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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.16767 |
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| _version_ | 1866915244386811904 |
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| author | Gesmundo, Fulvio Han, Young In Lovitz, Benjamin |
| author_facet | Gesmundo, Fulvio Han, Young In Lovitz, Benjamin |
| contents | We study the problem of characterizing linear preserver subgroups of algebraic varieties, with a particular emphasis on secant varieties and other varieties of tensors. We introduce a number of techniques built on different geometric properties of the varieties of interest. Our main result is a simple characterization of the linear preservers of secant varieties of Segre varieties in many cases, including $σ_r((\mathbb{P}^{n-1})^{\times k})$ for all $r \leq n^{\lfloor k/2 \rfloor}$. We also characterize the linear preservers of several other sets of tensors, including subspace varieties, the variety of slice rank one tensors, symmetric tensors of bounded Waring rank, the variety of biseparable tensors, and hyperdeterminantal surfaces. Computational techniques and applications in quantum information theory are discussed. We provide geometric proofs for several previously known results on linear preservers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16767 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear preservers of secant varieties and other varieties of tensors Gesmundo, Fulvio Han, Young In Lovitz, Benjamin Algebraic Geometry Representation Theory 15A69, 15A86, 14N07, 47B49 We study the problem of characterizing linear preserver subgroups of algebraic varieties, with a particular emphasis on secant varieties and other varieties of tensors. We introduce a number of techniques built on different geometric properties of the varieties of interest. Our main result is a simple characterization of the linear preservers of secant varieties of Segre varieties in many cases, including $σ_r((\mathbb{P}^{n-1})^{\times k})$ for all $r \leq n^{\lfloor k/2 \rfloor}$. We also characterize the linear preservers of several other sets of tensors, including subspace varieties, the variety of slice rank one tensors, symmetric tensors of bounded Waring rank, the variety of biseparable tensors, and hyperdeterminantal surfaces. Computational techniques and applications in quantum information theory are discussed. We provide geometric proofs for several previously known results on linear preservers. |
| title | Linear preservers of secant varieties and other varieties of tensors |
| topic | Algebraic Geometry Representation Theory 15A69, 15A86, 14N07, 47B49 |
| url | https://arxiv.org/abs/2407.16767 |