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arithmetic.g
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arithmetic.g
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# Functionality for arithmetic groups, based on papers by AD,DF,AH
ARITHVERSION:="1.13";
# October 2021
DeclareInfoClass("InfoArithSub");
SetInfoLevel(InfoArithSub,1);
#
if LoadPackage("matgrp")=fail then
Error(
"The ``matgrp'' package is required, but is currently not installed, or some of its dependencies\n",
"(recog,recogbase,genss,forms,orb,io) have not been installed or (orb,io) not been compiled.");
fi;
Print("Arithmetic group routines, Version ",ARITHVERSION,", Alexander Hulpke\n");
# Part 1: Density testing
IsTransvection:=mat->RankMat(mat-mat^0)=1 and IsZero((mat-mat^0)^2);
# find proper kind, using an `arg' type argument list, returning SL or SP
CheckProperKind:=function(arglist,len)
local kind,form,n;
if Length(arglist)>=len then
kind:=arglist[len];
if kind=0 then
kind:=SL; # SL
# determine proper kind
n:=Length(One(arglist[1]));
if IsEvenInt(n) then
form:=NullMat(n,n,0);
form{[1..n/2]}{[n/2+1..n]}:=IdentityMat(n/2,Rationals);
form{[n/2+1..n]}{[1..n/2]}:=-IdentityMat(n/2,Rationals);
if ForAll(GeneratorsOfGroup(arglist[1]),
x->TransposedMat(x)*form*x=form) then
Info(InfoArithSub,1,"Detected Form SP");
kind:=SP;
fi;
fi;
fi;
if kind<>1 and kind<>2 and kind<>SL and kind<>SP then
Error("Arithmetic groups of type ",kind," are not yet supported");
fi;
if kind=SL or kind=1 then
kind:=SL;
elif kind=2 or kind=SP then
kind:=SP;
fi;
else
kind:=SL;
fi;
# test degree
if DegreeOfMatrixGroup(arglist[1])<2 then
Error("The functions only work for degree >=3");
fi;
# test determinant
if ForAny(GeneratorsOfGroup(arglist[1]),x->not IsOne(DeterminantMat(x))) then
Error("The group is generated with a matrix of determinant <> 1");
fi;
return kind;
end;
ASPseudoRandom:=function(H)
local epi,e,r;
r:=ValueOption("radius");
if r=fail then r:=50;fi;
epi:=EpimorphismFromFreeGroup(H);
e:=PseudoRandom(Source(epi):radius:=r);
return Image(epi,e);
end;
GenericAbsirr:=function(G)
local nsq,bas,basm,b,g,a,pone,basmo,v,prime;
prime:=11;
pone:=Z(prime)^0;
nsq:=Length(One(G))^2;
bas:=[Flat(One(G))];
basmo:=[];
for b in bas do
v:=b*pone;
ConvertToVectorRep(v,prime);
Add(basmo,v);
od;
basm:=[One(G)];
b:=1;
while b<=Length(bas) do
for g in GeneratorsOfGroup(G) do
a:=basm[b]*g;
v:=Flat(a)*pone;
ConvertToVectorRep(v,prime);
# try to test mod prime first if dimension is larger
if SolutionMat(basmo,v)=fail or
SolutionMat(bas,Flat(a))=fail then
Add(basm,a);
Add(bas,Flat(a));
Add(basmo,Flat(a)*pone);
if Length(bas)=nsq then
return true;
fi;
fi;
od;
b:=b+1;
od;
return false;
end;
# kind is SL or SP
IsDenseIR1:=function(arg)
local G,kind,n,w1,w2,p,gal,r,t;
G:=arg[1];
kind:=CheckProperKind(arg,2);
n:=Length(One(G));
r:=0;
repeat
r:=r+1;
