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

We give non-trivial lower bounds for the border rank of families of $\mathbf{G}\mathbf{L}(V)$-invariant tensors in $U\otimes {\mathbf{S}}_{\lambda }V\otimes {\mathbf{S}}_{\mu }V$ where $U$ is $V$, ${\mathrm{S}\mathrm{y}\mathrm{m}}^{2}V$ or ${\bigwedge }^{2}V$. 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 ${\mathrm{S}\mathrm{y}\mathrm{m}}^{2}V$ and ${\bigwedge }^{2}V$, and extend the number of examples of $\mathbf{G}\mathbf{L}(V)$-invariant tensors that are not of minimal border rank using Kempf collapsing.