对称持续张量的克罗内克积
On the Kronecker Products of Symmetric Persistent Tensors
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中文总结 AI 辅助
本文证明对称持续张量的克罗内克积仍为对称持续张量，建立海森矩阵相关恒等式并推导其在迭代克罗内克积下的封闭性，丰富了张量秩下界相关理论。
中文摘要 AI 辅助
持续张量是一类递归定义的张量，适用于代入法，可给出张量秩的非平凡下界。尽管一般情况下持续性质在克罗内克积下不保持，但我们证明其在对称情形下保持：任意两个对称持续张量的克罗内克积仍是持续的。我们首先建立一个全局微分恒等式，表明任意齐次多项式克罗内克积的海森矩阵，是各因子海森矩阵的克罗内克积，仅需按还原约定进行归一化。对于持续因子，我们接着证明所有(n-2)阶部分极化的海森行列式满足极化完全幂恒等式。结合对称持续的海森矩阵刻画，这就得到了克罗内克积下的封闭性。作为推论，对称持续在迭代克罗内克积和克罗内克幂下也具有封闭性。
英文摘要
Persistent tensors form a recursively defined class adapted to the substitution method and provide nontrivial lower bounds on tensor rank. We study the behavior of symmetric persistent tensors under Kronecker products. We establish a global differentiation identity expressing the Hessian matrix of a Kronecker product of homogeneous polynomials in terms of the Hessian matrices of its factors; although no corresponding determinant identity holds globally, the Hessian polynomials factor exactly at decomposable points. This yields a multiplicative formula for the distinguished Hessian coefficients of isobaric forms and closure under Kronecker products for symmetric persistent tensors that are isobaric of the distinguished weight, including iterated products and powers. Combined with the classification in small dimensions, this implies closure when both factors have dimension at most three, and for persistent cubics when both have dimension at most four. We further give sufficient closure criteria via simultaneous strict triangularizability of normalized Hessian spaces, for cubics and then arbitrary degree, and show that the distinguished isobaric class satisfies this condition. Finally, we show that symmetric persistence is not preserved in general by constructing a persistent cubic $f\in\operatorname{Sym}^3\mathbb{C}^{12}$ such that $f\boxtimes f\in\operatorname{Sym}^3\mathbb{C}^{144}$ is not persistent. The obstruction occurs at a nondecomposable point, showing that the Hessian factorization on the Segre variety does not extend to the full tensor product space.
