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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
 Format: LaTeX MathJax PDF

## 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
 MSC Classifications: 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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