Some subspaces of the \(k\)th exterior power of a symplectic vector space
Let \(V\) be a \(2n\)-dimensional vector space over a field equipped with a nondegenerate alternating bilinear form \(f\). Let \(n, k, l\) be nonnegative integers satisfying \(n\geq 1\), \(1\leq k\leq 2n\), \(0\leq l\leq\min(n ,k)\). Let \(W_{k,l}\) denote the subspace of \(\Lambda ^kV\) generated by all vectors \(\bar{v}_1\wedge\cdots\wedge\bar{v}_k\), where \(\bar{v}_1, \dots ,\bar{v}_k\) are \(k\) linearly independent vectors of \(V\) such that the subspace \(\langle\bar{v}_1, \dots ,\bar{v}_k\rangle\) is totally isotropic with respect to \(f\). It is shown that \(\dim (W_{k,l})={2n\choose k}-{2n\choose 2l-k-2}\). Also, a recursive method for constructing a basis for \(W_{k,l}\) is given, as well as a decomposition of \(W_{k,l}\) relative to a given hyperbolic basis of \(V\). Finally, the author studies two linear mappings, one between the spaces \(W_{k,l}\) and \(W_{k-2,l-1}\) and another between \(W_{k,l}\) and \(W_{2n-k,n+l-k}\).
scientific article; zbMATH DE number 5561716