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

Abstract Let K be an arbitrary field, let n , k , l be nonnegative integers satisfying n ⩾ 1 , 1 ⩽ k ⩽ 2 n , 0 ⩽ l ⩽ min ( n , k ) , and let V be a 2 n -dimensional vector space over K equipped with a nondegenerate alternating bilinear form f . Let W k , l denote the subspace of ⋀ k V generated by all vectors v ¯ 1 ∧ ⋯ ∧ v ¯ k , where v ¯ 1 , … , v ¯ k are k linearly independent vectors of V such that 〈 v ¯ 1 , … , v ¯ l 〉 is totally isotropic with respect to f . We prove that dim ( W k , l ) = 2 n k - 2 n 2 l - k - 2 . We give a recursive method for constructing a basis of W k , l and give a decomposition of W k , l relative to a given hyperbolic basis of V . We also study two linear mappings, one between the spaces W k , l and W k - 2 , l - 1 and another one between W k , l and W 2 n - k , n + l - k .