Automorphisms Group and Radical Polynomial of Standard Trilinear Forms

Let V be a vector space of dimension n over a field K and let ω be a trivector of ∧³V. For such trivectors, we can associate three invariants, the automorphism group Aut(ω), the radical polynominal P(ω) and the commutant C(ω). We use the classification of trivectors of rank ≤9, we give a general rule for the standard trivector ω_{3k} in dimension 3k, the trivector with the transitive automorphism group and the trivector with an isotropic hyperplane ω_{2k+1} in dimension 2k+1. We compute their radical polynomials and the sizes of the groups automorphisms. We demonstrate that there exists a vector space V and a trivector ω of ∧³V where C(ω) is not a Frobenius algebra and dimV≤3dimC(ω). Finally, We give a classification of trivectors in dimension 8 over a finite field of characteristic 2 and its applications in the theory of codes.