On the geometry of tensor products over finite fields

TL;DR

This paper explores the geometric structure of tensor products over finite fields, introducing a new embedding that reveals invariants and classifications of tensors and algebras, and leads to novel constructions of quasi-hermitian varieties.

Contribution

It introduces an alternative geometric embedding of tensor spaces over finite fields, enabling new insights into tensor invariants, classifications, and the construction of quasi-hermitian varieties.

Findings

New geometric model for tensor contractions and invariants

Classification results for nonsingular threefold tensors

Construction of quasi-hermitian varieties in projective space

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