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**V. Jungic, L. Robinson**

This tutorial is designed to accompany result given by Arjeh M. Cohen and David B. Wales in [2] and [3]. The basic definitions and some details and examples necessary for understanding the result and its proof are given as well as a sketch of the proof. The reader will find a few suggestions as to how Maple could be used for proving and checking some parts of the proof.

Let **k** be an algebraically closed field of characteristic **3** and let
be the general linear group of degree **4** over **k** (i.e.
the group of invertible matricies of order **4** with entries in **k**). Let
be the set of all homogeneus polynomials in four variables
over **k**, of degree **n**. Thus

There are exactly tenG- orbits of vectors in . Representative vectors for these orbits are listed above.

We can formulate the Theorem in the following way:

Chen Zhijie [1] found eight of these orbits and conjuctured that
there were no other **G** - orbits. 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. Considering hypersurfaces in the same -orbit
as "equivalent" means working modulo a choice of coordinates.
For example, a view on the real points in the hyperplane **t=1** of the
cubic hypersurface

We will follow the idea of the proof given in [2].

Let be the general linear algebra (i.e. the algebra
of all matricies of order **4** with entries in **k**).We consider
**g** as a Lie algebra. A Lie algebra **g** is a vector space together
with a bilinear map

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 . Here
is matrix whose **ij** - th row consists of the
coefficients of written out on the basis of **V**
just given. In the array below, the row beginning with
represents the **ij** - th row of , that is the vector
such that

For example, Hence, and for , . From the forth row ( which begins with ) we have and, for , . Hence

For , the Lie stabilizer ofFor example:

is a Lie subalgebra ofThe proof of the fact that

is based on the following Lemma.Let and and leth=Yf. Then .

Hence, if **f** and **h** belong to the same orbit
then and are isomorphic as Lie algebras. To prove

To prove , Cohen and Wales consider two cases. If and 16 - dim then there is such that . An algorithm how to construct such that is given. Secondly, it is shown, in a rather complicated manner, that

This, together with the fact finishs the proof. In his talk at the Workshop on Organic Mathematics [3], A. M. Cohen suggested a different way of proving that The proof of the fact above can be reduced to thanks toIf the number ofNow it is sufficient to show that, for every where the elements of are the sets of polynomials with coefficients in . The right hand side is . The left hand side is computed by determining the group stabilizers in of each of the ten . The complete list of those group stabilizers could be found in [1] and [2].Gorbits over is finite, it is also finite over the extension fields of with bijective correspondence.

An idea of how to try to use Maple to check the result above follows. This idea is based on the fact that we KNOW that .

- Select,
- Compute: ,
- Compute: dim ,
- Pick: ; dim = dim ; (Only if dim we have a dilemma.)
- Form: ( We know that .)
- Find: ;(We know that exists.)

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