Border rank lower bounds for families of GL(V)-invariant tensors
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918130161287168 |
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| author | Gondi, Suhas Vadan |
| author_facet | Gondi, Suhas Vadan |
| contents | We give non-trivial lower bounds for the border rank of families of $\mathbf{GL}(V)$-invariant tensors in $U\otimes \mathbf{S}_λV\otimes \mathbf{S}_μV$ where $U$ is $V$, $\mathrm{Sym}^2V$ or $\bigwedge^2V$. We build on the techniques introduced by Wu, who used Young flattenings to obtain bounds for a family of tensors when $U$ is $V$. We complete this case by resolving a conjecture introduced by Wu, using certain pure resolutions constructed by Ford-Levinson-Sam. We then use a theorem of Kostant to generalise this to $\mathrm{Sym}^2 V$ and $\bigwedge^2 V$, and extend the number of examples of $\mathbf{GL}(V)$-invariant tensors that are not of minimal border rank using Kempf collapsing. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17845 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Border rank lower bounds for families of GL(V)-invariant tensors Gondi, Suhas Vadan Algebraic Geometry Commutative Algebra Representation Theory We give non-trivial lower bounds for the border rank of families of $\mathbf{GL}(V)$-invariant tensors in $U\otimes \mathbf{S}_λV\otimes \mathbf{S}_μV$ where $U$ is $V$, $\mathrm{Sym}^2V$ or $\bigwedge^2V$. We build on the techniques introduced by Wu, who used Young flattenings to obtain bounds for a family of tensors when $U$ is $V$. We complete this case by resolving a conjecture introduced by Wu, using certain pure resolutions constructed by Ford-Levinson-Sam. We then use a theorem of Kostant to generalise this to $\mathrm{Sym}^2 V$ and $\bigwedge^2 V$, and extend the number of examples of $\mathbf{GL}(V)$-invariant tensors that are not of minimal border rank using Kempf collapsing. |
| title | Border rank lower bounds for families of GL(V)-invariant tensors |
| topic | Algebraic Geometry Commutative Algebra Representation Theory |
| url | https://arxiv.org/abs/2508.17845 |