Real subrank of order-three tensors

Abstract

We study the subrank of real order-three tensors and give a lower bound to the subrank of a real tensor given its complex subrank. Using similar arguments to those used by Bernardi-Blekherman-Ottaviani, we show that all subranks between the minimal typical subrank and the maximal typical subrank, which equals the generic subrank, are also typical. We then study small tensor formats with more than one typical subrank. In particular, we construct a $3 \times 3 \times 5$-tensor with subrank $2$ and show that the subrank of the $4 \times 4 \times 4$-quaternion multiplication tensor is $2$. Finally, we consider the tensor associated to componentwise complex multiplication in $\mathbb{C}^n$ and show that this tensor has real subrank $n$---informally, {\em no more than $n$ real scalar multiplications can be carried out using a device that does $n$ complex scalar multiplications}. We also prove a version of this result for other real division algebras.