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Basic tools


Set , the Lie algebra of all endomorphisms of the vector space . We shall use its basis for , where stands for the matrix with 1 at entry i,j and 0 elsewhere.

For simplicity of computations, we choose a basis x,y,z,t of and let act on by the usual matrix action on column vectors from the left. Thus, for instance

This then determines the action of on and hence the action on V. For example,

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 -th monomial in the second row of the array below.

The action of on V can now be explicitly given in terms of the following matrix where is written as

Here is the 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 , we write to denote the Lie subalgebra of consisting of all with .

A major tool for discriminating orbits is the following family of varieties :

If f and are equivalent by an element y in G, then and so the dimensions of and are the same. Hence all forms equivalent to a given f are in the same . Suppose is the stabilizer in G of a form f and is its Lie algebra in . Then is a Lie subalgebra of . This gives that the dimension of is at most the dimension of . Therefore, the dimension of the orbit Gf of f is at least i if f is in .

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Contents Next: Forms Up: No Title Previous: Introduction