if r>5 then return false;fi;
w1:=ASPseudoRandom(G);
w2:=ASPseudoRandom(G);
until Order(w2)=infinity and w1*w2<>w2*w1;
if kind=SL then
t:=TransitiveIdentification(SymmetricGroup(n));
else
t:=TransitiveIdentification(WreathProduct(Group((1,2)),SymmetricGroup(n/2)));
fi;
p:=CharacteristicPolynomial(w1);
if (not IsIrreducible(p)) or GaloisType(p)<>t then
#Print("galtype w1=",GaloisType(p),"\n");
return false;
fi;
p:=CharacteristicPolynomial(w2);
if (not IsIrreducible(p)) or GaloisType(p)<>t then
#Print("galtype w2=",GaloisType(p),"\n");
return false;
fi;
# test irreducible
if GenericAbsirr(G) then
return true;
else
return false;
fi;
end;
# kind is SL or SP
IsDenseIR2:=function(arg)
local G,kind,n,unflat,bas,i,j,m,V,act,form,mats;
G:=arg[1];
kind:=CheckProperKind(arg,2);
unflat:=m->List([1..n],x->m{[1+n*(x-1)..n*x]});
n:=Length(One(G));
if IsFinite(G) then return false;fi;
if kind=SL then
#trace 0
bas:=[];
for i in [1..n] do
for j in [1..n] do
m:=NullMat(n,n);
m[i][j]:=1;
if i=j then
m[n][n]:=-1;
fi;
if i<n or j<n then
Add(bas,m);
fi;
od;
od;
else
# symmetric
bas:=[];
for i in [1..n] do
for j in [i..n] do
m:=NullMat(n,n);
m[i][j]:=1;
if i<>j then
m[j][i]:=1;
fi;
Add(bas,m);
od;
od;
form:=NullMat(n,n);
form{[1..n/2]}{[n/2+1..n]}:=IdentityMat(n/2);
form{[n/2+1..n]}{[1..n/2]}:=-IdentityMat(n/2);
bas:=List(bas,x->LeftQuotient(form,x));
fi;
bas:=List(bas,Flat);
act:=function(m,g) return Flat(unflat(m)^g);end;
mats:=List(GeneratorsOfGroup(G),x->List(bas,b->SolutionMat(bas,act(b,x))));
return GenericAbsirr(Group(mats));
end;
PrimesForDense:="forward declare";
NumberNotAbsIrr:="forward declare";
IsDenseDFH:=function(arg)
local G,kind;
G:=arg[1];
kind:=CheckProperKind(arg,2);
return PrimesForDense(G,arg[Length(arg)],kind:densitytest);
end;
# Part 2: Level and Index
# utility function
RegModZSpan:=function(gens,seed,oneirr)
local n,matsplit,b,cnt,HM,si,new,bc,i,j,d,process,stack,max,prime,
fp,fpgens,words,k,sol,det,odet,reducer,dens;
reducer:=function(M)
if dens then
return LLLReducedBasis(M).basis;
else
return HermiteNormalFormIntegerMat(M);
fi;
end;
dens:=not (ForAll(Flat(seed),IsInt) and ForAll(Flat(gens),IsInt));
if oneirr=false then
oneirr:=1;
fi;
n:=Length(gens[1]);
matsplit:=l->List([1..n],x->l{[n*(x-1)+1..n*x]});
# first work modulo `oneirr' to get full rank
fp:=FreeGroup(Length(gens));
fpgens:=GeneratorsOfGroup(fp);
bc:=ShallowCopy(seed)*oneirr;
i:=0;
words:=List(bc,x->[0,One(fp)]);
while Length(bc)<Length(bc[1]) do
i:=i+1;
if i>Length(bc) then
if oneirr=1 then
return bc;
fi;
Info(InfoArithSub,2,"not full rank mod irr");
return fail; # wrong generators -- not full rank
fi;
si:=matsplit(bc[i]);
for j in [1..Length(gens)] do
for k in [1,-1] do
new:=Concatenation(si*(gens[j]*oneirr)^k);
if SolutionMat(bc,new)=fail then
Add(bc,new);
words[Length(bc)]:=[i,fpgens[j]^k];
fi;
od;
od;
od;
Info(InfoArithSub,3,"Zspan");
