On the trivectors of a 6-dimensional symplectic vector space. II

Let V be a 6-dimensional vector space over a field IF, let f be a nondegenerate alternating bilinear form on V and let Sp(V, f) congruent to Sp(6) (F) denote the symplectic group associated with (V, f). The group GL(V) has a natural action on the third exterior power Lambda(3) V of V and this action defines five families of nonzero trivectors of V. Four of these families are orbits for any choice of the field IF. The orbits of the fifth family are in one-to-one correspondence with the quadratic extensions of IF that are contained in a fixed algebraic closure IF of IF. In this paper, we divide the orbits corresponding to the separable quadratic extensions into suborbits for the action of Sp(V,f) subset of GL(V) on Lambda(3) V. (C) 2012 Elsevier Inc. All rights reserved.