On polyvectors of vector spaces and hyperplanes of projective Grassmannians

We investigate relationships between polyvectors of a vector space V, alternating multilinear forms on V, hyperplanes of projective Grassmannians and regular spreads of projective spaces. Suppose V is an n-dimensional vector space over a field F and that An-1,k(F) is the Grassmannian of the (k − 1)-dimensional subspaces of PG(V) (1 ⩽ k ⩽ n − 1). With each hyperplane H of An-1,k(F), we associate an (n − k)-vector of V (i.e., a vector of ∧n−kV) which we will call a representative vector of H. One of the problems which we consider is the isomorphism problem of hyperplanes of An-1,k(F), i.e., how isomorphism of hyperplanes can be recognized in terms of their representative vectors. Special attention is paid here to the case n = 2k and to those isomorphisms which arise from dualities of PG(V). We also prove that with each regular spread of the projective space PG(2k-1,F), there is associated some class of isomorphic hyperplanes of the Grassmannian A2k-1,k(F), and we study some properties of these hyperplanes. The above investigations allow us to obtain a new proof for the classification, up to equivalence, of the trivectors of a 6-dimensional vector space over an arbitrary field F, and to obtain a classification, up to isomorphism, of all hyperplanes of A5,3(F).