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The partition algebra $\\mathsf{P}_k(n)$ maps surjectively onto the centralizer algebra $\\mathsf{End}_{\\mathsf{S}_n}(\\mathsf{M}_n^{\\otimes k})$ for all $k, n \\in \\mathbb{Z}_{\\ge 1}$ and isomorphically when $n \\ge 2k$. We describe the image of the surjection $\\Phi_{k,n}:\\mathsf{P}_k(n) \\to \\mathsf{End}_{\\mathsf{S}_n}(\\mathsf{M}_n^{\\otimes k})$ explicitly in terms of the orbit basis of $\\mathsf{P}_k(n)$ and show that when $2k > n$"},"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":"1707.01410","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-07-05T14:17:18Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"f9fc82090def1205dedc7f12d0c7b570ca9aebbbf45361e48d7d3c6d843c5d4e","abstract_canon_sha256":"a5254072c956d2e3f71edf03998e341433aeef6785253ea0b1ed7c8cb8fdd403"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:17.843832Z","signature_b64":"hQyf2t4fmDYXSrHhLsdYbV0Pn5xlwaSP9R9ytnw2NidcMxp39sr+vIVsTGPMUxjvCU9EXyT023KIXKsjk0uIAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e0b8bff4a3b65a753650bbaa8dc1f2a3fec964c6e551e89f3f932ff8cadb9629","last_reissued_at":"2026-05-18T00:04:17.843119Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:17.843119Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Partition algebras $\\mathsf{P}_k(n)$ with $2k>n$ and the fundamental theorems of invariant theory for the symmetric group $\\mathsf{S}_n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RT","authors_text":"Georgia Benkart, Tom Halverson","submitted_at":"2017-07-05T14:17:18Z","abstract_excerpt":"Assume $\\mathsf{M}_n$ is the $n$-dimensional permutation module for the symmetric group $\\mathsf{S}_n$, and let $\\mathsf{M}_n^{\\otimes k}$ be its $k$-fold tensor power. The partition algebra $\\mathsf{P}_k(n)$ maps surjectively onto the centralizer algebra $\\mathsf{End}_{\\mathsf{S}_n}(\\mathsf{M}_n^{\\otimes k})$ for all $k, n \\in \\mathbb{Z}_{\\ge 1}$ and isomorphically when $n \\ge 2k$. We describe the image of the surjection $\\Phi_{k,n}:\\mathsf{P}_k(n) \\to \\mathsf{End}_{\\mathsf{S}_n}(\\mathsf{M}_n^{\\otimes k})$ explicitly in terms of the orbit basis of $\\mathsf{P}_k(n)$ and show that when $2k > n$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.01410","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1707.01410","created_at":"2026-05-18T00:04:17.843232+00:00"},{"alias_kind":"arxiv_version","alias_value":"1707.01410v1","created_at":"2026-05-18T00:04:17.843232+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.01410","created_at":"2026-05-18T00:04:17.843232+00:00"},{"alias_kind":"pith_short_12","alias_value":"4C4L75FDWZNH","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_16","alias_value":"4C4L75FDWZNHKNSQ","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_8","alias_value":"4C4L75FD","created_at":"2026-05-18T12:30:58.224056+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4C4L75FDWZNHKNSQXOVI3QPSUP","json":"https://pith.science/pith/4C4L75FDWZNHKNSQXOVI3QPSUP.json","graph_json":"https://pith.science/api/pith-number/4C4L75FDWZNHKNSQXOVI3QPSUP/graph.json","events_json":"https://pith.science/api/pith-number/4C4L75FDWZNHKNSQXOVI3QPSUP/events.json","paper":"https://pith.science/paper/4C4L75FD"},"agent_actions":{"view_html":"https://pith.science/pith/4C4L75FDWZNHKNSQXOVI3QPSUP","download_json":"https://pith.science/pith/4C4L75FDWZNHKNSQXOVI3QPSUP.json","view_paper":"https://pith.science/paper/4C4L75FD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1707.01410&json=true","fetch_graph":"https://pith.science/api/pith-number/4C4L75FDWZNHKNSQXOVI3QPSUP/graph.json","fetch_events":"https://pith.science/api/pith-number/4C4L75FDWZNHKNSQXOVI3QPSUP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4C4L75FDWZNHKNSQXOVI3QPSUP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4C4L75FDWZNHKNSQXOVI3QPSUP/action/storage_attestation","attest_author":"https://pith.science/pith/4C4L75FDWZNHKNSQXOVI3QPSUP/action/author_attestation","sign_citation":"https://pith.science/pith/4C4L75FDWZNHKNSQXOVI3QPSUP/action/citation_signature","submit_replication":"https://pith.science/pith/4C4L75FDWZNHKNSQXOVI3QPSUP/action/replication_record"}},"created_at":"2026-05-18T00:04:17.843232+00:00","updated_at":"2026-05-18T00:04:17.843232+00:00"}