Canonical forms of small tensors over F_2

Abstract

We consider multidimensional arrays with at most 27 entries over the field with two elements, and their equivalence classes for the action of the direct product of general linear groups. The possible 3-dimensional formats are p x 2 x 2 (p = 2, ..., 6), p x 3 x 2 (p = 3, 4), and 3 x 3 x 3; the possible 4-dimensional formats are p x 2 x 2 x 2 (p = 2, 3). In each case, we compute the orbits for the group action, and then we determine the rank of each orbit. In particular, we determine the maximum rank for these arrays over F_2.