# next, form these images over Z.
b:=ShallowCopy(seed);
i:=Length(b);
while i<Length(bc) do
i:=i+1;
si:=matsplit(b[words[i][1]]);
new:=si*MappedWord(words[i][2],fpgens,gens);
Add(b,Concatenation(new));
od;
Info(InfoArithSub,3,"Detfix");
#Error("GNU");
# check that we got all images in
HM:=reducer(b);
#det:=Determinant(HM);
for i in [1..Length(b)] do
si:=matsplit(b[i]);
for j in [1..Length(gens)] do
for k in [1,-1] do
# did we do this image already -- if so unneeded
if not [i,fpgens[j]^k] in words then
new:=Concatenation(si*gens[j]^k);
sol:=SolutionMat(HM,new);
if not ForAll(sol,IsInt) then
Info(InfoArithSub,4,"Add Vector");
#odet:=det;
HM:=reducer(Concatenation(HM,[new])){[1..Length(b)]};
#det:=Determinant(HM);
#Print("not in ",odet/det,"\n");
fi;
fi;
od;
od;
od;
Info(InfoArithSub,3,"Zlatt Done");
return HM;
end;
# Part 3: Orbit/Stabilizer
OrbitStabRepsMultistage:=
function(pcgs,pcgsimgs,pcisom,solvsz,solvtriv,gens,imgs,fgens,
factorhom,gpsz,actfun,domain,obj)
local stabilizergen,st,stabrsub,stabrsubsz,ratio,subsz,vp,stabrad,
stabfacgens,s,orblock,orb,rep,p,repwords,i,j,k,repword,
imgsinv,img,stabfac,reps,stage,genum,orbstabs,stabfacimg,
fgrp,solsubsz,failcnt,stabstack,relo,dict;
stabilizergen:=function()
local im,i,fe,gpe;
if stage=1 then
Error("solv stage is gone");
else
# in radical factor, still it could be the identity
if Length(repword)>0 then
# build the factor group element
fe:=One(Image(factorhom));
for i in repword do
fe:=fe*fgens[i];
od;
for i in Reversed(repwords[p]) do
fe:=fe/fgens[i];
od;
if not fe in stabfac then
# not known -- add to generators
Add(stabfacimg,fe);
if IsRecord(st) then
if st.left<>fail then
Error("cannot happen");
st:=st.left/st.right;
else
gpe:=One(Source(factorhom));
for i in repwords[st.vp] do
gpe:=gpe*gens[i];
od;
gpe:=gpe*gens[st.genumr];
for i in Reversed(repwords[st.right]) do
gpe:=gpe/gens[i];
od;
# vector image under st
im:=orb[1];
for i in repwords[st.vp] do
im:=actfun(im,imgs[i]);
od;
im:=actfun(im,imgs[st.genumr]);
for i in Reversed(repwords[st.right]) do
im:=actfun(im,imgsinv[i]);
od;
fi;
# make sure st really stabilizes by dividing off solvable bit
st:=gpe/reps[LookupDictionary(dict,im)];
fi;
Add(stabfacgens,st);
stabfac:=ClosureGroup(stabfac,fe);
subsz:=stabrsubsz*Size(stabfac);
ratio:=gpsz/subsz/Length(orb);
if ratio=1 then vp:=Length(orb);fi;
Assert(1,GeneratorsOfGroup(stabfac)=stabfacimg);
fi;
fi;
fi;
# radical stabilizer element. TODO: Use PCGS to remove
# duplicates
end;
fgrp:=Group(fgens,One(Range(factorhom)));
imgsinv:=List(imgs,Inverse);
# now compute orbit, being careful to get stabilizers in steps
dict:=NewDictionary(obj,true,domain);
orb:=[obj];
AddDictionary(dict,obj,1);
reps:=[One(Source(factorhom))];
stabstack:=[];
stabrad:=[];
stabrsub:=solvtriv;
stabrsubsz:=Size(solvtriv);
stabfac:=TrivialSubgroup(fgrp);
subsz:=stabrsubsz*Size(stabfac);
stabfacgens:=[];
stabfacimg:=[];
repwords:=[[]];
