[GAP Forum] Stabilizers and orbits of subspaces of a vector space

[Dima Pasechnik]

Dear Doug,

On Fri, Dec 17, 2021 at 05:12:37AM -0600, D Brozovic wrote:
> The general question is: given a matrix group G acting on Row space V, how
> does one compute the stabilizer in G of a given subspace W of V?
> 
> It's not quite clear to me how to set up the G-action on subspaces.  For
> example, if
> 
> V:=FullRowSpace(GF(3),4) and
> 
> X:=Subspaces(V,2), how does one obtain the action of G on this collection
> of subspaces?  If that's done, presumably I can just compute Stablizer(G,
> W) with the usual gap command.
The action you need to use in this case is OnSubspacesByCanonicalBasis.

gap> G:=GL(5,3);
GL(5,3)
gap> Length(Orbit(G,CanonicalBasis(Subspace(GF(3)^5,Z(3)^0*[[1,-1,0,0,0],[0,1,1,-1,0]])),OnSubspacesByCanonicalBasis));
1210
gap> Stabilizer(G,CanonicalBasis(Subspace(GF(3)^5,Z(3)^0*[[1,-1,0,0,0],[0,1,1,-1,0]])),OnSubspacesByCanonicalBasis);
<matrix group of size 393030144 with 7 generators>

Hope this helps,
Dima