A Gap in the Subrank of Tensors

TL;DR

This paper determines the precise growth constants for the subrank of tensors of any order, revealing new gaps in tensor parameters and extending recent results on slice rank to a broader class of tensor measures.

Contribution

It precisely characterizes the growth constant for the subrank of tensors of any order and identifies a second gap for third-order tensors, extending prior work on slice rank.

Findings

Identifies the exact growth constant for subrank in tensors of any order.

Proves a second gap in growth rates for third-order tensors.

Extends gap results from slice rank to all normalized monotones.

Abstract

The subrank of tensors is a measure of how much a tensor can be ''diagonalized''. This parameter was introduced by Strassen to study fast matrix multiplication algorithms in algebraic complexity theory and is closely related to many central tensor parameters (e.g. slice rank, partition rank, analytic rank, geometric rank, G-stable rank) and problems in combinatorics, computer science and quantum information theory. Strassen (J. Reine Angew. Math., 1988) proved that there is a gap in the subrank when taking large powers under the tensor product: either the subrank of all powers is at most one, or it grows as a power of a constant strictly larger than one. In this paper, we precisely determine this constant for tensors of any order. Additionally, for tensors of order three, we prove that there is a second gap in the possible rates of growth. Our results strengthen the recent work of Costa and Dalai (J. Comb. Theory, Ser. A, 2021), who proved a similar gap for the slice rank. Our theorem on the subrank has wider applications by implying such gaps not only for the slice rank, but for any ``normalized monotone''. In order to prove the main result, we characterize when a tensor has a very structured tensor (the W-tensor) in its orbit closure. Our methods include degenerations in Grassmanians, which may be of independent interest.