a412cb0237b9fd7196d10e752fdf","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-16T15:58:59Z","title_canon_sha256":"3b07199169fce84a84664a597786e8da975b24bdac173898a72ee0f649fe9d39"},"schema_version":"1.0","source":{"id":"1605.04817","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.04817","created_at":"2026-05-18T01:14:46Z"},{"alias_kind":"arxiv_version","alias_value":"1605.04817v1","created_at":"2026-05-18T01:14:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.04817","created_at":"2026-05-18T01:14:46Z"},{"alias_kind":"pith_short_12","alias_value":"4GZRJL24BZTU","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4GZRJL24BZTUQMCM","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4GZRJL24","created_at":"2026-05-18T12:29:58Z"}],"graph_snapshots":[{"event_id":"sha256:188107f264b4d628a0b3bae4b5639c4fb527ce205e8264eeafbbf8358355c614","target":"graph","created_at":"2026-05-18T01:14:46Z","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 introduce two families of symmetric functions with an extra parameter t that specialize to Schubert representatives for cohomology and homology of the affine Grassmannian when t = 1. The families are defined by a statistic on combinatorial objects associated to the type-A affine Weyl group and their transition matrix with Hall-Littlewood polynomials is t-positive. We conjecture that one family is the set of k-atoms.","authors_text":"Avinash J. Dalal, Jennifer Morse","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-16T15:58:59Z","title":"A t-generalization for Schubert Representatives of the Affine Grassmannian"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.04817","kind":"arxiv","version":1},"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":"sha256:0953d7d55f45e9bf501b8038c73ca9cad15d289e4d1ba3df555ea5fd6aae944f","target":"record","created_at":"2026-05-18T01:14:46Z","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":"c9e40b6f7a50ccde76b4e74355b4f96ab138a412cb0237b9fd7196d10e752fdf","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-16T15:58:59Z","title_canon_sha256":"3b07199169fce84a84664a597786e8da975b24bdac173898a72ee0f649fe9d39"},"schema_version":"1.0","source":{"id":"1605.04817","kind":"arxiv","version":1}},"canonical_sha256":"e1b314af5c0e6748304c976880597d16cca931eb9a2edf76b23b070b84092e34","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e1b314af5c0e6748304c976880597d16cca931eb9a2edf76b23b070b84092e34","first_computed_at":"2026-05-18T01:14:46.200380Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:14:46.200380Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SXUA82uf1p3V5GKTrBZE6R0U5IJCUH1btBKBr+sCiCzlrmhfbpMW62JCNbXLxXuxcz1U2vlYmIrFSvNnMgTaDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:14:46.201215Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.04817","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0953d7d55f45e9bf501b8038c73ca9cad15d289e4d1ba3df555ea5fd6aae944f","sha256:188107f264b4d628a0b3bae4b5639c4fb527ce205e8264eeafbbf8358355c614"],"state_sha256":"6115994d4d4b35197a0a2008ac3f826169fe52c09237b5b4a0351c6739741591"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qcn6x9QRz5Z5dbrkt8cPpsMAw+pPu8ndZBT4CZfjaVmqlZNp55vhRA0bdoP3O3VQ21H1Bdo7EkKJovmqXSfxAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T09:30:20.917097Z","bundle_sha256":"3acf1616c387fecf582d2d6b6820340712cd46251db70ae4f5de878717d1da26"}}