A gap in the subrank of tensors
SIAM Journal on Applied Algebra and Geometry , Volume 7 - Issue 4
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., 375–376 (1988), pp. 406–443] 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. Combin. Theory, Ser. A, 177 (2021), 105335] 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.