{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:H7QNCNZICI4GIWFMBWA6PJUHQL","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":"2ed4228092aa9848f12a9687892292e27dd4194d01e71bca51ed0779359e2504","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-05-20T21:35:48Z","title_canon_sha256":"4bec9a61a12599c28893fc28bbfd1a96cf5c68bcd78f54bedacd64f0cdcb1480"},"schema_version":"1.0","source":{"id":"1605.06543","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.06543","created_at":"2026-05-18T01:13:10Z"},{"alias_kind":"arxiv_version","alias_value":"1605.06543v2","created_at":"2026-05-18T01:13:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.06543","created_at":"2026-05-18T01:13:10Z"},{"alias_kind":"pith_short_12","alias_value":"H7QNCNZICI4G","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"H7QNCNZICI4GIWFM","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"H7QNCNZI","created_at":"2026-05-18T12:30:19Z"}],"graph_snapshots":[{"event_id":"sha256:e3191c213346d7e0a3fc93de98024aac63a32db148b0cb644c8614dda152726a","target":"graph","created_at":"2026-05-18T01:13:10Z","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":"The partition algebra $\\mathsf{P}_k(n)$ and the symmetric group $\\mathsf{S}_n$ are in Schur-Weyl duality on the $k$-fold tensor power $\\mathsf{M}_n^{\\otimes k}$ of the permutation module $\\mathsf{M}_n$ of $\\mathsf{S}_n$, so there is a surjection $\\mathsf{P}_k(n) \\to \\mathsf{Z}_k(n) := \\mathsf{End}_{\\mathsf{S}_n}(\\mathsf{M}_n^{\\otimes k}),$ which is an isomorphism when $n \\ge 2k$. We prove a dimension formula for the irreducible modules of the centralizer algebra $\\mathsf{Z}_k(n)$ in terms of Stirling numbers of the second kind. Via Schur-Weyl duality, these dimensions equal the multiplicities ","authors_text":"Georgia Benkart, Nate Harman, Tom Halverson","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-05-20T21:35:48Z","title":"Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06543","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":"sha256:8764d7705a1c7c48a2064a6a80e54d06e9cf5f0cef961aefa229680c2d4bf433","target":"record","created_at":"2026-05-18T01:13:10Z","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":"2ed4228092aa9848f12a9687892292e27dd4194d01e71bca51ed0779359e2504","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-05-20T21:35:48Z","title_canon_sha256":"4bec9a61a12599c28893fc28bbfd1a96cf5c68bcd78f54bedacd64f0cdcb1480"},"schema_version":"1.0","source":{"id":"1605.06543","kind":"arxiv","version":2}},"canonical_sha256":"3fe0d1372812386458ac0d81e7a68782ca41ea8b0a089cab25820d78dbe5145e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3fe0d1372812386458ac0d81e7a68782ca41ea8b0a089cab25820d78dbe5145e","first_computed_at":"2026-05-18T01:13:10.878815Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:13:10.878815Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cChIpzWGHAJDpCxNXQpR68EeDfDzYtMkCBfLwSVIQvw56mwIQKyA/LHurmwfvWylOcXxoLI+E712YknDV3FCDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:13:10.879220Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.06543","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8764d7705a1c7c48a2064a6a80e54d06e9cf5f0cef961aefa229680c2d4bf433","sha256:e3191c213346d7e0a3fc93de98024aac63a32db148b0cb644c8614dda152726a"],"state_sha256":"b9e1ecff35876263f03fea83e2a17990fbbcc3ffd3650d794fb40ca8d0168716"}