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Product Ranks of the $3\times 3$ Determinant and Permanent

  Published:2016-01-21
 Printed: Jun 2016
  • Nathan Ilten,
    Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby BC V5A1S6
  • Zach Teitler,
    Department of Mathematics, Boise State University, 1910 University Drive Boise, ID 83725-1555, USA
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Abstract

We show that the product rank of the $3 \times 3$ determinant $\det_3$ is $5$, and the product rank of the $3 \times 3$ permanent $\operatorname{perm}_3$ is $4$. As a corollary, we obtain that the tensor rank of $\det_3$ is $5$ and the tensor rank of $\operatorname{perm}_3$ is $4$. We show moreover that the border product rank of $\operatorname{perm}_n$ is larger than $n$ for any $n\geq 3$.
Keywords: product rank, tensor rank, determinant, permanent, Fano schemes product rank, tensor rank, determinant, permanent, Fano schemes
MSC Classifications: 15A21, 15A69, 14M12, 14N15 show english descriptions Canonical forms, reductions, classification
Multilinear algebra, tensor products
Determinantal varieties [See also 13C40]
Classical problems, Schubert calculus
15A21 - Canonical forms, reductions, classification
15A69 - Multilinear algebra, tensor products
14M12 - Determinantal varieties [See also 13C40]
14N15 - Classical problems, Schubert calculus
 

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