TITLE: Fundamental domain for symmetric group $S_n$ acting on $\mathop{Gr}(k,n)$?
QUESTION [5 upvotes]: Is there a nice fundamental domain for the symmetric group $S_n$ acting on the Grassmannian of $k$-planes in $\mathbb{R}^n$?  (The action of $S_n$ is by permuting the coordinates, of course.)
I'm looking for a way to efficiently test whether two subspaces of $\mathbb{R}^n$ are related by permuting and/or negating the coordinates.  (So I really care not about $S_n$, but about $(\mathbb{Z}/2)^n \rtimes S_n$, the Weyl group of the B/C series.)  I'm somewhat skeptical this is easy, but has anyone thought about it?

REPLY [2 votes]: An observation: If you ask for an exact decision procedure for whether to points of $G(k,n)$ are in the same $S_n$ orbit, this problem includes the problem of bipartite graph isomorphism. Namely, if $A$ and $B$ are the adjacency matrices of bipartite graphs on $k$ white and $n-k$ black vertices, then the row spans of
$$\begin{pmatrix} 2 \cdot \mathrm{Id} & A \end{pmatrix} \ \mbox{and} \ \begin{pmatrix} 2 \cdot \mathrm{Id} & B \end{pmatrix}$$
are in the same orbit if and only if the graphs are isomorphic.
Since you really care about generic points of $G(k,n)$, and since graph isomorphism is quite fast in practice, this shouldn't be considered to rule out the possibility of a good practical algorithm, but it does rule out certain strategies.