# now do a two-stage orbit algorithm. first solvable, then via the
# factor group. Both times we can check that we have the correct orbit.
# ratio 1: full orbit/stab known, ratio <2 stab cannot grow any more.
ratio:=5;
vp:=1; # position in orbit to process
# solvable iteration
stage:=1;
for genum in [Length(pcgs),Length(pcgs)-1..1] do
relo:=RelativeOrders(pcisom!.sourcePcgs)[
DepthOfPcElement(pcisom!.sourcePcgs,pcgs[genum])];
img:=actfun(orb[1],pcgsimgs[genum]);
repword:=repwords[1];
p:=LookupDictionary(dict,img);
if p=fail then
# new orbit images
vp:=Length(orb)*(relo-1);
for j in [1..vp] do
img:=actfun(orb[j],pcgsimgs[genum]);
Add(orb,img);
AddDictionary(dict,img,Length(orb));
Add(reps,reps[j]*pcgs[genum]);
Add(repwords,repword);
od;
else
rep:=pcgs[genum]/reps[p];
Add(stabrad,rep);
stabrsubsz:=stabrsubsz*relo;
subsz:=stabrsubsz;
ratio:=gpsz/subsz/Length(orb);
fi;
od;
stabrad:=Reversed(stabrad);
subsz:=stabrsubsz;
if solvsz>subsz*Length(orb) then
Error("processing stabstack solvable ", Length(stabrad));
s:=1;
while solvsz<>subsz*Length(orb) do
vp:=stabstack[s][1];
genum:=stabstack[s][2];
img:=actfun(orb[vp],pcgsimgs[genum]);
rep:=reps[vp]*pcgs[genum];
repword:=repwords[vp];
p:=LookupDictionary(dict,img);
st:=rec(left:=rep,right:=reps[p]);
stabilizergen();
s:=s+1;
od;
Info(InfoHomClass,5,"processed solvable ",s," from ",Length(stabstack));
fi;
subsz:=stabrsubsz;
solsubsz:=subsz;
orblock:=Length(orb);
Info(InfoHomClass,5,"solvob=",orblock);
# nonsolvable iteration: We act on orbits
stage:=2;
# ratio 1: full orbit/stab known, ratio <2 stab cannot grow any more.
ratio:=5;
vp:=1;
while vp<=Length(orb) do
for genum in [1..Length(gens)] do
img:=actfun(orb[vp],imgs[genum]);
repword:=Concatenation(repwords[vp],[genum]);
p:=LookupDictionary(dict,img);
if p=fail then
# new orbit image
Add(orb,img);
AddDictionary(dict,img,Length(orb));
Add(repwords,repword);
for j in [1..orblock-1] do
img:=actfun(orb[vp+j],imgs[genum]);
Add(orb,img);
AddDictionary(dict,img,Length(orb));
Add(repwords,Concatenation(repwords[vp+j],[genum]));
od;
ratio:=gpsz/subsz/Length(orb);
if ratio=1 then vp:=Length(orb);fi;
elif ratio>=2 then
# old orbit element -- stabilizer generator
# if ratio <2 the stabilizer cannot grow any more
st:=rec(left:=fail,vp:=vp,genumr:=genum,right:=p);
stabilizergen();
fi;
od;
vp:=vp+orblock; # jump in steps
od;
s:=1;
subsz:=stabrsubsz*Size(stabfac);
if gpsz<>subsz*Length(orb) then
Error("should not happen nonslv stabstack");
fi;
Info(InfoHomClass,4,"orblen=",Length(orb)," blocked ",orblock,
" len=", Length(stabrad)," ",Length(stabfacgens));
s:=rec(rep:=orb[1],len:=Length(orb),stabradgens:=stabrad,
