Let $scrM$ be a $p$-dimensional subspace of $scrH$ and $scrN$ its orthogonal complement. Choosing $j$ vectors from $scrM$ and $k-j$ vectors from $scrN$ and forming the linear span of the antisymmetric tensor products of all such vectors, we get different subspaces of $wedge^kscrH$; for example, one of those is $vee^kscrM$. Determine all the subspaces thus obtained and their dimensionalities. Do the same for $vee^kscrH$.
Solution. (1). Let $e_1,cdots,e_p$ be the orthonormal basis of $scrM$, and $e_{p+1},cdots,e_k$ be the orthonormal basis of $scrN$. Then for $0leq jleq k$, the subspace we consider has a basis $$ex e_{i_1}wedge cdots wedge e_{i_j}wedge e_{i_{j+1}}wedgecdots wedge e_{i_k}, eex$$ where $$ex 1leq i_1<cdots<i_jleq p<p+1leq i_{j+1}<cdots<i_kleq n. eex$$ Thus its dimension is $$ex sex{patop j}cdot sex{n-patop k-j}. eex$$ (2). Now we consider the subspace of $vee^kscrH$. In this case, it has a basis $$ex e_{i_1}vee cdots vee e_{i_j}vee e_{i_{j+1}}vee cdots vee e_{i_k}, eex$$ where $$ex 1leq i_1leqcdotsleq i_jleq p<p+1leq i_{j+1}leqcdotsleq i_kleq n. eex$$ Thus its dimension is $$ex sex{p+j-1atop j}cdot sex{n-p+k-j+1atop k-j}. eex$$