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Actions of the form GX |
SVorbit,
SVlast and
SVgen. The VECTOR
SVorbit gives for each i∈ X the number of the
orbit ω such that i∈ ω. (In other words the number
of the orbit G(i) is the INTEGER object
S_V_I(SVorbit,i-1L).) The other two VECTORs of length
|X| are used for describing how an arbitrary element of the orbit
can be reached through a path labelled by generators of the group
when applied to the standard representative of the orbit. For a
standard representative i both S_V_I(SVlast,i-1L) and
S_V_I(SVgen,i-1L) have the value -1. If i is not a
standard representative of an orbit ω, then there is a
further element j∈ ω and a generator g of G such that
i=gj. In this case S_V_I(SVlast,i-1L) is the number
j∈ X (1≤ j≤ n) and S_V_I(SVgen,i-1L) is the
position of g in the VECTOR of generators of the acting group G. So
it takes values from 0 to S_V_LI(a)-1L.
Furthermore there are two VECTORs of size |G\\X|, namely
Ofirst and
Osize. The standard
representative of the orbit ω labelled by i∈
{1,...,|G\\X|} is given by S_V_I(Ofirst,i-1L), whereas
the number of elements in this orbit is given by
S_V_I(Osize,i-1L).
You can compute these VECTORs using the routine
INT all_orbits(g,SVorbits,SVlast,SVgen,Ofirst,Osize) OP g,SVorbits,SVlast,SVgen,Ofirst,Osize;where
g is a VECTOR consisting of generators of G.
These generators must be PERMUTATIONs all of the same degree.
If you are only interested in the orbit representatives you can use
INT allorbits(g,Ofirst) OP g,Ofirst;where
g is a VECTOR consisting of generators of G.
Having the Schreier vectors at hand we can determine an element of the group G which maps the standard representative of the orbit G(k) of an arbitrary element k∈ X to k, by using
INT trace_schreier_vectors(g,SVlast,SVgen,k,gtrace) OP g,SVlast,SVgen,k,gtrace;where
g is a VECTOR consisting of generators of G.
k is an INTEGER object 1≤ k≤ n and
gtrace is a group element which maps the standard
representative of the orbit G(k) to k.
Furthermore we can compute the stabilizer of the standard representative of the orbit labelled with k by
INT schreier_stabilizer(g,SVorbit,SVlast,SVgen,Ofirst,Osize,k,stab) OP g,SVorbit,SVlast,SVgen,Ofirst,Osize,k,stab;where
g is a VECTOR consisting of generators of G.
Please take care that k is the label of an orbit (i.e.
an element of {1,...,|G\\X|}) and not the standard representative
of an orbit. A VECTOR of generators of the stabilizer of the
standard representative of this orbit is computed as
stab. The stabilizer of any other element of this
orbit is conjugated to stab.
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Actions of the form GX |  |