Border subrank of higher order tensors and algebras

Abstract: We determine the border subrank of higher order structure tensors of several families of algebras, and in particular obtain the following results. (1) We determine tight bounds on the border subrank of $k$-fold matrix multiplication and $k$-fold upper triangular matrix multiplication for all $k$. (2) We determine the border subrank of the higher order structure tensors of truncated polynomial algebras, null algebras, and apolar algebras of a quadric. (3) We determine the border subrank of the higher order structure tensors of the Lie algebra $\mathfrak{sl}_2$ for all orders. (4) We prove that degeneration of structure tensors of algebras propagates from higher to lower order. Along the way, we investigate which upper bound methods (geometric rank, $G$-stable rank, socle degree) are effective in which settings, and how they relate. Our work extends the results of Strassen (J.~Reine Angew.~Math., 1987, 1991), who determined the asymptotic subrank of these algebras for tensors of order three, in two directions: we determine the border subrank itself rather than its asymptotic version, and we consider higher order structure tensors.

• The paper establishes tight bounds for the border subrank of higher-order tensors across associative, commutative, and Lie algebras.
• It employs explicit algebraic degeneration constructions and geometric invariant theory to systematically propagate subrank properties.
• Results impact matrix multiplication complexity and quantum information, providing concrete benchmarks for future research.

Border Subrank of Higher Order Tensors and Algebras

Introduction

The study of tensor invariants such as (border) subrank and geometric rank is pivotal in the intersection of algebra, complexity theory, and quantum information. The paper "Border subrank of higher order tensors and algebras" (2604.19872) focuses on the border subrank of higher-order structure tensors arising from families of algebras including associative algebras (matrix multiplication, upper triangular), commutative algebras (truncated polynomial, null, and apolar algebras), and Lie algebras ($\mathfrak{sl}_2$). The authors generalize and sharpen previous results by Strassen and later developments, extending them beyond tensors of order three and capturing the precise border subrank rather than its asymptotic value.

Principal Contributions and Results

Border Subrank of Iterated Matrix Multiplication

The border subrank of $k$-fold $n \times n$ matrix multiplication tensors, denoted ${\rm MaMu}_{(n, \dots, n)}$, is bounded tightly:

• Lower bound: $\frac{n^2}{k-1}$
• Upper bound: $\frac{k+1}{2k} n^2 + O(k)$

This refines classical results ($k=2$) and extends them to arbitrary $k \geq 3$. The upper bound leverages an exact calculation of geometric rank, while the lower bound arises from explicit degenerations using general-position orthogonal representations.

Iterated Upper Triangular Matrix Multiplication

For $k$-fold multiplication of $n \times n$ upper triangular matrices ($k$0), border subrank and geometric rank coincide:

$k$1

The precise value is $k$2, with $k$3.

Commutative Algebras: Truncated Polynomial, Null, and Apolar Algebras

The border subrank of higher-order structure tensors for several families of commutative algebras is addressed:

• Null algebra $k$4: Border subrank collapses for higher $k$5 (to 1 for $k$6).
• Truncated polynomial algebra $k$7: Novel lower and upper bounds for border subrank, with explicit degenerations to unit tensors and upper bounds via $k$8-stable rank.
• Apolar algebra of quadrics (generalized Coppersmith–Winograd): Full characterization for all $k$9, including border subrank collapse for $n \times n$0.

For all these commutative cases, the authors analyze the efficacy of upper bound techniques: geometric rank is often loose, while $n \times n$1-stable rank and socle degree often provide sharp or nearly sharp estimates (with explicit constructions verifying tightness in several cases).

Special Linear Lie Algebra $n \times n$2

The border subrank for the $n \times n$3-fold structure tensor for $n \times n$4 is computed for all $n \times n$5:

• For $n \times n$6, $n \times n$7. For general $n \times n$8, the lower bound for subrank is $n \times n$9, which is not always tight with the geometric upper bound, due to the semisimple structure.

General Propagation of Degenerations

A structural result is established: if a degeneration between higher-order structure tensors exists (${\rm MaMu}_{(n, \dots, n)}$0-fold), then it propagates to all lower orders ${\rm MaMu}_{(n, \dots, n)}$1. With equal dimensions, such degenerations propagate to all ${\rm MaMu}_{(n, \dots, n)}$2. This is shown via a Grassmannian-based flattening argument and enhances earlier results by removing technical conciseness constraints.

Technical Approach

The main technical advances combine explicit algebraic degeneration constructions, invariants from geometric invariant theory (including ${\rm MaMu}_{(n, \dots, n)}$3-stable rank and geometric rank), and the study of algebraic and combinatorial flags in quiver representation theory for the matrix multiplication case. For subrank propagation, the authors generalize flattening methods, extending to orbit closures on suitable Grassmannians.

New connections are established with representation theory (quivers for matrix multiplication components), combinatorics (average-free sets for subrank lower bounds), and classical algebra (socle degree as a subrank barrier).

Strong Numerical and Structural Claims

• Tight bounds for iterated matrix multiplication:

${\rm MaMu}_{(n, \dots, n)}$4

• Collapse for border subrank with high ${\rm MaMu}_{(n, \dots, n)}$5 for local commutative algebras with bounded socle degree.
• For upper triangular matrices:

${\rm MaMu}_{(n, \dots, n)}$6

• For ${\rm MaMu}_{(n, \dots, n)}$7: ${\rm MaMu}_{(n, \dots, n)}$8 for all ${\rm MaMu}_{(n, \dots, n)}$9.

These stark boundaries isolate cases where border subrank is strictly less than geometric rank or boarder rank, most notably for higher-order tensors.

Implications and Future Directions

Practical

Understanding border subrank underpins lower bounds for arithmetic complexity, with implications for the barrier results in the laser method and related matrix multiplication exponent ($\frac{n^2}{k-1}$0) studies. The results give explicit benchmarks and obstructions for approaches aiming to reduce $\frac{n^2}{k-1}$1 via higher-order tensor methods.

Border subrank is also directly linked to SLOCC entanglement classifications in quantum information theory, particularly for tensors corresponding to matrix product states. The collapse of border subrank for higher $\frac{n^2}{k-1}$2 signals limits for certain quantum protocols when generalized to more parties or higher-dimensional systems.

Theoretical

The propagation theorem implies structural rigidity in degeneration hierarchies, suggesting that high-order properties "trap" lower-order ones. This opens up pathways to study reduction of complexity invariants across algebra families, and frames new geometric perspectives on algebraic complexity barriers.

The explicit connections established between algebraic features (socle, maximal ideals) and border subrank illuminate avenues for classifying tensors and algebras by their "degeneration profiles," with possible new invariants arising from this perspective.

Outlook

• Generalizations: Extending the framework to more general non-associative algebras, non-unital cases, or modular settings (positive characteristic).
• Quantum Information: Leveraging these tensor subrank results to refine classifications of multipartite entanglement.
• Complexity Theory: Employing sharp subrank bounds to derive new lower bounds in circuit and arithmetic complexity, or to demarcate further laser method limitations.

Open questions include precise classification of border subrank stabilization cutoff $\frac{n^2}{k-1}$3 for matrix multiplication tensors and a deeper algebraic understanding of subrank in the context of tensor network states beyond the Gaussian regime.

Conclusion

This work systematically elucidates the border subrank for higher-order structure tensors across multiple algebraic and combinatorial families. The integration of explicit degeneration constructions, geometric invariant theory, and propagation theorems advances the quantitative and structural understanding of tensor complexity invariants, providing a robust set of tools and results for future exploration in algebraic complexity, combinatorics, and quantum information theory (2604.19872).