Symmetric Tensor Matroids, Dual Rigidity Matroids, and the Maximality Conjecture
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917960997666816 |
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| author | Jackson, Bill Tanigawa, Shin-ichi |
| author_facet | Jackson, Bill Tanigawa, Shin-ichi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14780 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symmetric Tensor Matroids, Dual Rigidity Matroids, and the Maximality Conjecture Jackson, Bill Tanigawa, Shin-ichi Combinatorics 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. |
| title | Symmetric Tensor Matroids, Dual Rigidity Matroids, and the Maximality Conjecture |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.14780 |