Canonical forms of 2 × 2 × 2 and 2 × 2 × 2 × 2 arrays over 𝔽₂ and 𝔽₃

Abstract

We consider arrays of size 2 × 2 × 2 and 2 × 2 × 2 × 2 over the fields with two and three elements. We use computer algebra to determine the canonical forms of these arrays with respect to the action of the semidirect product of general linear groups with the symmetric group. For each canonical form, we determine the size of its orbit and the rank of the arrays in its orbit.

Keywords:

• multidimensional arrays
• canonical forms
• finite fields
• group actions
• computer algebra

AMS Subject Classifications::

• Primary: 15A21
• Secondary: 15-04
• 15A69
• 15B33
• 20B25
• 20C20
• 20G40

Acknowledgements

The authors thank the referee for a very detailed report which included many additional references and made many useful suggestions for improving this article. We also thank Jiaxiong Hu for help with Maple programming. These results form part of the Master's thesis of the second author, written under the supervision of the first author, who was supported by a Discovery Grant from NSERC, the Natural Sciences and Engineering Research Council of Canada.