The computation of such a group stabilizer is a less straightforward matter than the computation of the Lie algebra stabilizer. Our methods are somewhat ad hoc, but have in common that they use the Lie algebra in two ways: both for finding linear tranformations fixing the form f by exponentiating nilpotent elements of the Lie algebra stabilizer and for finding -invariant linear subspaces of . We use computer algebra packages such as Mathematica and Maple to check in each case that certain matrices in preserve the form. Knowing the dimension of from Table 1 gives a bound for the dimension of the stabilizer. If this is attained by a subgroup it contains at least the connected component of the full stabilizer and group arguments involving the normalizer can be used. If not, we use the packages to show there are no further elements fixing the form.
The stabilizer of is (as in [Z]) a finite group isomorphic to . It consists of all permutation matrices together with all products PSQ where P and Q are permutation matrices and S is the matrixThis is the 4-dimensional constituent of the natural representation of on by taking for example the differences of 1 with the remaining integers. The stabilizer of is (as indicated in [Z])
The stabilizer of is
The stabilizer of is
The stabilizer of isIt is straightforward to check that this group fixes . It has dimension 4 which is the same as its Lie algebra. By considering the action on the first, third, and fourth coordinates, it is immediate that this is acting faithfully and irreducibly on the homogeneous degree two polynomials in the two coordinates. It must then be the connected component of the identity. But this action is self normalizing in and because it is a direct sum of a 1-dimensional constituent and this constituent, this is the full normalizer in . This establishes this is the full stabilizer of .
Note that the forms listed are all distinct. Indeed the dimensions of
the Lie algebras are distinct except for the two of dimension 12 and
the two of dimension 14. The ones of dimension 14, and ,
correspond to orbit dimensions 15
and 14 and so are distinct. The Lie algebras for and
are not isomorphic. Indeed is abelian but is
not. The forms and can be also distinguished because the tori
have different eigenvalues. Alternatively, exponentiating a
nilpotent element in the Lie algebra for fixes the form but does
not for .