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arithmetic_fieldext.g
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arithmetic_fieldext.g
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# Functionality for arithmetic groups, based on papers by AD,DF,AH
ARITHVERSION:="1.14";
# July 2023
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");
PrintToGlobals:=function(file,l)
local i;
if Length(Intersection(["file","i","l"],l))>0 then Error("variable names");fi;
PrintTo(file,"# Global Variable save ",l,"\n");
for i in l do
AppendTo(file,i,":=",EvalString(i));
AppendTo(file,";\n");
if IsGroup(EvalString(i)) and HasSize(EvalString(i))
and IsFinite(EvalString(i)) then
AppendTo(file,"SetSize(i,",Size(EvalString(i)),");\n");
fi;
od;
end;
MyListString:=function(l)
local s,i,q,first;
s:="[";
first:=true;
for i in l do
if first then first:=false;
else Add(s,',');fi;
Add(s,' ');
if IsRat(i) then Append(s,String(i));
else
q:=Quadratic(i,true);
if q=fail then
Append(s,String(i));
else
Append(s,q.display);
fi;
fi;
od;
Append(s," ]");
return s;
end;
SmallerGensetFpSub:=function(u)
local p,gens,sel,i,v,t;
p:=Parent(u);
gens:=GeneratorsOfGroup(u);
for i in [Length(gens),Length(gens)-1..1] do
sel:=gens{Difference([1..Length(gens)],[i])};
v:=Subgroup(p,sel);
t:=TryCosetTableInWholeGroup(v:silent:=true);
if t<>fail and Length(t[1])=IndexInWholeGroup(u) then
Print("Elim ",i,":",gens[i],"\n");
gens:=sel;
fi;
od;
return gens;
end;
# Part 0: Number Theory
Char0ScalarDomainMatrixList:=function(L)
local den,i,I,F,c;
L:=Set(Filtered(Flat(L),x->not IsInt(x)));
den:=1;
for i in Filtered(L,IsRat) do
den:=Lcm(den,DenominatorRat(i));
od;
L:=Filtered(L,x->not IsRat(L));
if Length(L)=0 then
if den=1 then
return [Integers,1];
else
return [Rationals,1];
fi;
fi;
I:=[];
F:=Rationals;
for i in L do
c:=CoefficientsOfUnivariatePolynomial(MinimalPolynomial(Rationals,i));
while not ForAll(c,IsInt) do
c:=First(Reversed(c),x->not IsInt(x));
c:=DenominatorRat(c);
den:=Lcm(den,c);
i:=i*c;
c:=CoefficientsOfUnivariatePolynomial(MinimalPolynomial(Rationals,i));
od;
if not i in F then
Add(I,i);
F:=Field(I);
fi;
od;
return [F,den];
end;
AllIntcoeffs:=function(c)
return c<>fail and ForAll(c,IsInt);
end;
# euclideanAlgorithm for Norm euclidean
NormEuclideanAlgorithm:=function(red,a,b)
local u,v,w,q,r;
u:=a;
v:=b;
while not IsZero(v) do
w:=v;
#v := EuclideanRemainder( R, u, v );
q:=u/v;
q:=Coefficients(red.bas,q);
if red.eucl=1 then
q:=List(q,RoundCyc);
q:=q*red.bas;
elif red.eucl=2 then
# recode in std base
q:=[q[1]+q[2]/2,q[2]/2];
r:=RoundCyc(2*q[2]);
q:=[RoundCyc(q[1]-r/2),r];
q:=q*red.bas;
else
Error("not usable");
fi;
v:=u-q*v;
u:=w;
od;
return u;
end;
# single generator in quadratic PID -- somewhat hack
QuadSingleIdealGen:=function(id,red)
local nid,sum,i,j,sgn,t,fa,common,bas;
bas:=red.bas;
# first try LLL
j:=List(id,x->Coefficients(bas,x));
j:=LLLReducedBasis(j).basis;
for i in j do
t:=i*bas;
if AllIntcoeffs(Coefficients(bas,id[1]/t))
and AllIntcoeffs(Coefficients(bas,id[2]/t)) then
#Print("found:",i,"\n");
