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#############################################################################
##
## This file is part of recog, a package for the GAP computer algebra system
## which provides a collection of methods for the constructive recognition
## of groups.
##
## This files's authors include Max Neunhöffer, Alice Niemeyer, Ákos Seress.
##
## Copyright of recog belongs to its developers whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-3.0-or-later
##
##
## Find an imprimitive action of a matrix group.
## This should better be (and is now) called "LowIndex".
##
#############################################################################
#############################################################################
##
#F OrbitSubspaceWithLimit( <grp>, <U>, <max> ) . . . . . . orbit of subspace
##
## Compute the orbit of a subspace but return fail if the orbit has
## more than <max> elements.
##
# RECOG.OrbitSubspaceWithLimit := function( grp, U, max )
#
# local orb, orbset, new, pnt, img, gen, gens;
#
# gens := GeneratorsOfGroup(grp);
#
# # start with the singleton orbit
# orb := [ U ];
# orbset := [ U ];
#
# # loop over all points found
# for pnt in orb do
#
# # apply all generators <gen>
# for gen in gens do
# img := OnSubspacesByCanonicalBasis( pnt, gen );
#
# # add the image <img> to the orbit if it is new
# # can perhaps be improved (is now)
# if not img in orbset then
# Add( orb, img );
# AddSet( orbset, img );
# if Length(orb) > max then return fail; fi;
# fi;
#
# od;
#
# od;
# return orbset;
# end;
#
# Test if the module hm already has a submodule with a
# short enough orbit.
# Note: We can tweak what ``short enough" means. I set it to 4d,
# but if we make it smaller it will speed up the code.
#
RECOG.IndexMaxSub := function( hm, grp, d )
# Call this function only with a reducible module hm!
local dim,dimnew,f,i,lastorb,lastsub,orb,sp,spnew,sub,subdim;
# Here is the invariant subspace
sub := MTX.ProperSubmoduleBasis(hm);
sub := MutableCopyMat( sub ); # make mutable copy
# FIXME: this will be unnecessary:
ConvertToMatrixRep(sub,hm.field);
lastsub := fail;
lastorb := fail;
repeat
TriangulizeMat(sub); # Find Hermite Normal Form
#orb := RECOG.OrbitSubspaceWithLimit(grp, sub, 4 * d );
orb := Orb(grp,sub,OnSubspacesByCanonicalBasis,
rec(storenumbers := true, hashlen := NextPrimeInt(8*d)));
Enumerate(orb,4*d);
if not IsClosed(orb) then
Info(InfoRecog,2,"Did not find nice orbit.");
if lastsub = fail then
return fail;
fi;
return rec( orb := lastorb,
hom := OrbActionHomomorphism(grp,lastorb) );
fi;
Info(InfoRecog,2,"Found orbit of length ",Length(orb),
" of subspaces of dimension ",Length(orb[1]),".");
subdim := Length(orb[1]);
if subdim * Length(orb) = d or # have block system!
subdim = 1 then # no hope in this case
return
rec(orb := orb,
hom := OrbActionHomomorphism(grp,orb));
fi;
# we try intersecting the subspaces in the orbit:
Info(InfoRecog,2,"Calculating intersections...");
f := hm.field;
sp := VectorSpace(f,orb[1]);
dim := Dimension(sp);
for i in [2..Length(orb)] do
spnew := Intersection(sp,VectorSpace(f,orb[i]));
dimnew := Dimension(spnew);
if dimnew > 0 and dimnew < dim then
sp := spnew;
dim := dimnew;
fi;
od;
Info(InfoRecog,2,"Got subspace of dimension ",Dimension(sp));
if dim = Length(sub) then # we got nothing new
Info(InfoRecog,2,"That we already knew, giving up.");
return
rec(orb := orb,
hom := OrbActionHomomorphism(grp,orb));
fi;
lastsub := sub;
lastorb := orb;
sub := MutableCopyMat(AsList(Basis(sp)));
# FIXME: This will vanish:
ConvertToMatrixRep(sub,hm.field);
until false;
end;
RECOG.SmallHomomorphicImageProjectiveGroup := function ( grp )
local hm, findred, ans, fld, d, i, gens;
fld := DefaultFieldOfMatrixGroup(grp);
d := DimensionOfMatrixGroup(grp);
findred := function( gens )
local hm,i,j,res;
i := Length(gens)+1;
for j in [1..d] do
gens[i] := PseudoRandom(grp);
hm := GModuleByMats(gens,fld);
if not MTX.IsIrreducible(hm) then
res := RECOG.IndexMaxSub( hm, grp, d );
if res <> fail then
return [res, gens];
fi;
if i < LogInt(d,2) then
res := findred(gens);
if res = false then
return false;
fi;
if res <> fail then
return res;
fi;
fi;
fi;
od;
return false; # go out all the way without success
end;
Info(InfoRecog,2,"LowIndex: Trying 10 first elements...");
for i in [1..10] do # this is just heuristics!
