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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| Accès en ligne: | https://arxiv.org/abs/2311.17896 |
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| _version_ | 1866913263832268800 |
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| author | Lia, Stefano Sheekey, John |
| author_facet | Lia, Stefano Sheekey, John |
| contents | In this paper we study finite dimensional algebras, in particular finite semifields, through their correspondence with nonsingular threefold tensors. We introduce a alternative embedding of the tensor product space into a projective space. This model allows us to understand tensors and their contractions in a new geometric way, relating the contraction of a tensor with a natural subspace of a subgeometry. This leads us to new results on invariants and classifications of tensors and algebras and on nonsingular fourfold tensors. A detailed study of the geometry of this setup for the case of the threefold tensor power of a vector space of dimension two over a finite field surprisingly leads to a new construction of quasi-hermitian varieties in $\mathrm{PG}(3,q^2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_17896 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the geometry of tensor products over finite fields Lia, Stefano Sheekey, John Combinatorics 12K10, 14N07, 11E39 In this paper we study finite dimensional algebras, in particular finite semifields, through their correspondence with nonsingular threefold tensors. We introduce a alternative embedding of the tensor product space into a projective space. This model allows us to understand tensors and their contractions in a new geometric way, relating the contraction of a tensor with a natural subspace of a subgeometry. This leads us to new results on invariants and classifications of tensors and algebras and on nonsingular fourfold tensors. A detailed study of the geometry of this setup for the case of the threefold tensor power of a vector space of dimension two over a finite field surprisingly leads to a new construction of quasi-hermitian varieties in $\mathrm{PG}(3,q^2)$. |
| title | On the geometry of tensor products over finite fields |
| topic | Combinatorics 12K10, 14N07, 11E39 |
| url | https://arxiv.org/abs/2311.17896 |