张量的对称子秩及其应用
Symmetric Subrank of Tensors and Applications
摘要：
Strassen（Strassen，J. Reine Angew.数学，375/376，1987）引入了张量的子秩作为矩阵秩到张量的自然扩展。子秩度量可以通过对张量的不同索引（分支）应用线性运算获得的最大对角张量（就像矩阵秩度量可以使用行和列运算获得的最大对角矩阵）。在组合学和复杂性理论的问题的启发下，我们引入了张量的对称子秩的新概念，通过限制这些线性运算对每个索引都是相同的。我们证明精确的关系和分离之间的子秩和对称子秩。我们证明了对称张量的子秩和对称子秩是渐近相等的。这证明了张量理论中一个称为Comon猜想的猜想的渐近次秩类比。这一结果使我们能够证明一个强有力的连接之间的一般和对称版本的渐近对偶定理的Strassen。我们引入了一种表示论方法来渐近约束对称子秩，称为对称量子泛函，类似于量子泛函（Christandl，Vrana，Zuiddam，J. Amer. Math. Soc.，2021），我们研究了这些泛函之间的关系。
英文摘要：
Strassen (Strassen, J. Reine Angew. Math., 375/376, 1987) introduced the subrank of a tensor as a natural extension of matrix rank to tensors. Subrank measures the largest diagonal tensor that can be obtained by applying linear operations to the different indices (legs) of the tensor (just like the matrix rank measures the largest diagonal matrix that can be obtained using row and column operations). Motivated by problems in combinatorics and complexity theory we introduce the new notion of symmetric subrank of tensors by restricting these linear operations to be the same for each index. We prove precise relations and separations between subrank and symmetric subrank. We prove that for symmetric tensors the subrank and the symmetric subrank are asymptotically equal. This proves the asymptotic subrank analogon of a conjecture known as Comon's conjecture in the theory of tensors. This result allows us to prove a strong connection between the general and symmetric version of an asymptotic duality theorem of Strassen. We introduce a representation-theoretic method to asymptotically bound the symmetric subrank called the symmetric quantum functional in analogy with the quantum functionals (Christandl, Vrana, Zuiddam, J. Amer. Math. Soc., 2021), and we study the relations between these functionals.
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