A differential k-form 
of degree 
in an exterior algebra 
is decomposable if there exist 
one-forms 
such that
 |
(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 
-form
has a form
envelope of dimension 
then it is decomposable. In fact, the one-forms
in the (dual) basis to the envelope can be used as the 
above.
Plücker's equations form a system of quadratic equations on the 
in
 |
(2)
|
which is equivalent to 
being decomposable. Since a decomposable 
-form corresponds to a 
-dimensional subspace, these quadratic equations show that
the Grassmannian is a projective
algebraic variety. In particular, 
is decomposable if for every 
,
 |
(3)
|
where 
denotes tensor contraction and 
is the dual vector space
to 
.
