论文标题
张量的子班轮的间隙
A Gap in the Subrank of Tensors
论文作者
论文摘要
张量的子班轮是对张量可以“对角线化”的量度的量度。 Strassen引入了该参数，以研究代数复杂性理论中的快速矩阵乘法算法，并且与许多中央张量参数（例如，切片等级，分区等级，几何级别，G-Stable等级，G-Stable等级）和组合术，计算机科学和定量信息理论密切相关。 Strassen（J. ReineAngew。Math。，1988）证明，在张量产品下采用大能力时，子班级存在差距：要么所有权力的子级别最多是一种，要么它会随着常数严格大于一个的力量而增长。在本文中，我们精确地确定了任何顺序的张量。此外，对于第三阶的张量，我们证明可能的增长率存在第二个差距。我们的结果加强了Costa和Dalai的最新工作（J.Comb。Theoly，Ser。A，2021），他们证明了Slice等级的差距类似。我们在子班级上的定理具有更广泛的应用程序，这不仅暗示了切片等级的差距，而且还暗示了任何“归一化单调”的差距。为了证明主要结果，我们表征了张量何时在其轨道闭合中具有非常结构化的张量（W量）。我们的方法包括格拉马尼亚人的退化，这可能具有独立的兴趣。
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.
