Recursive Koszul flattenings of determinant and permanent tensors
Jong In Han, Jeong-Hoon Ju, Yeongrak Kim
TL;DR
The paper develops and applies the recursive Koszul flattening technique to determinant and permanent tensors to obtain strong tensor-rank lower bounds. By leveraging $\operatorname{SL}_n$-representation theory, it proves a universal lower bound $\underline{\mathbf{R}}(\det_n) \ge \dfrac{n^{n-1}}{(n-1)!}$, which separates determinant from permanent since $\mathbf{R}(\operatorname{perm}_n) \le 2^{n-1}$. It also yields the exact tensor ranks for small cases, notably $\mathbf{R}(\det_4)=12$ and $\mathbf{R}(\operatorname{perm}_4)=8$ over fields with characteristic not equal to $2$, and provides computational lower bounds for $n\le 7$ in the permanent case. The results highlight a rapid growth gap for determinants versus permanents in tensor rank and showcase the utility of recursive Koszul flattenings for high-order tensors.
Abstract
We investigate new lower bounds on the tensor rank of the determinant and the permanent tensors via recursive usage of the Koszul flattening method introduced by Landsberg-Ottaviani and Hauenstein-Oeding-Ottaviani-Sommese. Our lower bounds on $\mathbf{R} (\det_n)$ completely separate the determinant and the permanent tensors by their tensor ranks. Furthermore, we determine the exact tensor ranks $\mathbf{R} (\det_4) = 12$ and $\mathbf{R} (\operatorname{perm}_4) = 8$ over arbitrary field of characteristic $\neq 2$.