Border subrank via a generalised Hilbert-Mumford criterion

TL;DR

The paper investigates the generic border subrank of order-$d$ tensors and proves an upper bound of $\mathcal{O}(n^{1/(d-1)})$ as the ambient dimension $n$ grows, aligning with the known generic subrank growth and highlighting a divergence between border subrank and subrank at large $n$. The approach hinges on a generalised Hilbert-Mumford criterion derived via a Cartan-Iwahori-Matsumoto decomposition in loop groups, combined with constructibility arguments and a dimension-counting framework based on a weight-incidence variety. The authors establish constructibility of border-subrank loci, derive a precise dimension bound (in terms of $s=\lfloor r/d\rfloor$) for $X_{\ge r}$, and then deduce the $O(n^{1/(d-1)})$ growth. For $d=3$ they further prove a matching lower bound up to constants and show the generic border subrank strictly exceeds the generic subrank for large $n$, underscoring non-coincidence of these invariants. The work advances invariant-theoretic methods for tensor complexity and introduces tools potentially useful beyond border subrank, including a broader Hilbert-Mumford-type criterion.

Abstract

We show that the border subrank of a sufficiently general tensor in $(\mathbb{C}^n)^{\otimes d}$ is $\mathcal{O}(n^{1/(d-1)})$ for $n \to \infty$. Since this matches the growth rate $Θ(n^{1/(d-1)})$ for the generic (non-border) subrank recently established by Derksen-Makam-Zuiddam, we find that the generic border subrank has the same growth rate. In our proof, we use a generalisation of the Hilbert-Mumford criterion that we believe will be of independent interest.

Figures (2)

• Figure 1: The pyramid $P$ seen from above along the $l$-axis. On the orange corners we have $a_{1j}+a_{2k}+a_{3l}=0$, and on the red positions (and below these and the orange corners) we have $a_{1j}+a_{2k}+a_{3l}<0$.
• Figure 2: The pyramid $P$ in red and orange for $r=4$, and the support of the additional full-rank matrices in $\tilde{T}$ in green.

Theorems & Definitions (20)

• Proposition 1
• Theorem 2: Main Theorem
• Theorem 3
• Theorem 4
• proof : Proof of Proposition \ref{['prop:Constructible']}
• Theorem 5
• proof
• Proposition 6
• proof
• Example 7