The interplay between bounded ranks of tensors arising from partitions

Abstract

Let $d\ge 2,h\ge 1$ be integers. Using a fragmentation technique, we characterise $(h+1)$-tuples $(R_1,\dots ,R_h,R)$ of non-empty families of partitions of $\{1,\dots ,d\}$ such that it suffices for an order-$d$ tensor to have bounded $R_i$-rank for each $i=1,\dots ,h$ for it to have bounded $R$-rank. On the way, we prove power lower bounds on products of identity tensors that do not have rank $1$, providing a qualitative answer to a question of Naslund.

25 pages