Contents

For Galois cohomology, one might want to know , for

In characteristic 0 we have that decomposes into -irreducible as follows, where the square brackets denote high weight modules whose high weight they embrace and exponents indicate multiplicities (bigger than one):

This computation has been done using LiE, cf. [LiE]. The first factor shows that there is a form of degree 8 which is left invariant by for any field
A second method we call the method of * determinants*. It is used when
a form is in but not in . The submatrix would be a
nonsingular matrix so the rank of **A** would be **13**. This
means the rows indexed by are linear combinations of the
rows of **A** included in the submatrix. Suppose the submatrix is
. Let be the vector of rows not
excluded by and the corresponding columns of **A** not
excluded by . In particular for **b** above, would be
and would be
. It is straightforward to
check that the condition that **A** has rank **13** is

Contents