stabfacgens:=stabfacgens,stabfacimgs:=stabfacimg,
stabrsub:=stabrsub,stabrsubsz:=stabrsubsz,subsz:=subsz
);
s.gens:=gens;
s.fgens:=fgens;
s.orbit:=orb;
s.orblock:=orblock;
s.reps:=reps;
s.repwords:=repwords;
return s;
end;
ModularImageMatrixGroup:=function(g,m)
local r,l,h;
r:=Integers mod m;
l:=List(GeneratorsOfGroup(g),x->List(x,r->List(r,Int)));
l:=List(l,x->x*One(r));
h:=Group(l);
return h;
end;
# word decomposition functions
###########################################################
#
# Compute HNF and keep word of the transforms
#
# based on library DoNFIM
#
# on generators [ [ 1, 1 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 1, 1 ] ]
SL2WORD:=function(gens,mat)
local w,q;
#if not IsMatrix(gens[1]) then
# Assert(1,SL2WORD([[ [ 1, 1 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 1, 1 ] ]],mat)=mat);
#fi;
mat:=List(mat,ShallowCopy);
w:=One(gens[1]);
#euclid column 1
while not IsZero(mat[2][1]) do
if IsZero(mat[1][1]) then
# add 2 to 1, subtract 1 from 2
AddRowVector(mat[1],mat[2],1);
w:=gens[1]*w;
AddRowVector(mat[2],mat[1],-1);
w:=gens[2]^-1*w;
elif AbsInt(mat[2][1])>=AbsInt(mat[1][1]) then
q:=QuoInt(AbsInt(mat[2][1]),AbsInt(mat[1][1]));
if SignInt(mat[2][1])=SignInt(mat[1][1]) then
AddRowVector(mat[2],mat[1],-q);
w:=gens[2]^-q*w;
else
AddRowVector(mat[2],mat[1],q);
w:=gens[2]^q*w;
fi;
else
q:=QuoInt(AbsInt(mat[1][1]),AbsInt(mat[2][1]));
if SignInt(mat[2][1])=SignInt(mat[1][1]) then
AddRowVector(mat[1],mat[2],-q);
w:=gens[1]^-q*w;
else
AddRowVector(mat[1],mat[2],q);
w:=gens[1]^q*w;
fi;
fi;
#Print(mat,"\n");
od;
if mat[1][1]<0 then
mat:=-mat;
w:=(gens[2]^-1*gens[1]^2)^2*w;
fi;
w:=gens[1]^-mat[1][2]*w;
#mat:=gens[1]^-mat[1][2]*mat;
return w^-1;
end;
HNFWord:=function(hom,mat)
local opt, sig, n, m, A, r, c2, rp, c1, j, k, N, L, b, a, g, c,
t, tmp, i, q, R,tIndex,gens,w,small,trunc;
trunc:=ValueOption("always")=true;
mat:=List(mat,ShallowCopy);
tIndex:=hom!.tIndex;
gens:=MappingGeneratorsImages(hom)[1];
w:=One(gens[1]);
#Parse options
opt:=[ false, true, true, false, false ];
#Embed arg[1] in 2 larger "id" matrix
n:=Length(mat)+2;
m:=Length(mat[1])+2;
k:=ListWithIdenticalEntries(m,0);
A:=[];
for i in [2..n-1] do
#A[i]:=[0];
#Append(A[i],arg[1][i-1]);
#A[i][m]:=0;
A[i]:=ShallowCopy(k);
A[i]{[2..m-1]}:=mat[i-1];
od;
A[1]:=ShallowCopy(k);
A[n]:=k;
A[1][1]:=1;
A[n][m]:=1;
#list:=[];
r:=0;
c2:=1;
rp:=[];
while m>c2 do
Info(InfoMatInt,2,"DoNFIM - reached column ",c2," of ",m);
r:=r+1;
c1:=c2;
rp[r]:=c1;
j:=c1+1;
while j<=m do
k:=r+1;
while k<=n and A[r][c1]*A[k][j]=A[k][c1]*A[r][j] do k:=k+1; od;
if k<=n then c2:=j; j:=m; fi;
j:=j+1;
od;
