Border rank bounds for $GL(V)$-invariant tensors arising from matrices of constant rank

Abstract

We prove border rank bounds for a class of $GL(V)$-invariant tensors in $V^{\ast }\otimes U\otimes W$, where $U$ and $W$ are $GL(V)$-modules. These tensors correspond to spaces of matrices of constant rank. In particular we prove lower bounds for tensors in ${\mathbb{C}}^l\otimes {\mathbb{C}}^m\otimes {\mathbb{C}}^n$ that are not $1_A$-generic, where no nontrivial bounds were known, and also when $l,m\ll n$, where previously only bounds for unbalanced matrix multiplication tensors were known. We give the first explicit use of Young flattenings for tensors beyond Koszul to obtain border rank lower bounds, and determine the border rank of three tensors.