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For Galois cohomology, one might want to know
,
for L the group of order 120 Zhijie mentions on page 25.
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 k (not necessarily of characteristic
3).
We would like to know of a geometrical explanation of this invariant
in terms of cubic surfaces in projective space.
Also, it seems reasonable to ask whether,
after modding out the invariant 4-dimensional subspace of
cubes in the case of characteristic 3, this invariant
describes the degree 8 invariant whose existence Chen derived
in [Z].
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
where, for
and
vectors, the notation
is
the submatrix of A keeping the rows in
and the columns
in
. We denote by D the
matrix above. When we
make conclusions using the method of determinants we indicate
the entry of D which gives us the information. In our cases,
the inverse can always be calculated by cofactors.

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