c:=MATINTmgcdex(AbsInt(A[r][c1]),A[r+1][c1],A{[r+2..n]}[c1]);
for i in [r+2..n] do
if c[i-r-1]<>0 then
AddRowVector(A[r+1],A[i],c[i-r-1]);
#Add(list,["add",r,i-1,c[i-r-1]]);
w:=gens[tIndex[r][i-1]]^c[i-r-1]*w;
#C[r+1][i]:=c[i-r-1];
#AddRowVector(Q[r+1],Q[i],c[i-r-1]);
fi;
od;
i:=r+1;
while A[r][c1]*A[i][c2]=A[i][c1]*A[r][c2] do i:=i+1; od;
if i>r+1 then
c:=MATINTmgcdex(AbsInt(A[r][c1]),A[r+1][c1]+A[i][c1],[A[i][c1]])[1]+1;;
AddRowVector(A[r+1],A[i],c);
if not IsZero(c) then
#Add(list,["add",r,i-1,c]);
w:=gens[tIndex[r][i-1]]^c*w;
fi;
#C[r+1][i]:=C[r+1][i]+c;
#AddRowVector(Q[r+1],Q[i],c);
fi;
g:=MATINTbezout(A[r][c1],A[r][c2],A[r+1][c1],A[r+1][c2]);
small:=[[g.coeff1,g.coeff2],[g.coeff3,g.coeff4]];
if DeterminantMat(small)=-1 then
small[2]:=-small[2];
fi;
# isbound is only relevant if matrix does not have determinant 1
if not IsOne(small) and IsBound(tIndex[r]) then
A{[r,r+1]}:=small*A{[r,r+1]};
#Add(list,["2mult",r-1,r,small]);
w:=SL2WORD([gens[tIndex[r-1][r]],gens[tIndex[r][r-1]]],small)*w;
fi;
#Q{[r,r+1]}:=[[g.coeff1,g.coeff2],[g.coeff3,g.coeff4]]*Q{[r,r+1]};
for i in [r+2..n] do
q:=QuoInt(A[i][c1],A[r][c1]);
AddRowVector(A[i],A[r],-q);
if not IsZero(q) then
#Add(list,["add",i-1,r-1,-q]);
w:=gens[tIndex[i-1][r-1]]^-q*w;
fi;
#AddRowVector(Q[i],Q[r],-q);
q:=QuoInt(A[i][c2],A[r+1][c2]);
AddRowVector(A[i],A[r+1],-q);
if not IsZero(q) then
#Add(list,["add",i-1,r,-q]);
w:=gens[tIndex[i-1][r]]^-q*w;
fi;
#AddRowVector(Q[i],Q[r+1],-q);
od;
od;
rp[r+1]:=m;
Info(InfoMatInt,2,"DoNFIM - r,m,n=",r,m,n);
# hermite or (smith w/ column transforms)
for i in [r, r-1 .. 1] do
Info(InfoMatInt,2,"DoNFIM - reducing row ",i);
for j in [i+1 .. r+1] do
q:=QuoInt(A[i][rp[j]]-(A[i][rp[j]] mod A[j][rp[j]]),A[j][rp[j]]);
AddRowVector(A[i],A[j],-q);
if not IsZero(q) then
#Add(list,["add",i-1,j-1,-q]);
w:=gens[tIndex[i-1][j-1]]^-q*w;
fi;
#AddRowVector(Q[i],Q[j],-q);
od;
od;
# #row transforms finisher
# for i in [r+2..n] do Q[i][i]:= 1; od;
for i in [2..n-1] do
A[i-1]:=A[i]{[2..m-1]};
od;
Unbind(A[n-1]);
Unbind(A[n]);
if trunc then return w^-1;
elif not IsOne(A) then Error("not one!");fi;
return w^-1;
end;
NormDriver:=function(hom,phom,mat,norm)
local n,bn,b,bs,i,sign,a,left,right,words,lwords,lgens,sel,lsel,gens,ln;
gens:=MappingGeneratorsImages(hom);
words:=gens[1];
gens:=gens[2];
# remove duplicates
if IsBound(hom!.unduplisel) then
sel:=hom!.unduplisel;
else
sel:=[];
for i in [1..Length(gens)] do
if not ForAny(sel,x->gens[i]=gens[x] or gens[i]=gens[x]^-1)
and gens[i]<>One(gens[i]) then
Add(sel,i);
fi;
od;
hom!.unduplisel:=sel;
fi;
left:=One(words[1]);
right:=left;
repeat
n:=norm(mat);
lwords:=words;
lgens:=gens;
