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:粒度的下等值是测量“ 解析” 。 这个参数是由斯特拉斯森引入的, 用于在代数复杂理论中研究快速矩阵乘法乘数算法, 并且与许多中枢参数( 如切片级、 分区级、 分析级、 几何级、 G- 稳定级) 以及组合式、 计算机科学和量子信息理论中的问题密切相关。 Strassen (J. Reine Angew. Math., 1988) 证明了在采用高压产品下的巨大权力时, 下级存在差距: 要么所有权力的下级最多是其中之一, 要么它作为恒定值严格大于一个的动力而增长。 在本文中, 我们精确地决定了这个常数值的常数。 此外, 对于组合的三, 我们证明在可能的增长率上还有第二个差距。 我们的结果加强了科斯塔和达赖最近的工作( J. Comb. Thery, Ser. A, 2021), 在采用高压等级时, 我们的底级应用中有一个相似的空隙差距, 只能以正等为主级。