On polyvectors of vector spaces and hyperplanes of projective Grassmannians

Abstract

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 A n - 1 , k ( F ) is the Grassmannian of the ( k − 1)-dimensional subspaces of PG( V) (1 ⩽ k ⩽ n − 1). With each hyperplane H of A n - 1 , k ( F ) , we associate an ( n − k)-vector of V (i.e., a vector of ∧ n− k V) which we will call a representative vector of H. One of the problems which we consider is the isomorphism problem of hyperplanes of A n - 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 = 2 k and to those isomorphisms which arise from dualities of PG( V). We also prove that with each regular spread of the projective space PG ( 2 k - 1 , F ) , there is associated some class of isomorphic hyperplanes of the Grassmannian A 2 k - 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 A 5 , 3 ( F ) .