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 
.
