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Symmetric Tensor Matroids, Dual Rigidity Matroids, and the Maximality Conjecture

Bill Jackson, Shin-ichi Tanigawa

TL;DR

The paper develops abstract symmetric $t$-tensor matroids as the matroidal duals of abstract rigidity matroids and proves a precise duality: an abstract symmetric $t$-tensor matroid on $K_n$ is characterized by its dual being an abstract $(n-t-1)$-rigidity matroid. This duality yields alternative, often simpler, rigidity characterizations for small parameters (notably when $n-d-1=t$ and $n-d\le 6$), and confirms Graver's maximality conjecture in this dual setting for $t\le 5$. The work further analyzes $K_{1,t+1}$-matroids, showing a unique maximal element only for $t\le 3$, while indicating non-uniqueness for larger $t$ and large $n$, with detailed construction in the appendix. A central technical contribution is a graph operation dual to rigidity extensions that preserves ${\cal S}_t$-independence, enabling a constructive route to a complete, computer-free characterization of ${\cal S}_t$-independence for $t\le 5$ via the cyclic-flats framework ${\cal C}_{n,t}$. Together, these results illuminate the rigidity-tensor duality landscape and delineate the limits of unique maximality phenomena in symmetric tensor matroid families.

Abstract

Inspired by a recent result of Brakensiek et al. that symmetric tensor matroids and rigidity matroids are linked by matroid duality, we define abstract symmetric tensor matroids as a dual concept to abstract rigidity matroids and establish their basic properties. We then exploit this duality to obtain an alternative characterisation of the generic $d$-dimensional rigidity on $K_n$ for $n-d\leq 6$ to that given by Grasseger et al. Our results imply that Graver's maximality conjecture holds for these matroids. We also consider the related family of $K_{1,t+1}$-matroids on $K_n$ and show that this family has a unique maximal element only when $t\leq 3$. This implies that the family of second quasi symmetric powers of the uniform matroid $U_{t,n}$ does not have a unique maximal matroid if $t\geq 4$ and $n$ is sufficiently large.

Symmetric Tensor Matroids, Dual Rigidity Matroids, and the Maximality Conjecture

TL;DR

The paper develops abstract symmetric -tensor matroids as the matroidal duals of abstract rigidity matroids and proves a precise duality: an abstract symmetric -tensor matroid on is characterized by its dual being an abstract -rigidity matroid. This duality yields alternative, often simpler, rigidity characterizations for small parameters (notably when and ), and confirms Graver's maximality conjecture in this dual setting for . The work further analyzes -matroids, showing a unique maximal element only for , while indicating non-uniqueness for larger and large , with detailed construction in the appendix. A central technical contribution is a graph operation dual to rigidity extensions that preserves -independence, enabling a constructive route to a complete, computer-free characterization of -independence for via the cyclic-flats framework . Together, these results illuminate the rigidity-tensor duality landscape and delineate the limits of unique maximality phenomena in symmetric tensor matroid families.

Abstract

Inspired by a recent result of Brakensiek et al. that symmetric tensor matroids and rigidity matroids are linked by matroid duality, we define abstract symmetric tensor matroids as a dual concept to abstract rigidity matroids and establish their basic properties. We then exploit this duality to obtain an alternative characterisation of the generic -dimensional rigidity on for to that given by Grasseger et al. Our results imply that Graver's maximality conjecture holds for these matroids. We also consider the related family of -matroids on and show that this family has a unique maximal element only when . This implies that the family of second quasi symmetric powers of the uniform matroid does not have a unique maximal matroid if and is sufficiently large.

Paper Structure

This paper contains 15 sections, 28 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Abstract Rigidity Matroids and Duality
  3. Abstract rigidity matroids
  4. Abstract Symmetric Tensor Matroids
  5. Maximality conjectures
  6. Exploring Duality between ${\cal R}_d$ and ${\cal S}_t$
  7. An operation for preserving ${\cal S}_t$-independence
  8. Symmetric-tensor matroids of dimension at most five
  9. Appendix
  10. Matroid Preliminaries
  11. Algorithm 1
  12. Maximal $K_{1,t}$-matroids on $K_n$
  13. Case 1: $t\equiv 2 \pmod{3}$.
  14. Case 2: $t\equiv 0 \pmod{3}$.
  15. Case 3: $t\equiv 1 \pmod{3}$.

Key Result

Theorem 1.1

Let $n, n_1, n_2, d, d_1, d_2$ be integers.

Figures (1)

  • Figure 1: Examples of valid labellings for $t=5$.

Theorems & Definitions (55)

  • Theorem 1.1: Brakensiek et al. brakensiek
  • Theorem 2.1: Graver-Servatius-ServatiusGSS93 and Nguyen N10
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 45 more