Classification of 8-Dimensional Trilinear Alternating Forms over GF(2)

Abstract

Let V be an n-dimensional vector space over a finite field and let f be a trilinear alternating form over V. For such forms we introduce a new invariant called radical polynomial and investigate its behaviour, in particular in the case of the 2-element field. We show that it is compatible with direct products of forms and how it is related to its values on dimension n − 1. Moreover, it turns out that it is full up to dimension 7. On the other hand, on higher dimensions it is no more full and it is necessary to generalize it to obtain (using computer) a classification of forms on dimension 8 over the 2-element field. This classification is provided, together with the sizes of stabilizers of the corresponding forms.

Key Words:

• Classification
• Radical polynomial
• Trilinear alternating form

2010 Mathematics Subject Classification:

• 15A69

Notes

Communicated by I. Shestakov.