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

TL;DR

This paper establishes new lower bounds on the border rank of certain GL(V)-invariant tensors, resolving a conjecture and extending the results to symmetric and wedge powers, with implications for tensor complexity.

Contribution

It resolves Wu's conjecture on border rank bounds for V-invariant tensors and generalizes the results to symmetric and wedge powers using advanced algebraic techniques.

Findings

Resolved Wu's conjecture on border rank bounds.

Extended bounds to symmetric and wedge power tensors.

Identified non-minimal border rank tensors using Kempf collapsing.

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.