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 $V$, ${\mathrm{Sym}}^{2}V$ or ${\bigwedge }^{2}V$. In particular, we provide a family of tensors with border rank reaching arbitrarily close to $2\ell$ in the unbalanced case, where $\ell$ is the largest ambient vector space dimension. We do this by resolving a conjecture introduced by Wu, and obtaining new results on $6j$-symbols as a byproduct. We then generalize our results to ${\mathrm{Sym}}^{2}V$ and ${\bigwedge }^{2}V$ using novel techniques based on an application of a theorem of Kostant and Kempf collapsing.

33 pages. Added Explicit bounds and connection to $6j$-symbols in version 2