For , the G - orbit of f is the set . There are exactly ten G - orbits of vectors in . Representative vectors for these orbits are listed above. The remaining two, and , were found by Cohen and Wales. The above kG - modul V was one of the open cases in a prospective classification of irreducible modules for almost simple algebraic groups over an algebraically closed field of positive characteristic for which there are a finite number of points.
P is the space of all cubic hypersurfaces on projective space.
We shall use as basis , where and, for , . If and , the action of g on is given by where For example, since we have This then determines the action of g on , and hence the action on V , in the following way. Then Hence, and its representative is Notice that and, in general, if , As a basis for V , we shall use the 16 monomials in x, y, z, t which are not cubes. Let us denote by the l - th monomial in the second row of the array below. The action of g on V can now be explicitly given in terms of the following matrix where is written as .