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?



Fundamental domain for symmetric group $S_n$ acting on $\mathop{Gr}(k,n)$?
Here is a suggestion that ought to work generically.


A general point on $\operatorname{Gr}(k,n)$ corresponds to the graph of a linear transformation $\mathbb{R}^k \to \mathbb{R}^{n-k}$, and hence to an $(n-k) \times k$ matrix. In its $S\_n$-orbit, there is one point that is distinguished by the following requirements (assuming that ties do not occur):


1) The maximum of the absolute values of the entries of the matrix is as large as possible (this determines a $k$-element subset of distinguished coordinates).


2) If $a\_i$ is the maximum of the absolute values of the entries in the $i$-th row, then $a\_1<a\_2<\cdots<a\_{n-k}$.


3) If $b\_j$ is the maximum of the absolute values of the entries in the $j$-th column, then $b\_1<b\_2<\cdots<b\_k$.


You can compute it by first trying all $k$-element subsets, and then permuting the rows and columns of the $(n-k) \times k$ matrix as needed. (Whether this is efficient enough for you will depend on the size of $n$ and $k$.)


If you also want to allow negating the coordinates, then you can choose the sign changes in a unique way (up to an overall sign change) so that all entries in the first row and first column are positive (assuming that the original point is general enough that these entries are nonzero).


