Border rank bounds for $GL(V)$-invariant tensors arising from matrices of constant rank

TL;DR

This paper establishes new border rank lower bounds for $GL(V)$-invariant tensors associated with constant rank matrices, using Young flattenings, and determines the border rank of three specific tensors.

Contribution

It introduces the first explicit use of Young flattenings beyond Koszul for border rank bounds and provides new bounds for non-$1_A$-generic tensors and unbalanced matrix multiplication tensors.

Findings

Proved border rank bounds for certain $GL(V)$-invariant tensors.

Determined the border rank of three specific tensors.

Applied Young flattenings beyond Koszul to tensor border rank estimation.

Abstract

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.