{"paper":{"title":"Planar Rook Algebras and Tensor Representations of gl(1|1)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"D. Moon, G. Benkart","submitted_at":"2012-01-12T06:12:32Z","abstract_excerpt":"We establish a connection between planar rook algebras and tensor representations $\\VV^{\\otimes k}$ of the natural two-dimensional representation $\\VV$ of the general linear Lie superalgebra $\\gl$. In particular, we show that the centralizer algebra $\\maths{End}_{\\gl}(\\VV^{\\otimes k})$ is the planar rook algebra $\\CC \\mathsf{P}_{k-1}$ for all $k \\geq 1$, and we exhibit an explicit decomposition of $\\VV^{\\otimes k}$ into irreducible $\\gl$-modules. We obtain similar results for the quantum enveloping algebra $\\UU_\\qq(\\gl)$ and its natural two-dimensional module $\\VV_\\qq$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.2482","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"}