{"paper":{"title":"Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RT","authors_text":"Georgia Benkart, Nate Harman, Tom Halverson","submitted_at":"2016-05-20T21:35:48Z","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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06543","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"}