{"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"}