  We prove border rank bounds for a class of $GL(V)$-invariant tensors in
$V^*\otimes U\otimes W$, 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.
