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##########################################################################
#0
#F FactorizationNParts
##
## Decompose a positive integer d into product of n integers
##
## Input: An pair of positive integers (d,n)
##
## Output: A list of all possible ways to decompose d into
## product of n integers
##
InstallGlobalFunction(FactorizationNParts,
function(d,n)
local
t,faclst,x,w,y,j;
if n=1 then
return [[d]];
fi;
t:=DivisorsInt(d);
faclst:=[];
for x in t do
y:=FactorizationNParts(d/x,n-1);
for j in y do
Add(faclst,AddFirst(j,x));
od;
od;
return faclst;
end);
################### end of FactorizationNParts ###########################
##########################################################################
#0
#F CrystGFullBasis
##
## Search for a G-fullbasis of a G-lattice
## for the given crystallographic G generated by
## the set of generators.
##
## Input: a crystallographic group G
##
## Output: If $G$ admits a parallelepiped fundamental region
## it outputs a $G$-full basis for a $G$-lattice $L$.
## If $G$ goes through the check for not admitting a
## parallelepiped fundamental region it outputs "false".
## Otherwise it outputs "fail".
##
##
InstallGlobalFunction(CrystGFullBasis,
function(arg)
local
G,P,Bt,d,n,vect,
i,j,a,faclst,S,gens,L,c,
B_delta,ctr,v,
x,SbGrp,T,coef,SubgroupsOfAutCube;
# Create a list of subgroup of the automorphism group of an n-cube
SubgroupsOfAutCube:=[
[ "1", "C2", "C2 x C2", "C4", "D8" ],
[ "1", "C2", "C3", "C2 x C2", "C4", "C6", "S3", "C2 x C2 x C2",
"D8", "C4 x C2", "A4", "D12", "C2 x D8", "C2 x A4", "S4", "C2 x S4" ],
[ "1", "C2", "C3", "C2 x C2", "C4", "S3", "C6", "C2 x C2 x C2", "D8",
"C4 x C2", "C8", "Q8", "D12", "A4", "C6 x C2", "(C4 x C2) : C2",
"C2 x D8", "C2 x C2 x C2 x C2", "C4 x C4", "C8 : C2", "C4 : C4",
"D16", "QD16", "C4 x C2 x C2", "C2 x A4", "S4", "SL(2,3)",
"C2 x C2 x S3", "C2 x C2 x D8", "(C4 x C2 x C2) : C2", "C4 x D8",
"((C4 x C2) : C2) : C2", "(C2 x D8) : C2", "(C2 x C2 x C2 x C2) : C2",
"(C8 : C2) : C2", "(C4 x C4) : C2", "C2 x S4", "C2 x C2 x A4",
"GL(2,3)", "((C4 x C4) : C2) : C2", "(((C4 x C2) : C2) : C2) : C2",
"D8 x D8", "((C8 : C2) : C2) : C2", "C2 x C2 x S4",
"((C2 x D8) : C2) : C3",
"(D8 x D8) : C2", "(((C2 x D8) : C2) : C3) : C2",
"((((C2 x D8) : C2) : C3) : C2) : C2" ]
];
####################end of Subgroup of Aut(cube)##########################
gens:=GeneratorsOfGroup(arg[1]);
G:=AffineCrystGroup(gens);
SbGrp:=TranslationSubGroup(G);
Bt:=G!.TranslationBasis;
n:=G!.DimensionOfMatrixGroup-1;
P:=PointGroup(G);
if not StructureDescription(P) in SubgroupsOfAutCube[n-1] then
return false;
fi;
if not GramianOfAverageScalarProductFromFiniteMatrixGroup(P)=
IdentityMat(n) then
return "Gramian matrix is not identity matrix";
fi;
if Length(arg)=1 then
L:=CrystCubicalTiling(n);
Add(L,Bt,1);
for i in [1..Length(L)] do
B_delta:=CrystGFullBasis(G,[L[i],Sum(L[i])/2]);
if IsList(B_delta) then
Add(B_delta,L[i]); #Added October 2024
return B_delta;
fi;
od;
return fail;
else #begin of length 2 input
L:=arg[2][1];
c:=arg[2][2];
vect:=Sum(L)/2-c;
S:=RightTransversal(G,SbGrp);
d:=DivisorsInt(Order(P));
i:=Length(d);
while i>0 do
faclst:=FactorizationNParts(d[i],n);
for x in faclst do
B_delta:=List([1..n],i->x[i]^-1*L[i]);
if IsCrystSufficientLattice(B_delta,S,SbGrp) then
#test if one vertex of fundamental domain is origin
ctr:=Sum(B_delta)/2-vect;
coef:=CombinationDisjointSets(x);
j:=1;
while j <= d[i] do
v:=ctr+coef[j]*B_delta;
if IsCrystSameOrbit(G,Bt,S,ctr,v)=false then
break;
fi;
j:=j+1;
od;
if j=d[i]+1 then
return [B_delta,ctr];
fi;
#test if center of fundamental domain is origin
ctr:=0*vect;
j:=1;
while j <= d[i] do
v:=ctr+coef[j]*B_delta;
if IsCrystSameOrbit(G,Bt,S,ctr,v)=false then
break;
fi;
j:=j+1;
od;
if j=d[i]+1 then
return [B_delta,ctr];
fi;
fi;
od;
i:=i-1;
od;
return fail;
fi; #end of length 2 input
end);
################### end of CrystGFullBasis ###############################
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