gens := [PseudoRandom(grp)];
hm := GModuleByMats(gens,fld);
if not MTX.IsIrreducible(hm) then
ans := findred( gens );
if ans <> fail and ans <> false then
return ans;
fi;
fi;
if InfoLevel(InfoRecog) >= 2 then Print(".\c"); fi;
od;
if InfoLevel(InfoRecog) >= 2 then Print("\n"); fi;
return fail;
end;
#! @BeginChunk LowIndex
#! This method is designed for the handling of the Aschbacher class C2
#! (stabiliser of a decomposition of the underlying vector space), but may
#! succeed on other types of input as well. Given <A>G</A> <M> \le PGL(d,q)</M>,
#! the output is either the permutation action of <A>G</A> on a short
#! orbit of subspaces or <K>fail</K>. In the current setup, <Q>short orbit</Q>
#! is defined to have length at most <M>4d</M>.
#! @EndChunk
BindRecogMethod(FindHomMethodsProjective, "LowIndex",
"find an (imprimitive) action on subspaces",
function(ri,G)
local res;
RECOG.SetPseudoRandomStamp(G,"LowIndex");
res := RECOG.SmallHomomorphicImageProjectiveGroup(G);
if res = fail then
return fail; # FIXME: fail = TemporaryFailure here really correct?
else
res := res[1];
# Now distinguish between a block system and just an orbit:
if Length(res.orb) * Length(res.orb[1]) <> ri!.dimension then
Info(InfoRecog,2,"Found orbit of length ",Length(res.orb),
" - not a block system.");
else
Info(InfoRecog,2,"Found block system with ",Length(res.orb),
" blocks.");
# A block system: We do a base change isomorphism:
InitialDataForKernelRecogNode(ri).t := Concatenation(res.orb);
InitialDataForKernelRecogNode(ri).blocksize := Length(res.orb[1]);
AddMethod(InitialDataForKernelRecogNode(ri).hints, FindHomMethodsProjective.DoBaseChangeForBlocks, 2000);
Setimmediateverification(ri,true);
findgensNmeth(ri).args[1] := Length(res.orb)+3;
findgensNmeth(ri).args[2] := 5;
fi;
# we are done, report the hom:
SetHomom(ri,res.hom);
Setmethodsforimage(ri,FindHomDbPerm);
return Success;
fi;
end);
#! @BeginChunk DoBaseChangeForBlocks
#! TODO
#! @EndChunk
BindRecogMethod(FindHomMethodsProjective, "DoBaseChangeForBlocks",
"Hint TODO",
function(ri,G)
# Do the base change:
local H,iso,newgens,ti;
ti := ri!.t^-1;
newgens := List(GeneratorsOfGroup(G),x->ri!.t*x*ti);
H := GroupWithGenerators(newgens);
iso := GroupHomByFuncWithData(G,H,RECOG.HomDoBaseChange,
rec(t := ri!.t,ti := ti));
# Now report back:
SetHomom(ri,iso);
findgensNmeth(ri).method := FindKernelDoNothing;
# Inform authorities that the image can be recognised easily:
InitialDataForImageRecogNode(ri).blocksize := ri!.blocksize;
AddMethod(InitialDataForImageRecogNode(ri).hints, FindHomMethodsProjective.Blocks, 2000);
return Success;
end);
#! @BeginChunk Blocks
#! TODO
#! @EndChunk
BindRecogMethod(FindHomMethodsProjective, "Blocks",
"Hint TODO",
function(ri,G)
# Here we use BlocksModScalars and then get a kernel of scalar blocks
# altogether mod scalars.
local blocks,d,hom,i;
hom := IdentityMapping(G);
SetHomom(ri,hom);
blocks := [];
d := ri!.dimension;
for i in [1..d/ri!.blocksize] do
Add(blocks,[(i-1)*ri!.blocksize+1..i*ri!.blocksize]);
od;
# For the image:
InitialDataForImageRecogNode(ri).blocks := blocks;
AddMethod(InitialDataForImageRecogNode(ri).hints, FindHomMethodsProjective.BlocksModScalars, 2000);
# For the kernel:
InitialDataForKernelRecogNode(ri).blocks := blocks;
AddMethod(InitialDataForKernelRecogNode(ri).hints, FindHomMethodsProjective.BlocksBackToMats, 2000);
return Success;
end);
RECOG.HomBackToMats := function(el)
