Border subrank of higher order tensors and algebras

TL;DR

This paper determines the border subrank of various higher order algebraic tensors, extending prior work on asymptotic subrank for order-three tensors to exact border subrank and higher orders.

Contribution

It provides exact border subrank bounds for multiple algebraic tensor families and explores degeneration propagation and effectiveness of upper bound methods.

Findings

Border subrank bounds for k-fold matrix multiplication tensors.

Border subrank of tensors of truncated polynomial, null, and apolar algebras.

Propagation of algebra tensor degeneration from higher to lower order.

Abstract

We determine the border subrank of higher order structure tensors of several families of algebras, and in particular obtain the following results. (1) We determine tight bounds on the border subrank of $k$-fold matrix multiplication and $k$-fold upper triangular matrix multiplication for all $k$. (2) We determine the border subrank of the higher order structure tensors of truncated polynomial algebras, null algebras, and apolar algebras of a quadric. (3) We determine the border subrank of the higher order structure tensors of the Lie algebra $\mathfrak{sl}_2$ for all orders. (4) We prove that degeneration of structure tensors of algebras propagates from higher to lower order. Along the way, we investigate which upper bound methods (geometric rank, $G$-stable rank, socle degree) are effective in which settings, and how they relate. Our work extends the results of Strassen (J.~Reine Angew.~Math., 1987, 1991), who determined the asymptotic subrank of these algebras for tensors of order three, in two directions: we determine the border subrank itself rather than its asymptotic version, and we consider higher order structure tensors.