Decomposable

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 .

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