Border rank lower bounds for families of GL(V)-invariant tensors

Abstract

We give non-trivial lower bounds for the border rank of families of $\mathbf{GL}(V)$-invariant tensors in $U\otimes \mathbf{S}_\lambda V\otimes \mathbf{S}_\mu V$ where $U$ is $V$, $\mathrm{Sym}^2V$ or $\bigwedge^2V$. We build on the techniques introduced by Wu, who used Young flattenings to obtain bounds for a family of tensors when $U$ is $V$. We complete this case by resolving a conjecture introduced by Wu, using certain pure resolutions constructed by Ford-Levinson-Sam. We then use a theorem of Kostant to generalise this to $\mathrm{Sym}^2 V$ and $\bigwedge^2 V$, and extend the number of examples of $\mathbf{GL}(V)$-invariant tensors that are not of minimal border rank using Kempf collapsing.