In [Z], Chen Zhijie produced eight G-orbits of nonzero vectors in V (one of which is dense) and conjectured there are no others. In this paper we find two more (of dimensions 12 and 14) and show that these are all.
Theorem. There are exactly ten G-orbits of vectors in . Representative vectors for these orbits are listed in Table 1.
Our proof uses Chen's techniques, but has three additional features. First, we use comparison with the finite field case so as to conclude that there are no further orbits of dimension 14. This part of the proof, including a full description of orbits in the case where k is finite, is given in . Second, we deploy Lie algebras to find the vector stabilizers in an efficient way. Third, and perhaps most crucial, we use computer algebra to automate the computations involved. In particular, the computer computations of the image of a given vector under a given group element, of the dimension of the space of images of a vector under the algebra of all -matrices over k, and of the full stabilizer of a given vector have been of great use. We have double checked these computations, by having one author perform the calculations in Maple, and the other (independently) in Mathematica. One purpose of these computations is to find an algorithm for transforming any vector in an orbit of dimension 13 or less into its representative in Table 1. This part of the proof is given in .
In a lecture at the AMS/LMS meeting in July 1992, Martin Liebeck mentioned the above G-module V as 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 orbits on points. This classification problem extends work of Kac, Popov and Vinberg in characteristic 0 (cf. [KPV]) to arbitrary characteristic. We are grateful to R. Guralnick for showing us a preliminary version of the preprint [GLMS] and for help with other parts of the paper particularly section 6. Apart from the examples that also arise in characteristic 0, a short list of examples typical for certain positive characteristics appears in [GLMS]. The above pair is among them. For most examples, [GLMS] contains a proof that the groups have finitely many orbits in these modules indeed, but for our particular example, it refers to the present paper for a proof. The search may be put in the wider context of finding all pairs of subgroups in almost simple algebraic groups over algebraically closed fields for which there are only finitely many double cosets. For our case in there are finitely many double cosets over the subgroups and the stabilizer of a point in the 16 dimensional vector space. This is an example where one subgroup is parabolic and the other subgroup is not contained in a parabolic and so acts irreducibly.
To finish this introduction, we would like to note that the
relative invariant of degree 8 does not seem to be as exceptional as
one might think.
For, if is a field of arbitrary characteristic distinct from 2,
there still is a nonzero relative invariant of degree 8 on
the 20-dimensional irreducible module ,
as has been noted by Salmon (cf. [S]) and
verified by a straightforward computation using LiE
(cf. [LiE]). Since the projective space of the module can be viewed
as the variety of all cubic surfaces in projective space, we
would be interested in an explanation for the existence of the
relative invariant in this context.