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#############################################################################
##
#W hybridst.gi AutPGrp package Bettina Eick
##
#H @(#)$Id: hybrstab.gi,v 1.10 2009/03/09 07:26:55 gap Exp $
##
Revision.("autpgrp/gap/hybridst_gi") :=
"@(#)$Id: hybrstab.gi,v 1.10 2009/03/09 07:26:55 gap Exp $";
#############################################################################
##
#F CollectToWord( list )
##
CollectToWord := function( list )
local coll, t, i;
coll := [];
t := [list[1], 1];
for i in [2..Length(list)] do
if t[1] <> list[i] then
Add( coll, t );
t := [list[i], 1];
else
t[2] := t[2] + 1;
fi;
od;
Add( coll, t );
return coll;
end;
#############################################################################
##
#F TransformPG( get, list, id ) . . . . . . . . . . . .convert get to element
##
TransformPG := function( get, list, id )
local coll, res, i;
# catch the special case
if Length( get ) = 0 then return id; fi;
# otherwise compute
coll := CollectToWord( get );
if coll[1][1] > 0 then
res := [PGPower( coll[1][2],list[coll[1][1]] )];
else
res := [PGPower( coll[1][2], PGInverse(list[-coll[1][1]]))];
fi;
for i in [2..Length(coll)] do
if coll[i][1] > 0 then
Add( res, PGPower(coll[i][2],list[coll[i][1]] ) );
else
Add( res, PGPower(coll[i][2], PGInverse(list[-coll[i][1]]) ) );
fi;
od;
return PGMultList( res );
end;
#############################################################################
##
#F Transform( get, list, id ) . . . . . . . . . . . . .convert get to element
##
Transform := function( get, list, id )
local res, i;
if Length( get ) = 0 then return id; fi;
if get[1] > 0 then res := list[get[1]];
else res := list[-get[1]]^-1;
fi;
for i in [2..Length( get )] do
if get[i] > 0 then res := res * list[get[i]];
else res := res * list[-get[i]]^-1;
fi;
od;
return res;
end;
#############################################################################
##
#F ReduceGet( ords, get ) . . . . . . . . . . . . . . .reduce get with orders
##
ReduceGet := function( ords, get )
local found, i, j, o;
found := true;
while found do
# first reduce by inverses
i := 1;
found := false;
while i <= Length( get ) - 1 do
if not IsBool( get[i+1] ) and get[i] = - get[i+1] then
get[i] := false;
get[i+1] := false;
found := true;
i := i + 1;
fi;
i := i + 1;
od;
# now reduce by orders
i := 1;
while i <= Length( get ) do
if not IsBool( get[i] ) then
if get[i] > 0 then
o := ords[ get[i] ];
else
o := ords[ -get[i] ];
fi;
if i+o-1 <= Length( get ) and
ForAll( get{[i..i+o-1]}, x -> x = get[i] ) then
for j in [i..i+o-1] do
get[j] := false;
od;
found := true;
i := i + o - 1;
fi;
fi;
i := i + 1;
od;
od;
return Filtered( get, x -> not IsBool( x ) );
end;
#############################################################################
##
#F OSTransversalInverse( j, trans, trels, id )
##
OSTransversalInverse := function( j, trans, trels, id )
local l, g, s, p, t;
if j = 1 then return id; fi;
l := Product( trels );
j := j - 1;
g := id;
for s in Reversed( [1..Length( trans )] ) do
p := trels[s];
l := l/p;
t := QuoInt( j, l );
j := RemInt( j, l );
if t > 0 then
g := PGMult( g, PGInverse(trans[s])^t );
fi;
od;
return g;
end;
#############################################################################
##
#F PcgsOrbitStabilizer( A, oper, pt, fpt, info )
##
PcgsOrbitStabilizer := function( A, oper, pt, fpt, info )
local pcgs, rels, stabl, srels, trans, trels, orbit, i, y, j, p, l, s,
k, t, h, g;
pcgs := A.agAutos;
rels := A.agOrder;
# catch trivial case
if Length( pcgs ) = 0 then
return rec( stabl := pcgs,
srels := rels,
orbit := [pt],
trans := [],
trels := [] );
fi;
# initialise orbit, stabiliser and transversal
stabl := [];
srels := [];
trans := [];
trels := [];
orbit := [pt];
