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

Abstract

We show that the product rank of the $3\times 3$ determinant is $5$, and the product rank of the $3\times 3$ permanent is $4$. As a corollary, we obtain that the tensor ranks of the $3\times 3$ determinant and permanent are $5$ and $4$, respectively. We show moreover that the border product rank of the $n\times n$ permanent is larger than $n$ for any $n\ge 3$.

9 pages. v2: added bound for border product rank of permanent

References in corpus (1)

• Fano schemes of determinants and permanents

Cited by in corpus (4)

• Rank and border rank of Kronecker powers of tensors and Strassen's laser method
• Koszul-Young Flattenings and Symmetric Border Rank of the Determinant
• On Fano Schemes of Toric Varieties
• Lower Bound on the Representation Complexity of Antisymmetric Tensor Product Functions