The subspaces of the second
order Grassmann product space consisting of products of a fixed irreducible length k
and zero are interesting not only for their own sake and their usefulness when
determining the structure of linear transformations on the product space
into itself which preserve the irreducible length k, but also because they are
isomorphic to subspaces of skew-symmetric matrices of fixed rank 2k. The
structure of these subspaces and the corresponding preservers are known
for k = 1, when the underlying field F is algebraically closed. This paper
gives a complete characterization of these subspaces when k = 2 and F is
algebraically closed. When F is not algebraically closed, these subspaces can be
different.
