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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 $\\VV_\\qq$."},"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":"1201.2482","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-01-12T06:12:32Z","cross_cats_sorted":[],"title_canon_sha256":"89c4ee768e3aa895e6dd231a4f219c828f2f3cca0f43bf57aa0cc858989feb64","abstract_canon_sha256":"9f5e3847e35d8c7139228d4cab71234993917bc91a0bb848684087df5c14b3f1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:04:44.312398Z","signature_b64":"Ix1HKGyqSuhIddfUwfPH4ypFgcp9qyFYw41svjglPNLscZ4/8ba7iVy3cjh02ZAPu2FYck9Qr9ffs3QKB3irBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7c7e0df37831f0cd93b26861d70c1a817fcca17abbf0bc282e873588318cf3fc","last_reissued_at":"2026-05-18T04:04:44.311925Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:04:44.311925Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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