{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:WTJZDK6D5BUJR4O6YOCOE4ZJF4","short_pith_number":"pith:WTJZDK6D","schema_version":"1.0","canonical_sha256":"b4d391abc3e86898f1dec384e273292f15d58cee8d21764a82e6a02ff17a33d5","source":{"kind":"arxiv","id":"1409.8356","version":7},"attestation_state":"computed","paper":{"title":"Hopf Algebras in Combinatorics","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.CO","authors_text":"Darij Grinberg, Victor Reiner","submitted_at":"2014-09-30T00:38:48Z","abstract_excerpt":"These notes -- originating from a one-semester class by their second author at the University of Minnesota -- survey some of the most important Hopf algebras appearing in combinatorics. After introducing coalgebras, bialgebras and Hopf algebras in general, we study the Hopf algebra of symmetric functions, including Zelevinsky's axiomatic characterization of it as a \"positive self-adjoint Hopf algebra\" and its application to the representation theory of symmetric and (briefly) finite general linear groups. The notes then continue with the quasisymmetric and the noncommutative symmetric function"},"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":"1409.8356","kind":"arxiv","version":7},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2014-09-30T00:38:48Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"8e069ee5643bf99862d7b20bf93411aab9dc308705f7b1c66f790dd69b77508a","abstract_canon_sha256":"71fdefc531cddf8d8d1875006c7c7995ff5cd0b292b947f8d40baf88c6b1f327"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:22:27.283113Z","signature_b64":"uac7C3+Kxabrerlu8YfAKkrx2LqdBd602s5ZHvojz/pUlIXaMe8g8bRhZdk1Z+2jgqlIfzIRSgRYW/C/nqkhCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b4d391abc3e86898f1dec384e273292f15d58cee8d21764a82e6a02ff17a33d5","last_reissued_at":"2026-07-05T01:22:27.282624Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:22:27.282624Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hopf Algebras in Combinatorics","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.CO","authors_text":"Darij Grinberg, Victor Reiner","submitted_at":"2014-09-30T00:38:48Z","abstract_excerpt":"These notes -- originating from a one-semester class by their second author at the University of Minnesota -- survey some of the most important Hopf algebras appearing in combinatorics. After introducing coalgebras, bialgebras and Hopf algebras in general, we study the Hopf algebra of symmetric functions, including Zelevinsky's axiomatic characterization of it as a \"positive self-adjoint Hopf algebra\" and its application to the representation theory of symmetric and (briefly) finite general linear groups. The notes then continue with the quasisymmetric and the noncommutative symmetric function"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.8356","kind":"arxiv","version":7},"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/1409.8356/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":"1409.8356","created_at":"2026-07-05T01:22:27.282682+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.8356v7","created_at":"2026-07-05T01:22:27.282682+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.8356","created_at":"2026-07-05T01:22:27.282682+00:00"},{"alias_kind":"pith_short_12","alias_value":"WTJZDK6D5BUJ","created_at":"2026-07-05T01:22:27.282682+00:00"},{"alias_kind":"pith_short_16","alias_value":"WTJZDK6D5BUJR4O6","created_at":"2026-07-05T01:22:27.282682+00:00"},{"alias_kind":"pith_short_8","alias_value":"WTJZDK6D","created_at":"2026-07-05T01:22:27.282682+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":8,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2607.07870","citing_title":"The Antipodes of $q$-Quasi-Symmetric Functions and Non-Commutative Quasi-Symmetric Functions","ref_index":19,"is_internal_anchor":true},{"citing_arxiv_id":"2606.23863","citing_title":"The Goncharov Lie coalgebra of a field","ref_index":61,"is_internal_anchor":false},{"citing_arxiv_id":"2605.30390","citing_title":"A Boundary--Residue Incidence Coalgebra for Associahedral Scattering Forms","ref_index":32,"is_internal_anchor":false},{"citing_arxiv_id":"1907.09975","citing_title":"Hopf algebra structure of symmetric and quasisymmetric functions in superspace","ref_index":11,"is_internal_anchor":false},{"citing_arxiv_id":"2509.18625","citing_title":"A formula for the Jack super nabla operator","ref_index":20,"is_internal_anchor":false},{"citing_arxiv_id":"2511.02649","citing_title":"A geometric and generating function approach to plethysm","ref_index":7,"is_internal_anchor":false},{"citing_arxiv_id":"2604.06431","citing_title":"On the quasisymmetric functions in superspace","ref_index":16,"is_internal_anchor":false},{"citing_arxiv_id":"2604.05534","citing_title":"A Pardon Algebra for Zero-cycles","ref_index":5,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WTJZDK6D5BUJR4O6YOCOE4ZJF4","json":"https://pith.science/pith/WTJZDK6D5BUJR4O6YOCOE4ZJF4.json","graph_json":"https://pith.science/api/pith-number/WTJZDK6D5BUJR4O6YOCOE4ZJF4/graph.json","events_json":"https://pith.science/api/pith-number/WTJZDK6D5BUJR4O6YOCOE4ZJF4/events.json","paper":"https://pith.science/paper/WTJZDK6D"},"agent_actions":{"view_html":"https://pith.science/pith/WTJZDK6D5BUJR4O6YOCOE4ZJF4","download_json":"https://pith.science/pith/WTJZDK6D5BUJR4O6YOCOE4ZJF4.json","view_paper":"https://pith.science/paper/WTJZDK6D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.8356&json=true","fetch_graph":"https://pith.science/api/pith-number/WTJZDK6D5BUJR4O6YOCOE4ZJF4/graph.json","fetch_events":"https://pith.science/api/pith-number/WTJZDK6D5BUJR4O6YOCOE4ZJF4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WTJZDK6D5BUJR4O6YOCOE4ZJF4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WTJZDK6D5BUJR4O6YOCOE4ZJF4/action/storage_attestation","attest_author":"https://pith.science/pith/WTJZDK6D5BUJR4O6YOCOE4ZJF4/action/author_attestation","sign_citation":"https://pith.science/pith/WTJZDK6D5BUJR4O6YOCOE4ZJF4/action/citation_signature","submit_replication":"https://pith.science/pith/WTJZDK6D5BUJR4O6YOCOE4ZJF4/action/replication_record"}},"created_at":"2026-07-05T01:22:27.282682+00:00","updated_at":"2026-07-05T01:22:27.282682+00:00"}