return t;
fi;
od;
# short Vectors
fa:=10;
fa:=Minimum(List(j,x->x*x));
j:=List(id,x->Reversed(Coefficients(bas,x)));
j:=Filtered(HermiteNormalFormIntegerMat(j),x->not IsZero(x));
j:=List(j,Reversed);
repeat
fa:=fa*10;
sgn:=ShortestVectors(j*TransposedMat(j),fa).vectors;
for i in sgn do
t:=i*j*bas;
if AllIntcoeffs(Coefficients(bas,id[1]/t))
and AllIntcoeffs(Coefficients(bas,id[2]/t)) then
return t;
fi;
od;
until Length(sgn)>10^4;
# now take primes we get from the norm
nid:=List(j,x->x*bas);
fa:= Gcd(List(nid,x->Norm(Field(x),x)));
fa:=Filtered(Set(PartialFactorization(fa)),x->x<=10^100 and IsPrimeInt(x));
common:=[];
for i in fa do
for j in red.factorsinnumbers(i) do
while ForAll(nid,x->AllIntcoeffs(Coefficients(bas,x/j))) do
Add(common,j);
nid:=List(nid,x->x/j);
od;
od;
od;
if Length(common)>0 then
fa:=Product(common); # already known common factors
sgn:=Gcd(List(nid,x->Norm(Field(x),x)));
j:=QuadSingleIdealGen(Concatenation([sgn],nid),red);
if IsList(j) and j[1]=fail then return [fail,j[2]*fa];fi;
return j*fa;
else
return [fail,Gcd(List(nid,x->Norm(Field(x),x)))];
fi;
Error("UUU");
for sum in [1..Maximum(id[1],10000)] do
for i in [0..sum] do
for sgn in [1,-1] do
j:=sum-i;
t:=sgn*i*id[1]+j*id[2];
if ForAll(id,x->AllIntcoeffs(Coefficients(bas,x/t))) then
#Print("found",[i,j],"\n");
return t;
fi;
od;
od;
od;
Error("not found");
end;
# so far only for 2-dim
IdealReductionMat:=function(field,zbas,ideal)
local m,n,mo;
Error("Steinbruch");
# get ideal generators
m:=List(ideal,x->Reversed(Coefficients(zbas,x)));
m:=List(Filtered(HermiteNormalFormIntegerMat(m),
x->not IsZero(x)),Reversed);
if m[1][2]=0 then
n:=m[1][1];
else
n:=Gcd(List(ideal,x->Norm(Field(x),x)));
fi;
mo:=[n];
# reduction of alpha?
if ForAny(m,x->x[1]<>0 and x[2]<>0) then
else
Error("no alpha");
fi;
end;
# general setup for reduction mod p. API should extend to rings.
PrepareReductionInfo:=function(ring)
local d,den,c;
if IsList(ring) then
den:=ring[2];
ring:=ring[1];
else
den:=1;
fi;
d:=rec(primes:=[],pinfo:=[],redfuns:=[],liftfuns:=[]); # data record, carrying the info
if ring=Integers or ring=Rationals then
d.case:=1; # integers
d.discriminant:=1;
d.makereductionfunc:=function(prime)
local one,pos;
pos:=Position(d.primes,prime);
if pos=fail then Add(d.primes,prime); pos:=Length(d.primes); fi;
if not IsBound(d.redfuns[pos]) then
one:=Z(prime)^0;
d.redfuns[pos]:=a->a*one;
fi;
return d.redfuns[pos];
end;
d.convertparam:=prime->prime;
d.bas:=[1];
d.makeliftfunc:=function(prime)
return Int;
end;
d.denominatorfun:=DenominatorRat;
d.associateduplicates:=l->Set(List(l,AbsInt));
d.factorsinnumbers:=function(nums)
if not IsList(nums) then nums:=[nums];fi;
if Length(nums)=0 then return [];fi;
nums:=AbsInt(Lcm(nums));
nums:=Set(Factors(nums));
nums:=Difference(nums,[1]);
return nums;
end;
d.partialFactorization:=x->PartialFactorization(x,6);
d.primalityTest:=IsPrimeInt;
d.commonPrimesPlus:=Gcd;
elif ring=GaussianIntegers or ring=GaussianRationals then
d.bas:=Basis(CF(4),[1,E(4)]);
d.case:=2;
d.discriminant:=2;
