Kronecker Products of Symmetric Persistent Tensors

Technical Summary: Kronecker Products of Symmetric Persistent Tensors

Problem Statement
Persistent tensors, introduced by Landsberg and Gharahi [GL, Gh], form a recursively defined class of tensors designed to yield nontrivial lower bounds on tensor rank via the substitution method. While persistence is known to coincide with conciseness for bipartite tensors, the behavior of this property under Kronecker products for higher-order tensors remained an open question. It was previously conjectured that the Kronecker product of persistent tensors is always persistent; however, Shitov [Sh] provided a counterexample to this unrestricted conjecture. Since Shitov's counterexample is nonsymmetric, the status of the conjecture within the symmetric setting was unresolved. This paper addresses whether the class of symmetric persistent tensors is closed under Kronecker products.

Methodology and Key Technical Tools
The paper operates within the framework of symmetric tensors identified with homogeneous polynomials ($f \in \text{Sym}^n V$). The methodology relies on a complete Hessian characterization of symmetric persistence established in prior work [GO, Theorem 2]. Specifically, a symmetric tensor is persistent if and only if the Hessian determinant of its $(n-2)$-fold partial polarizations satisfies a specific "perfect-power" condition.

The core technical approach involves three main steps:

1. Global Differentiation Identity: The author establishes an identity for the Hessian matrices of Kronecker products of arbitrary homogeneous polynomials. For $f \in \text{Sym}^n \mathbb{C}^{d_1}$ and $g \in \text{Sym}^n \mathbb{C}^{d_2}$, the Hessian matrix of the Kronecker product $f \boxtimes g$ satisfies:
   $$H_{f \boxtimes g} = \frac{1}{n(n-1)} H_f \boxtimes H_g$$
   where the Kronecker product on the right-hand side is understood entrywise. The paper notes that while this matrix identity holds globally, the corresponding identity for Hessian polynomials (determinants) generally fails due to the presence of the Segre ideal.

2. Polarized Perfect-Power Identity: To overcome the failure of the general determinant identity, the paper proves a specialized result for persistent factors. If $f$ and $g$ are persistent, there exists a nonzero multihomogeneous polynomial $P_{f \boxtimes g}$ of multidegree $(1, \dots, 1)$ such that the Hessian determinant of the $(n-2)$-fold polarized Kronecker product is the $(d_1 d_2)$-th power of this polynomial:
   $$\text{Hess}\big((f \boxtimes g)_{U^{(1)}, \dots, U^{(n-2)}}(z)\big) = \big[P_{f \boxtimes g}(U^{(1)}, \dots, U^{(n-2)})\big]^{d_1 d_2}$$
   This is achieved by relating the polarized Hessian of the product to the Kronecker product of the polarized Hessians of the factors, utilizing the fact that for persistent tensors, the individual Hessian determinants are perfect powers of multilinear forms.

3. Application of the Hessian Criterion: By combining the polarized identity with the Hessian characterization of persistence (Theorem 3), the author demonstrates that the Kronecker product $f \boxtimes g$ satisfies the necessary condition for persistence.

Key Results

• Theorem 11 (Closure under Kronecker Products): The class of symmetric persistent tensors is closed under Kronecker products. If $f \in \text{Sym}^n \mathbb{C}^{d_1}$ and $g \in \text{Sym}^n \mathbb{C}^{d_2}$ are persistent, then $f \boxtimes g \in \text{Sym}^n (\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2})$ is persistent.
• Corollaries: As immediate consequences, the paper establishes that symmetric persistence is preserved under iterated Kronecker products (Corollary 14) and Kronecker powers (Corollary 15).
• Example Application: The paper applies these results to the binary W-tensor ($W_n = x_0^{n-1}x_1$), confirming that all its Kronecker powers form an explicit infinite family of persistent tensors in $\text{Sym}^n \mathbb{C}^{2^k}$.

Significance and Claims
The paper claims to resolve the open question regarding the preservation of persistence under Kronecker products specifically within the symmetric setting, a case where the general conjecture had failed. The significance lies in:

1. Structural Insight: It provides a rigorous proof that the symmetric setting avoids the pathologies found in the general tensor case, largely due to the specific structure of Hessian determinants for persistent forms.
2. Methodological Contribution: The derivation of the global differentiation identity for Hessian matrices and the subsequent polarized perfect-power identity offers new tools for analyzing tensor ranks and persistence.
3. Expansion of Known Classes: The results expand the known families of tensors with provable lower bounds on rank, specifically generating infinite families of persistent tensors via Kronecker powers.

The author dedicates the work to Giorgio Ottaviani, acknowledging his feedback on the global differentiation identities and the Hessian characterization. The paper concludes by noting a complementary matrix-space viewpoint for persistent cubic forms in Appendix A, raising a structural question regarding the simultaneous triangularizability of their normalized Hessian spaces.