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arXiv · 2311.17896

On the geometry of tensor products over finite fields

Stefano Lia·John Sheekey
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Abstract

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 $\mathrm{PG}(3,q^2)$.

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Stefano Lia, John Sheekey. 2023-11-29. On the geometry of tensor products over finite fields. https://arxiv.org/abs/2311.17896

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