d.makereductionfunc:=function(prime)
local pos,one,pol,x,root;
pos:=Position(d.primes,prime);
if pos=fail then
Add(d.primes,prime);
pos:=Length(d.primes);
# build reduction info
if IsInt(prime) then
x:=X(GF(prime),1);
pol:=x^2+1;
if not IsIrreducible(pol) then Error("not prime!");fi;
root:=RootsOfUPol(GF(prime^2),pol)[1];
d.pinfo[pos]:=[root,d.bas,[One(GF(prime)),root],prime^2];
d.redfuns[pos]:=function(a)
return Coefficients(d.bas,a)*d.pinfo[pos][3];
end;
elif IsCyclotomic(prime) then
x:=Norm(prime);
root:=Coefficients(d.bas,prime);
root:=-root[1]/root[2] mod x;
root:=root*One(GF(x));
d.pinfo[pos]:=[root,d.bas,[One(GF(x)),root],x];
d.redfuns[pos]:=function(a)
return Coefficients(d.bas,a)*d.pinfo[pos][3];
end;
else
Error("different ideal");
fi;
fi;
return d.redfuns[pos];
end;
d.convertparam:=function(prime)
d.makereductionfunc(prime); # set up
return d.pinfo[Position(d.primes,prime)][4];
end;
d.denominatorfun:=function(a)
return Lcm(List(Coefficients(d.bas,a),DenominatorRat));
end;
d.factorsinnumbers:=function(num)
local a,an,b,c,n,i;
if not IsList(num) then num:=[num];fi;
if Length(num)=0 then return [];fi;
a:=Set(Concatenation(List(num,x->Factors(GaussianIntegers,x))));
# find associated ones. Use norm to avoid many pairs
# Norm 1 is unit
an:=List(a,Norm);
b:=[];
for n in Difference(an,[1]) do
c:=[];
for i in a{Filtered([1..Length(an)],x->an[x]=n)} do # ones of right norm
if not ForAny(c,x->i/x in GaussianIntegers) then
Add(b,i);
Add(c,i);
fi;
od;
od;
#Print("Factor ",num," -> ",b,"\n");
return b;
end;
d.partialFactorization:=function(a)
#Print("Not partial but full factor\n");
return d.factorsinnumbers(a);
end;
d.primalityTest:=x->1= Length(Factors(GaussianIntegers,x));
d.commonPrimesPlus:=function(arg)
local f,a,i,b,l;
#Print("commonPrimes :",a," ",b,"\n");
if Length(arg)=1 then l:=arg[1]; else l:=arg;fi;
a:=l[1];
i:=1;
while i<Length(l) do
i:=i+1;
b:=l[i];
if IsInt(a) and IsInt(b) then
a:=Gcd(a,b);
elif a=0 then a:=b;
elif b<>0 then
f:=d.factorsinnumbers([Gcd(Norm(a),Norm(b))]);
f:=Filtered(f,x->not IsZero(x));
f:=Filtered(f,x->a/x in GaussianIntegers and b/x in GaussianIntegers);
a:=Product(f,1);
fi;
od;
return a;
end;
elif IsAbelianNumberField(ring) and DegreeOverPrimeField(ring)=2 then
d.case:=3;
# quadratic field. Find out which root
c:=Conductor(ring);
if ForAll(GeneratorsOfField(ring),x->GaloisCyc(x,-1)=x) then
# real, root of positive
if c mod 4=0 then c:=c/4; fi;
else
# complex, root of negative
if c mod 4=0 then c:=c/4; fi;
c:=-c;
fi;
# now the field is sqrt(c)
if c mod 4=1 then
d.bas:=Basis(ring,[1,(1+ER(c))/2]);
d.eucl:=2;
else
d.bas:=Basis(ring,[1,ER(c)]);
d.eucl:=1;
fi;
if not c in [-1,-2,-3,-7,-11] then
d.eucl:=0;
fi;
d.minpol:=MinimalPolynomial(Rationals,d.bas[2]);
d.discriminant:=Discriminant(d.minpol);
d.omega:=(d.discriminant+ER(d.discriminant))/2;
d.field:=Field(d.omega);
d.primefacs:=[];
d.factorprime:=function(p)
local pos,id,f,b;
pos:=PositionSorted(d.primefacs,[p,-infinity]);
if pos<=Length(d.primefacs) and d.primefacs[pos][1]=p then
