Some subspaces of the kth exterior power of a symplectic vector space

Let K be an arbitrary field, let n, k, l be nonnegative integers satisfying n >= 1, 1 <= k <= 2n, 0 <= l <= min(n, k), and let V be a 2n-dimensional vector space over K equipped with a nondegenerate alternating bilinear form f. Let W-k,W-l denote the subspace of boolean AND V-k generated by all vectors (v) over bar (1) boolean AND ... boolean AND (v) over bar (k), where (v) over bar (1), ... , (v) over bar (k) are k linearly independent vectors of V such that ((v) over bar (1),..., (v) over bar (1)) is totally isotropic with respect to f. We prove that dim(W-k,W-l ) =(2n k) - (2n 2l - k - 2). We give a recursive method for constructing a basis ofW(k,l) and give a decomposition of W-k,W-l relative to a given hyperbolic basis of V. We also study two linear mappings, one between the spaces W-k,W-l and W-k-2,W-l-1 and another one between W-k,W-l and W-2n-k,W-n+l-k. (C) 2009 Elsevier Inc. All rights reserved.