lsel:=sel;
bn:=0;
bs:=0;
b:=0;
while b=0 do
for i in lsel do
for sign in [1,-1] do
a:=lgens[i]^sign*mat;
ln:=(n-norm(a))/Length(lwords[i]);
if ln>bn then
bn:=ln;
b:=sign*i;
bs:=1;
fi;
a:=mat*lgens[i]^sign;
ln:=(n-norm(a))/Length(lwords[i]);
if ln>bn then
bn:=ln;
b:=sign*i;
bs:=2;
fi;
od;
od;
#Print("DID ",bn,"\n");
if bs=1 then
mat:=lgens[AbsInt(b)]^SignInt(b)*mat;
left:=lwords[AbsInt(b)]^SignInt(b)*left;
elif bs=2 then
mat:=mat*lgens[AbsInt(b)]^SignInt(b);
right:=right*lwords[AbsInt(b)]^SignInt(b);
# now the cases that it failed, hom and phom
elif IsIdenticalObj(lgens,gens) then
lgens:=MappingGeneratorsImages(phom);
lwords:=lgens[1];
lgens:=lgens[2];
lsel:=Filtered([1..Length(lwords)],x->Length(lwords[x])>0);
else
b:=1; # to break while loop
fi;
od;
until bn=0;
return rec(left:=left,mat:=mat,right:=right);
end;
SlSPDecompExtraGenHom:=function(hom,lev)
local gens,words,sel,i,pw,pm,new,nem,j,k,sign;
gens:=MappingGeneratorsImages(hom);
words:=gens[1];
gens:=gens[2];
# remove duplicates
sel:=[];
for i in [1..Length(gens)] do
if not ForAny(sel,x->gens[i]=gens[x] or gens[i]=gens[x]^-1) then
Add(sel,i);
fi;
od;
gens:=gens{sel};
words:=words{sel};
pw:=[One(Source(hom))];
pm:=[One(Range(hom))];
for i in [1..lev] do
for j in [1..Length(pw)] do
for k in [1..Length(gens)] do
for sign in [1,-1] do
nem:=pm[j]*gens[k]^sign;
if not (nem in pm or nem^-1 in pm) then
new:=pw[j]*words[k]^sign;
Add(pm,nem);
Add(pw,new);
fi;
od;
od;
od;
od;
Info(InfoArithSub,1,"Built pool of size ",Length(pm));
return GroupHomomorphismByImagesNC(Source(hom),Range(hom),pw,pm);
end;
WordSL:=function(hom,elm)
local phom,left,right,one,norm2dist,red,norm,n,i,slhom,a,b,factorrad;
n:=Length(elm);
if IsBound(hom!.phom) then
phom:=hom!.phom;
else
factorrad:=ValueOption("factorrad");
if factorrad=fail then factorrad:=4;fi;
phom:=SlSPDecompExtraGenHom(hom,factorrad);
hom!.phom:=phom;
fi;
left:=One(Source(hom));
right:=left;
one:=One(elm);
norm:=m->Sum(m-one,x->x*x);
red:=NormDriver(hom,phom,elm,norm);
left:=red.left*left;right:=right*red.right;elm:=red.mat;
if norm(elm)<>0 then
Info(InfoWarning,1,"not perfect cleanout");
red:=HNFWord(hom,elm);
else
red:=One(Source(hom));
fi;
return left^-1*red/right;
end;
SLNZFP:=function(n)
local t,geni,m,gens,f,rels,i,j,k,l,mat,mats,id,hom,inv;
# for 2, get the nice pres
if n=2 then
f:=FreeGroup("S","T");
rels:=ParseRelators(f, "S^4, (S^3*T)^3, S^2*T*S^-2*T^-1");
t:=f/rels;
gens:=ShallowCopy(GeneratorsOfGroup(t));
Add(gens,t.1^2*t.2/t.1*t.2);
mats:=[[[0,-1],[1,0]],[[1,1],[0,1]],[[1,0],[1, 1]]];
geni:=[[,2],[3]];
else
geni:=List([1..n],x->[]);
t:=function(i,j)
return gens[geni[i][j]];