return d.primefacs[pos][2];
fi;
# Prop. 5.1.4 in Cohen
if d.discriminant mod p=0 then
if p=2 and d.discriminant mod 16=12 then
id:=[p,1+d.omega];
else
id:=[p,d.omega];
fi;
id:=QuadSingleIdealGen(id,d);
f:=[id,id];
elif Jacobi(d.discriminant,p)=-1 then
f:=[p];
else # Legendre=1
f:=d.discriminant mod (4*p);
#b:=First([1..4*p],x->x^2 mod (4*p)=f);
id:=Cartesian(List(RootsOfUPol(X(GF(p))^2-(f mod p)),Int),
Filtered([0..4],x->x^2 mod 4=(f mod 4)));
id:=List(id,x->ChineseRem([p,4],x));
b:=First(id,x->x^2 mod (4*p)=f);
if b=fail then Error("bfail");fi;
id:=[p,d.omega-(d.discriminant+b)/2];
id:=QuadSingleIdealGen(id,d);
f:=[id];
id:=[p,d.omega-(d.discriminant-b)/2];
id:=QuadSingleIdealGen(id,d);
Add(f,id);
fi;
AddSet(d.primefacs,[p,f]);
return f;
end;
d.makereductionfunc:=function(prime)
local pos,one,pol,x,root;
pos:=Position(d.primes,prime);
if pos=fail then
Add(d.primes,prime);
pos:=Length(d.primes);
# build reduction info
if IsInt(prime) then
pol:=PolynomialModP(d.minpol,prime);
if not IsIrreducible(pol) then Error("not prime!");fi;
root:=RootsOfUPol(GF(prime^2),pol)[1];
d.pinfo[pos]:=[root,d.bas,[One(GF(prime)),root],prime^2];
d.redfuns[pos]:=function(a)
return Coefficients(d.bas,a)*d.pinfo[pos][3];
end;
elif IsCyclotomic(prime) then
x:=Norm(Field(prime),prime);
root:=Coefficients(d.bas,prime);
root:=-root[1]/root[2] mod x;
root:=root*One(GF(x));
d.pinfo[pos]:=[root,d.bas,[One(GF(x)),root],x];
d.redfuns[pos]:=function(a)
return Coefficients(d.bas,a)*d.pinfo[pos][3];
end;
else
Error("different ideal");
fi;
fi;
return d.redfuns[pos];
end;
d.convertparam:=function(prime)
d.makereductionfunc(prime); # set up
return d.pinfo[Position(d.primes,prime)][4];
end;
d.denominatorfun:=function(a)
return Lcm(List(Coefficients(d.bas,a),DenominatorRat));
end;
d.factorsinnumbers:=function(num)
local a,an,b,c,n,i,j,try,oi;
if not IsList(num) then num:=[num];fi;
if Length(num)=0 then return [];fi;
a:=[];
for i in Filtered(num,x->Norm(x)<>1) do
# factor over Z and then split primes and check which ones do
c:=i;
if not IsRat(c) then
c:=Norm(d.field,c);
try:=true;
else
try:=false;
fi;
c:=List(Factors(c),AbsInt); # Z-primes
if try then
c:=Union(List(Set(c),d.factorprime));
b:=[];
oi:=i;
for j in c do
while AllIntcoeffs(Coefficients(d.bas,i/j)) do
i:=i/j;
Add(b,j);
od;
od;
#Print(i," factors ",b,"\nproduct:",Product(b),", QN:",
# Norm(d.field,oi/Product(b)),"\n");
Append(a,b);
else
c:=Concatenation(List(c,d.factorprime));
#Print(i," factors ",c,"\nproduct:",Product(c),", QN:",
# Norm(d.field,i/Product(c)),"\n");
Append(a,c);
fi;
od;
Sort(a);
if ForAll(a,IsCyclotomic) then
# find associated ones. Use norm to avoid many pairs
# Norm 1 is unit
an:=List(a,x->Norm(d.field,x));
b:=[];
for n in Difference(an,[1]) do
c:=[];
for i in a{Filtered([1..Length(an)],x->an[x]=n)} do # ones of right norm
if not ForAny(List(c,x->i/x),x->AllIntcoeffs(Coefficients(d.bas,x))) then
Add(b,i);
Add(c,i);
fi;
od;
od;
#Print("Factor ",num," -> ",b,"\n");
else
b:=a;
fi;
return b;
end;
d.partialFactorization:=function(a)
