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 $\wedge^k\scrH$; for example, one of those is $\vee^k\scrM$. Determine all the subspaces thus obtained and their dimensionalities. Do the same for $\vee^k\scrH$.
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 $0\leq j\leq k$, the subspace we consider has a basis $$\bex e_{i_1}\wedge \cdots \wedge e_{i_j}\wedge e_{i_{j+1}}\wedge\cdots \wedge e_{i_k}, \eex$$ where $$\bex 1\leq i_1<\cdots<i_j\leq p<p+1\leq i_{j+1}<\cdots<i_k\leq n. \eex$$ Thus its dimension is $$\bex \sex{p\atop j}\cdot \sex{n-p\atop k-j}. \eex$$ (2). Now we consider the subspace of $\vee^k\scrH$. In this case, it has a basis $$\bex e_{i_1}\vee \cdots \vee e_{i_j}\vee e_{i_{j+1}}\vee \cdots \vee e_{i_k}, \eex$$ where $$\bex 1\leq i_1\leq\cdots\leq i_j\leq p<p+1\leq i_{j+1}\leq\cdots\leq i_k\leq n. \eex$$ Thus its dimension is $$\bex \sex{p+j-1\atop j}\cdot \sex{n-p+k-j+1\atop k-j}. \eex$$
