{"paper":{"title":"On quandle representations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A finite-dimensional representation of a finite quandle over the complex numbers decomposes into irreducibles precisely when every matrix in its image is diagonalizable.","cross_cats":[],"primary_cat":"math.RT","authors_text":"Mohamad Maassarani","submitted_at":"2026-05-12T19:40:50Z","abstract_excerpt":"A unitary finite dimensional quandle representation is decomposable into a direct sum of irreducible represenations. Not all quandle representations satisfy this property. We prove that a finite dimensional quandle represenation $\\rho :Q \\to GL(V) $ of a finite quandle $Q$ over $\\mathbb{C}$ is decomposable into a direct sum of irreducibles if and only if every element in the image of $\\rho$ is diagonlizable. We show that an irreducible representation $\\rho :Q \\to GL(V)$ of a finite quandle over $\\mathbb{C}$ is unitary for some inner product if and only if every element of the image of $\\rho$ h"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"A finite dimensional quandle representation ρ : Q → GL(V) of a finite quandle Q over ℂ is decomposable into a direct sum of irreducibles if and only if every element in the image of ρ is diagonalizable.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The quandle Q is finite and the representation is finite-dimensional over ℂ; these hypotheses are used to invoke diagonalizability and determinant properties that may fail for infinite quandles or other fields.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Finite-dimensional representations of finite quandles over ℂ decompose into irreducibles iff image elements are diagonalizable, and irreducibles are unitary for some inner product iff determinants have modulus 1.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A finite-dimensional representation of a finite quandle over the complex numbers decomposes into irreducibles precisely when every matrix in its image is diagonalizable.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c8cb1d593e0836d2973d3f54fa3844f87b81f2a40ad835c4fde4945a86c3bc42"},"source":{"id":"2605.12692","kind":"arxiv","version":1},"verdict":{"id":"4c956c53-ac37-4253-a6da-c6bcb7c4eb13","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T19:58:41.988096Z","strongest_claim":"A finite dimensional quandle representation ρ : Q → GL(V) of a finite quandle Q over ℂ is decomposable into a direct sum of irreducibles if and only if every element in the image of ρ is diagonalizable.","one_line_summary":"Finite-dimensional representations of finite quandles over ℂ decompose into irreducibles iff image elements are diagonalizable, and irreducibles are unitary for some inner product iff determinants have modulus 1.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The quandle Q is finite and the representation is finite-dimensional over ℂ; these hypotheses are used to invoke diagonalizability and determinant properties that may fail for infinite quandles or other fields.","pith_extraction_headline":"A finite-dimensional representation of a finite quandle over the complex numbers decomposes into irreducibles precisely when every matrix in its image is diagonalizable."},"references":{"count":6,"sample":[{"doi":"","year":2014,"title":"Eisermann, Michael, Quandle Coverings and Their Galois Correspondence. Fundamenta Mathematicae, vol. 225, no. 1, pp. 103–67, 2014","work_id":"9b72519c-8513-460f-adf6-6759da38f75e","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"Elhamdadi, M. and Moutuou, E. kaïoum M.. Finitely stable racks and rack representations. 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