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Classification and degenerations of small minimal border rank tensors via modules

Jakub Jagiełła, Joachim Jelisiejew

TL;DR

This work provides a complete classification of minimal border rank tensors in $\mathbb{C}^m\otimes\mathbb{C}^m\otimes\mathbb{C}^m$ for $m\le5$, distinguishing $1_*$-generic from $1$-degenerate cases. Central to the approach is a module-theoretic correspondence: $1_A$-generic tensors correspond to End-closed End-closed $S$-modules, enabling a classification via the degree-$m$ $S$-modules (with $S=\Bbbk[x_1,...,x_{m-1}]$) and their End-closedness, together with apolarity for modules. The paper introduces and exploits the 111-algebra framework to connect tensors to bilinear maps between modules, and uses this to derive both a refined classification and a detailed degeneration diagram (66 minimal degenerations, plus obstructions). For $m\le4$ there are no $1$-degenerate minimal border rank tensors, while for $m=5$ a complete degeneration/dimension analysis yields $107$ isomorphism classes (and $37$ up to permutations), with a precise account of which indecomposable tensors arise as degenerations. The results have implications for understanding border rank structure and for constructing explicit degeneration data useful in complexity-theoretic contexts.

Abstract

We give a self-contained classification of $1_*$-generic minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m$ for $m \leq 5$. Together with previous results, this gives a classification of all minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m$ for $m \leq 5$: there are $107$ isomorphism classes (only $37$ up to permuting factors). We fully describe possible degenerations among the tensors. We prove that there are no $1$-degenerate minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m $ for $m \leq 4$.

Classification and degenerations of small minimal border rank tensors via modules

TL;DR

This work provides a complete classification of minimal border rank tensors in for , distinguishing -generic from -degenerate cases. Central to the approach is a module-theoretic correspondence: -generic tensors correspond to End-closed End-closed -modules, enabling a classification via the degree- -modules (with ) and their End-closedness, together with apolarity for modules. The paper introduces and exploits the 111-algebra framework to connect tensors to bilinear maps between modules, and uses this to derive both a refined classification and a detailed degeneration diagram (66 minimal degenerations, plus obstructions). For there are no -degenerate minimal border rank tensors, while for a complete degeneration/dimension analysis yields isomorphism classes (and up to permutations), with a precise account of which indecomposable tensors arise as degenerations. The results have implications for understanding border rank structure and for constructing explicit degeneration data useful in complexity-theoretic contexts.

Abstract

We give a self-contained classification of -generic minimal border rank tensors in for . Together with previous results, this gives a classification of all minimal border rank tensors in for : there are isomorphism classes (only up to permuting factors). We fully describe possible degenerations among the tensors. We prove that there are no -degenerate minimal border rank tensors in for .
Paper Structure (53 sections, 40 theorems, 68 equations, 2 figures)

This paper contains 53 sections, 40 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Methods
  4. Modules
  5. 111-algebras
  6. Previous work
  7. Acknowledgements
  8. Preliminaries
  9. Tensors
  10. Modules I
  11. Modules and $1_A$-generic tensors
  12. Modules II
  13. Apolarity for modules
  14. Maps between modules and 111-equations
  15. A special case of Kronecker's normal form
  16. ...and 38 more sections

Key Result

Theorem 1.1

Up to isomorphism, there are exactly $1, 2, 6, 21, 107$ minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m$ for $m = 1,2,3,4,5$. Up to permutations, the numbers are $1,2,4,11,37$. An explicit list is given in §classification_tensors.

Figures (2)

  • Figure 4.1: Degenerations of minimal border rank tensors in $\Bbbk^5\otimes \Bbbk^5\otimes \Bbbk^5$
  • Figure 4.2: Non-degenerations of minimal border rank tensors in $\Bbbk^5\otimes \Bbbk^5\otimes \Bbbk^5$

Theorems & Definitions (86)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Example 1.7
  • Remark 1.8
  • Example 2.1
  • Example 2.2
  • ...and 76 more