# Start constructing orbit.
i := Length( pcgs );
while i >= 1 do
if oper[i] = 1 then
Add( stabl, pcgs[i] );
Add( srels, rels[i] );
else
y := fpt( pt, oper[i], info );
j := Position( orbit, y );
if IsBool( j ) then
# enlarge transversal
Add( trans, pcgs[i] );
Add( trels, rels[i] );
# enlarge orbit
p := rels[i];
l := Length( orbit );
orbit[p*l] := true;
s := 0;
for k in [ 1 .. p - 1 ] do
t := s + l;
for h in [ 1 .. l ] do
orbit[h + t] := fpt( orbit[h + s], oper[i], info );
od;
s := t;
od;
else
# enlarge stabilizer
if j > 1 then
g := OSTransversalInverse(j, trans, trels, A.one);
Add( stabl, PGMult( pcgs[i], g ) );
else
Add( stabl, pcgs[i] );
fi;
Add( srels, rels[i] );
fi;
fi;
i := i - 1;
od;
return rec( stabl := Reversed( stabl ),
srels := Reversed( srels ),
orbit := orbit,
trans := trans,
trels := trels );
end;
#############################################################################
##
#F BlockPosition( orbit, pt )
##
BlockPosition := function( orbit, pt )
local h, j;
for j in [1..Length(orbit)] do
h := Position( orbit[j], pt );
if not IsBool( h ) then return [j, h]; fi;
od;
return false;
end;
#############################################################################
##
#F BlockOrbitStabilizer( B, oper, os, fpt, info )
##
BlockOrbitStabilizer := function( B, oper, os, fpt, info )
local bl, l, li, orbit, trans, stabl, pstab, mats, auts, ords, pers,
k, pt, i, y, j, new, get, aut, g, per, s;
# the block and limit for orbit length
bl := os.orbit;
l := Length( bl );
li := B.glOrder / Factors( B.glOrder )[1];
# set up orbit, transversal and stab
orbit := [ bl ];
trans := [ [] ];
stabl := [];
pstab := [];
# get acting elements
auts := B.glAutos;
ords := List( auts, Order );
if IsBound( B.glOper ) then pers := B.glOper; fi;
# loop
k := 1;
while k <= Length( orbit ) do
pt := orbit[k][1];
for i in [ 1..Length(oper) ] do
# compute the image of a point
y := fpt( pt, oper[i], info );
j := BlockPosition( orbit, y );
if IsBool( j ) then
# enlarge orbit and transversal
new := List( [1..l], x -> true );
for s in [1..l] do
new[s] := fpt( orbit[k][s], oper[i], info );
od;
Add( orbit, new );
get := Concatenation( trans[k], [i] );
Add( trans, get );
else
# enlarge stabilizer
get := Concatenation(trans[k], [i], Reversed(-trans[j[1]]));
get := ReduceGet( ords, get );
aut := TransformPG( get, auts, B.one );
# reduce from block-stab to point-stab
if j[2] > 1 then
g := OSTransversalInverse(j[2], os.trans, os.trels, B.one);
aut := PGMult( aut, g );
fi;
Add( stabl, aut );
# add permutations if known
if IsBound( B.glOper ) then
Add( pstab, Transform( get, pers, () ) );
fi;
fi;
od;
if Length( orbit ) > li then
return rec( stabl := [], pstab := [], length := B.glOrder );
else
k := k + 1;
fi;
od;
return rec( stabl := stabl, pstab := pstab,
length := Length(orbit), trans := trans );
end;
#############################################################################
##
#F PGHybridOrbitStabilizer( A, glMats, agMats, pt, oper, info )
##
PGHybridOrbitStabilizer := function( A, glMats, agMats, pt, oper, info )
local os, OS, B;
# compute ag orbit stabilizier
if Length( glMats ) = 0 and Length( agMats ) = 0 then return; fi;
os := PcgsOrbitStabilizer( A, agMats, pt, oper, info );
Info( InfoAutGrp, 4, " ag-orbit -- length ",Length(os.orbit));
# add info to A
A.agAutos := os.stabl;
A.agOrder := os.srels;
# compute block orbit and stabiliser
if Length( glMats ) = 0 then return; fi;
OS := BlockOrbitStabilizer( A, glMats, os, oper, info );
Info( InfoAutGrp, 4, " gl-orbit -- length ", OS.length,
" -- gens ",Length(OS.stabl));
# set up new aut grp
A.glAutos := OS.stabl;
A.glOrder := A.glOrder / OS.length;
Assert(1,IsInt(A.glOrder));
if IsBound( A.glOper ) then A.glOper := OS.pstab; fi;
# nice the glAutos if necessary
if NICE_STAB and OS.length > 1 then NiceHybridGroup( A ); fi;
end;
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