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On the geometry of tensor products over finite fields

Stefano Lia, John Sheekey

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

On the geometry of tensor products over finite fields

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 .
Paper Structure (20 sections, 38 theorems, 62 equations)

This paper contains 20 sections, 38 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. A model for threefold tensors
  3. One- and twofold tensors
  4. Threefold tensors and algebras
  5. Contractions and Nonsingularity
  6. Group actions, isotopisms, and collineations
  7. Action of $Sym(3)$
  8. Finalising our setup
  9. Contractions in the cyclic model
  10. Interpreting the third action of $GL(V)$ in the cyclic model
  11. An upper-bound for the BEL rank of a semifield
  12. The geometry of nonsinglar tensors for $n=2$
  13. Analysis of the System \ref{['system']}
  14. The Geometry of Nonsingular Fourfold Tensors
  15. Quasi-Hermitian surfaces
  16. ...and 5 more sections

Key Result

Theorem 2.2

The contraction space $C_3(M_T)$ of a tensor with matrix representation $M_T$ is the unique subspace of $\Sigma$ of minimal dimension whose extension to $\Sigma^*$ contains $P=\langle M_T\rangle$.

Theorems & Definitions (67)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 57 more