end;
mats:=[];
id:=IdentityMat(n,1);
m:=0;
gens:=[];
for i in [1..n] do
for j in [1..n] do
if j<>i then
m:=m+1;
geni[i][j]:=m;
Add(gens,Concatenation("t",String(i),String(j)));
mat:=List(id,ShallowCopy);
mat[i][j]:=1;
Add(mats,mat);
fi;
od;
od;
f:=FreeGroup(gens);
gens:=GeneratorsOfGroup(f);
rels:=[];
for i in [1..n] do
for j in Difference([1..n],[i]) do
for k in Difference([1..n],[j]) do
for l in Difference([1..n],[i,k]) do
if i>k or j>l then
Add(rels,Comm(t(i,j),t(k,l)));
fi;
od;
if k<>i then
Add(rels,Comm(t(i,j),t(j,k))/t(i,k));
fi;
od;
od;
od;
Add(rels,(t(1,2)/t(2,1)*t(1,2))^4);
t:=f/rels;
gens:=GeneratorsOfGroup(t);
fi;
m:=Group(mats);
hom:=GroupHomomorphismByImages(t,m,gens,mats);
hom!.tIndex:=geni;
inv:=GroupHomomorphismByFunction(m,t,function(mat)
return WordSL(hom,mat);
end);
SetIsBijective(hom,true);
SetIsBijective(inv,true);
SetInverseGeneralMapping(hom,inv);
SetInverseGeneralMapping(inv,hom);
return hom;
end;
WordSP:=function(hom,elm)
local phom,left,right,one,norm2dist,red,norm,n,i,slhom,a,b,factorrad;
factorrad:=ValueOption("factorrad");
if factorrad=fail then factorrad:=4;fi;
n:=Length(elm);
#phom:=SlSPDecompExtraGenHom(hom,3);
a:=MappingGeneratorsImages(hom);
b:=ShallowCopy(a[2]);
a:=ShallowCopy(a[1]);
for i in hom!.wrds do
Add(a,i[2]);
Add(b,Image(hom,i[2]));
od;
if IsBound(hom!.phom) then
phom:=hom!.phom;
else
phom:=GroupHomomorphismByImagesNC(Source(hom),Range(hom),a,b);
phom:=SlSPDecompExtraGenHom(phom,factorrad);
hom!.phom:=phom;
fi;
left:=One(Source(hom));
right:=left;
one:=One(elm);
norm2dist:=m->Sum(m-one,x->x*x); #2-norm distance to identity
norm:=m->Sum(m-one,x->x*x);
red:=NormDriver(hom,phom,elm,norm);
left:=red.left*left;right:=right*red.right;elm:=red.mat;
# upper right block
for i in [1..n/2] do
norm:=m->Sum(m{[1..i]}{[n/2+1..n]},x->x*x);
#Display(elm);Print(norm(elm)," \n");
if norm(elm)>0 then
red:=NormDriver(hom,phom,elm,norm);
left:=red.left*left;right:=right*red.right;elm:=red.mat;
fi;
od;
# lower left block
for i in [n/2+1..n] do
norm:=m->Sum(m{[n/2+1..i]}{[1..n/2]},x->x*x)+Sum(m{[1..n/2]}{[n/2+1..n]},x->x*x);
#Display(elm);Print(norm(elm)," \n");
if norm(elm)>0 then
red:=NormDriver(hom,phom,elm,norm);
left:=red.left*left;right:=right*red.right;elm:=red.mat;
fi;
od;
if norm(elm)<>0 then Error("not perfect cleanout");fi;
if not IsBound(hom!.halfslhom) then
slhom:=SLNZFP(n/2:factorrad:=2*factorrad);
hom!.halfslhom:=slhom;
else
slhom:=hom!.halfslhom;
fi;
a:=elm{[1..n/2]}{[1..n/2]};
if DeterminantMat(a)=-1 then
a:=MappingGeneratorsImages(hom)[1];
#(Y1^2U1)^2 is -1...1 blcoks
a:=(First(a,x->String(x)="Y1")^2*First(a,x->String(x)="U1"))^2;