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., 1988) 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. Comb. Theory, Ser. A, 2021), 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.

翻译：张量子秩是衡量张量可“对角化”程度的参数，由斯特拉森在代数复杂度理论中为研究快速矩阵乘法算法而提出，与许多核心张量参数（例如切片秩、分割秩、解析秩、几何秩、G-稳定秩）以及组合学、计算机科学和量子信息论中的问题密切相关。斯特拉森（《纯粹与应用数学杂志》，1988年）证明，在对张量积取大幂次时，子秩存在一个间隙：要么所有幂次的子秩至多为1，要么其增长速度严格大于1的常数幂次。本文精确确定了任意阶张量的这一常数。此外，对于三阶张量，我们证明了增长速率存在第二个间隙。我们的结果强化了科斯塔和达拉伊（《组合理论杂志A辑》，2021年）近期关于切片秩存在类似间隙的工作。我们的子秩定理具有更广泛的应用：它不仅能推导出切片秩的此类间隙，还可适用于任何“归一化单调”参数。为了证明主要结论，我们刻画了当张量在其轨道闭包中具有高度结构化张量（W-张量）时的条件。我们的方法包括格拉斯曼流形中的退化，这可能具有独立的研究价值。