# We assume that el is block diagonal with the last block being scalar.
# This just norms this last block to 1.
local d;
d := Length(el);
return (el[d,d]^-1)*el;
end;
#! @BeginChunk BlocksBackToMats
#! TODO
#! @EndChunk
BindRecogMethod(FindHomMethodsProjective, "BlocksBackToMats",
"Hint TODO",
function(ri,G)
# This is only called as hint from Blocks, so we know that we in fact
# have scalar blocks along the diagonal and nothing else.
local H,hom,newgens;
newgens := List(GeneratorsOfGroup(G),RECOG.HomBackToMats);
H := Group(newgens);
hom := GroupHomomorphismByFunction(G,H,RECOG.HomBackToMats);
SetHomom(ri,hom);
# hints for the image:
Setmethodsforimage(ri,FindHomDbMatrix);
InitialDataForImageRecogNode(ri).blocks := ri!.blocks{[1..Length(ri!.blocks)-1]};
AddMethod(InitialDataForImageRecogNode(ri).hints, FindHomMethodsMatrix.BlockScalar, 2000 );
# This is an isomorphism:
findgensNmeth(ri).method := FindKernelDoNothing;
return Success;
end);
#FindHomMethodsProjective.BalTreeForBlocks := function(ri,G)
# local H,cut,dim,hom,newgens,nrblocks,subdim,gen;
# dim := ri!.dimension;
# nrblocks := dim / ri!.blocksize; # this we know from above
# if nrblocks = 1 then # this might happen during the descent into the tree
# return false;
# fi;
# cut := QuoInt(nrblocks,2); # this is now at least 1
# subdim := cut * ri!.blocksize;
#
# # Project onto image:
# newgens := List(GeneratorsOfGroup(G),
# x->ExtractSubMatrix(x,[subdim+1..dim],[subdim+1..dim]));
# for gen in newgens do
# ConvertToMatrixRep(gen);
# od;
# H := GroupWithGenerators(newgens);
# hom := GroupHomByFuncWithData(G,H,RECOG.HomInducedOnFactor,
# rec(subdim := subdim));
# SetHomom(ri,hom);
#
# # Create some more kernel generators:
# findgensNmeth(ri).args[1] := 20 + nrblocks;
#
# # Pass the block information on to the image:
# InitialDataForImageRecogNode(ri).blocksize := ri!.blocksize;
# AddMethod(InitialDataForImageRecogNode(ri).hints, FindHomMethodsProjective.BalTreeForBlocks, 2000);
#
# # Inform authorities that the kernel can be recognised easily:
# InitialDataForKernelRecogNode(ri).subdim := subdim;
# InitialDataForKernelRecogNode(ri).blocksize := ri!.blocksize;
# AddMethod(InitialDataForKernelRecogNode(ri).hints, FindHomMethodsProjective.BalTreeForBlocksProjKernel, 2000);
#
# # Verify the kernel immediately after its recognition:
# Setimmediateverification(ri,true);
#
# return Success;
#end;
#
#RECOG.HomInducedOnSubspace := function(data,el)
# local m;
# m := ExtractSubMatrix(el,[1..data.subdim],[1..data.subdim]);
# # FIXME: no longer necessary, as soon as the new interface is in place
# ConvertToMatrixRep(m);
# return m;
#end;
#FindHomMethodsMatrix.InducedOnSubspace := function(ri,G)
# local H,dim,gens,hom,newgens,gen,data;
# # Are we applicable?
# if not IsBound(ri!.subdim) then
# return NotEnoughInformation;
# fi;
#
# # Project onto subspace:
# gens := GeneratorsOfGroup(G);
# data := rec(subdim := ri!.subdim);
# newgens := List(gens, x->RECOG.HomInducedOnSubspace(data,x));
# H := Group(newgens);
# hom := GroupHomByFuncWithData(G,H,RECOG.HomInducedOnSubspace,data);
#
# # Now report back:
# SetHomom(ri,hom);
#
# # We know that the kernel is a GF(p) vectorspace and thus may need
# # quite some generators. Therefore we generate them in a non-standard
# # way such that we can be reasonably sure that we got enough:
# findgensNmeth(ri).method := FindKernelLowerLeftPGroup;
# findgensNmeth(ri).args := [];
#
# # Inform authorities that the kernel can be recognised easily:
# InitialDataForKernelRecogNode(ri).subdim := ri!.subdim;
# AddMethod(InitialDataForKernelRecogNode(ri).hints, FindHomMethodsMatrix.LowerLeftPGroup, 2000);
#
# return Success;
#end;
#FindHomMethodsProjective.BalTreeForBlocksProjKernel := function(ri,G)
# # We just have to project onto the upper left corner, see
# local H,gen,hom,newgens;
#
# newgens := List(GeneratorsOfGroup(G),x->x{[1..ri!.subdim]}{[1..ri!.subdim]});
# for gen in newgens do ConvertToMatrixRep(gen); od;
# H := Group(newgens);
# hom := GroupHomByFuncWithData(G,H,RECOG.HomInducedOnSubspace,
# rec(subdim := ri!.subdim));
# SetHomom(ri,hom);
#
# findgensNmeth(ri).method := FindKernelDoNothing;
#
# # But pass on the information on blocks:
# InitialDataForImageRecogNode(ri).blocksize := ri!.blocksize;
# AddMethod(InitialDataForImageRecogNode(ri).hints, FindHomMethodsProjective.BalTreeForBlocks, 2000);
#
# return Success;
#end;
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