On the Trivectors of a 6-Dimensional Symplectic Vector Space. II

Let V be a 6-dimensional vector space over a field F, let f be a nondegenerate alternating bilinear form on V and let Sp(V,f)≅Sp6(F) denote the symplectic group associated with (V,f). The group GL(V) has a natural action on the third exterior power ⋀3V 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 F. The orbits of the fifth family are in one-to-one correspondence with the quadratic extensions of F that are contained in a fixed algebraic closure F¯ of F. In this paper, we divide the orbits corresponding to the separable quadratic extensions into suborbits for the action of Sp(V,f)⊆GL(V) on ⋀3V.