| | | Actions of the form GX |
SVorbit,
SVlast and SVgen.
The VECTOR SVorbit
gives for each iÎX the number of the orbit w such that
iÎw. (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 w, then there is a further element jÎw
and a generator g of G such that i=gj. In this case
S_V_I(SVlast,i-1L) is the number jÎX (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 w labelled by
iÎ{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 kÎX 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.
| | | Actions of the form GX |