Strassen's rank additivity for small tensors, including tensors of rank less or equal 7

Abstract

The article is concerned with the problem of additivity of the tensor rank. That is, for two independent tensors, we study when the rank of their direct sum is equal to the sum of their individual ranks. The statement saying that additivity always holds was previously known as Strassen's conjecture (1969) until Shitov proposed counterexamples (2019). They are not explicit and only known to exist asymptotically for very large tensor spaces. In this article, we present families of pairs of small three-way tensors for which the additivity holds. For instance, over the base field $C$, it is the case if both tensors are of rank less or equal 7. This proves that a pair of $2\times 2$ matrix multiplication tensors has the rank additivity property. We show also that the Alexeev-Forbes-Tsimerman substitution method preserves the structure of a direct sum of tensors.