It is possible to study the
structure of rank preservers on n-square skew-symmetric matrices over an
algebraically closed field F by considering instead the linear transformations on the
second Grassmann Product Space ∧2𝒰 (𝒰 an n-dimensional vector space) over F into
itself, which preserve the irreducible lengths of the products. In this paper, it is
shown that preservers of irreducible length 2 are also preservers of all irreducible
lengths of the products. Correspondingly, rank 4 preservers are rank 2k
preservers for all positive integer values of k. The structure of the preservers in
each case is deduced from the fact that these preservers are in particular
irreducible length 1 and rank 2 preservers respectively, whose structures are
known.
