Decomposable describes a differential p-form 
in an exterior
algebra 
that is a scalar multiple of a wedge
product of 
one-forms. Thus, for a scalar 
and one-forms 
,
 |
(1)
|
where 
denotes a wedge product. Forms of degree 0, 1, 
, and 
are always decomposable. Hence the first instance of indecomposable forms occurs in 
, in which case 
is indecomposable.
If a p-form 
has a form envelope of
dimension 
then it is decomposable. In fact, the one-forms
in a basis of the form
envelope can be used as the 
above.
Plücker relations form a system of quadratic equations on the 
in
 |
(2)
|
which is equivalent to 
being decomposable. Since a nonzero decomposable 
-form, considered up to a nonzero scalar multiple, corresponds to a 
-dimensional subspace,
these quadratic equations show that the Grassmannian is a projective algebraic
variety. In particular, 
is decomposable if for all 
,
 |
(3)
|
where 
denotes tensor contraction and 
is the dual vector space
to 
.