#Print("Not partial but full factor\n");
return d.factorsinnumbers(a);
end;
d.primalityTest:=function(num)
local f;
f:=d.factorsinnumbers(num);
if Length(f)>1 then return false;fi;
return Norm(d.field,num/f[1])=1;
end;
d.commonPrimesPlus:=function(arg)
local f,a,i,b,l,m;
if Length(arg)=1 then l:=arg[1]; else l:=arg;fi;
m:=List(l,x->Norm(Field(x),x));
l:=Concatenation([Gcd(m)],l);
if d.eucl>0 then
a:=l[1];
for i in [2..Length(l)] do
a:=NormEuclideanAlgorithm(d,a,l[i]);
od;
return a;
fi;
return QuadSingleIdealGen(l,d);
# m:=Filtered(l,x->not IsInt(x));
# if Length(m)>0 then
# f:=Field(m[1]);
# m:=List(m,x->Norm(f,x));
# m:=Gcd(m);
# else
# m:=1;
# fi;
#
# f:=Gcd(Concatenation([m],Filtered(l,IsInt)));
# f:=d.factorsinnumbers([f]);
#
# a:=[];
# for i in f do
# if ForAll(l,x->AllIntcoeffs(Coefficients(d.bas,x/i))) then
# Add(a,i);
# fi;
# od;
# return a;
end;
else
Error("cannot yet do");
fi;
return d;
end;
# 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,basmo,v,prime,red,modfun,cvparam;
red:=PrepareReductionInfo(Char0ScalarDomainMatrixList(
GeneratorsOfGroup(G)));
prime:=11;
prime:=red.factorsinnumbers([prime])[1];
modfun:=red.makereductionfunc(prime);
cvparam:=red.convertparam(prime);
nsq:=Length(One(G))^2;
bas:=[Flat(One(G))];
basmo:=[];
for b in bas do
v:=List(b,modfun);
ConvertToVectorRep(v,cvparam);
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:=List(Flat(a),modfun);
ConvertToVectorRep(v,cvparam);
# 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,List(Flat(a),modfun));
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,omodfun,bas)
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,modfun,matfuse;
if Length(bas)=1 then bas:=fail;fi; # no extension needed
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));
# reduction only if rational
if not (ForAll(Flat(seed),IsRat) and ForAll(Flat(gens),IsRat)) then
reducer:=M->M;
fi;
if omodfun=false then
modfun:=x->x;
else
modfun:=omodfun;
fi;
n:=Length(gens[1]);
matsplit:=l->List([1..n],x->l{[n*(x-1)+1..n*x]});
# first work modulo to get full rank
fp:=FreeGroup(Length(gens));
fpgens:=GeneratorsOfGroup(fp);
bc:=List(seed,r->List(r,modfun));
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 omodfun=false 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);
new:=Concatenation(si*List(gens[j],r->List(r,modfun))^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");
matfuse:=function(mat)
mat:=Concatenation(mat);
if bas<>fail then
mat:=Concatenation(List(mat,x->Coefficients(bas,x)));
fi;
return mat;
end;
matsplit:=function(l)
local a,i;
if bas<>fail then
a:=l;
l:=[];
for i in [1,1+Length(bas)..Length(a)+1-Length(bas)] do
Add(l,a{[i..i+Length(bas)-1]}*bas);
od;
fi;
return List([1..n],x->l{[n*(x-1)+1..n*x]});
end;
# next, form these images over Z.
#b:=ShallowCopy(seed);
b:=[matfuse(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,matfuse(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:=matfuse(si*gens[j]^k);
sol:=SolutionMat(HM,new);
if sol=fail or 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;