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#######################################################################
#0
#F Counting cells in a cell complex formatted in the Gcomplex datatype
## Input: A cell complex formatted in Gcomplex datatype
##
## Output: The structure in which we can extract the list of k-cells
## or the number of k-cells
##
InstallGlobalFunction(CountingCellsOfACellComplex,
function(C)
local Cells,N,i,Dims,pos,j,x,y,w,id,t,k,ck,c,s,a,v,g,b,
Elts,Rep,mult,ListUnion, AddReturn,NCells,
Orbit, nrCells,R, PWsbgrp;
Elts:=C!.elts;
##################################################################
# If g in Elts return the position of g in the list,
# otherwise, add g to Elts and return the position.
pos:=function(g)
local posit;
posit:=Position(Elts,g);
if posit=fail then
Add(Elts,g);
return Length(Elts);
else
return posit;
fi;
end;
##################################################################
# returns a "canonical" representative of the right coset
# Elts[g]*Stab[i+1][j]
Rep:=function(i,j,g)
return pos(CanonicalRightCountableCosetElement
(C!.stabilizer(i,j),Elts[g]^-1)^-1);
end;
##################################################################
# AddReturn:=function(a,g)
# local b;
# b:=StructuralCopy(a);
# Add(b,g);
# return b;
# end;
##################################################################
mult:=function(L,g)
return List(L,a->[a[1],pos(Elts[g]*Elts[a[2]])]);
end;
##################################################################
ListUnion:=function(x,y)
local a;
for a in y do
if not a in x then
Add(x,a);
fi;
od;
end;
##################################################################
Dims:=[];
for i in [0..Length(C)] do
if C!.dimension(i)=0 then N:=i-1; break; fi;
Dims[i+1]:=C!.dimension(i);
od;
Cells:=[];
id:=pos(One(C!.group));
for i in [1..N+1] do
Cells[i]:=[];
od;
if IsBound(C!.Partition) and (not C!.Partition=fail) then
Cells[N+1]:=StructuralCopy(C!.Partition);
else
for j in [1..Dims[N+1]] do
Add(Cells[N+1],[j,id]);
od;
fi;
# Construct the list of cells and the corresponding coboundary of those cells
i:=N;
while i>0 do
for k in [1..Length(Cells[i+1])] do
x:=Cells[i+1][k];
w:=StructuralCopy(C!.boundary(i,AbsInt(x[1])));
w:=mult(w,x[2]);
w:=List(w,a->[AbsInt(a[1]),Rep(i-1,AbsInt(a[1]),a[2])]);
ListUnion(Cells[i],w);
od;
i:=i-1;
od;
##################################################################
nrCells:=List([1..N+1],i->Length(Cells[i]));
##################################################################
return nrCells;
end);
################### end of ControlledSubdivision ############################
InstallGlobalFunction(CountingControlledSubdividedCells,
function(C)
local Cells,N,i,Dims,pos,j,x,y,w,id,t,k,ck,c,s,a,v,g,b,L,intst,
Elts,Rep,mult,ListUnion, AddReturn,NCells,d,PWsbgrp,bdry,
Orbit, nrCells,R;
Elts:=C!.elts;
##################################################################
# If g in Elts return the position of g in the list,
# otherwise, add g to Elts and return the position.
pos:=function(g)
local posit;
posit:=Position(Elts,g);
if posit=fail then
Add(Elts,g);
return Length(Elts);
else
return posit;
fi;
end;
##################################################################
# returns a "canonical" representative of the right coset
# Elts[g]*Stab[i+1][j]
Rep:=function(i,j,g)
return pos(CanonicalRightCountableCosetElement
(C!.stabilizer(i,j),Elts[g]^-1)^-1);
end;
##################################################################
# AddReturn:=function(a,g)
# local b;
# b:=StructuralCopy(a);
# Add(b,g);
# return b;
# end;
##################################################################
mult:=function(L,g)
return List(L,a->[a[1],pos(Elts[g]*Elts[a[2]])]);
end;
##################################################################
ListUnion:=function(x,y)
local a;
for a in y do
if not a in x then
Add(x,a);
fi;
od;
end;
##################################################################
Dims:=[];
for i in [0..Length(C)] do
if C!.dimension(i)=0 then N:=i-1; break; fi;
Dims[i+1]:=C!.dimension(i);
od;
Cells:=[];
id:=pos(One(C!.group));
for i in [1..N+1] do
Cells[i]:=[];
od;
if IsBound(C!.Partition) then
Cells[N+1]:=StructuralCopy(C!.Partition);
else
for j in [1..Dims[N+1]] do
Add(Cells[N+1],[j,id]);
od;
fi;
# Construct the list of cells and the corresponding coboundary of those cells
i:=N;
while i>0 do
for k in [1..Length(Cells[i+1])] do
x:=Cells[i+1][k];
w:=StructuralCopy(C!.boundary(i,AbsInt(x[1])));
w:=mult(w,x[2]);
w:=List(w,a->[AbsInt(a[1]),Rep(i-1,AbsInt(a[1]),a[2])]);
ListUnion(Cells[i],w);
od;
i:=i-1;
od;
##################################################################
PWsbgrp:=[];
for i in [1..N] do
PWsbgrp[i]:=[];
for j in [1..Dims[i+1]] do
bdry:=C!.boundary(i,j);
L:=List(bdry,w->Elements(ConjugateGroup(C!.stabilizer(i-1,AbsInt(w[1])),Elts[w[2]]^-1)));
intst:=Intersection(L);
Add(PWsbgrp[i],Size(C!.stabilizer(i,j))/Size(intst));
od;
od;
d:=[];
for i in [1..N] do
d[i]:=0;
for x in Cells[i+1] do
d[i]:=d[i]+PWsbgrp[i][AbsInt(x[1])];
od;
od;
return d;
end);
#######################################################################################
InstallGlobalFunction(CountingBaryCentricSubdividedCells,
function(C)
local Cells,N,i,Dims,pos,j,x,y,w,id,t,k,ck,c,s,a,v,g,b,L,intst,
Elts,Rep,mult,ListUnion, AddReturn,NCells,d,nrCells,bdry,
Orbit, R;
Elts:=C!.elts;
##################################################################
# If g in Elts return the position of g in the list,
# otherwise, add g to Elts and return the position.
pos:=function(g)
local posit;
posit:=Position(Elts,g);
if posit=fail then
Add(Elts,g);
return Length(Elts);
else
return posit;
fi;
end;
##################################################################
# returns a "canonical" representative of the right coset
# Elts[g]*Stab[i+1][j]
Rep:=function(i,j,g)
return pos(CanonicalRightCountableCosetElement
(C!.stabilizer(i,j),Elts[g]^-1)^-1);
end;
##################################################################
# AddReturn:=function(a,g)
# local b;
# b:=StructuralCopy(a);
# Add(b,g);
# return b;
# end;
##################################################################
mult:=function(L,g)
return List(L,a->[a[1],pos(Elts[g]*Elts[a[2]])]);
end;
##################################################################
ListUnion:=function(x,y)
local a;
for a in y do
if not a in x then
Add(x,a);
fi;
od;
end;
##################################################################
Dims:=[];
for i in [0..Length(C)] do
if C!.dimension(i)=0 then N:=i-1; break; fi;
Dims[i+1]:=C!.dimension(i);
od;
Cells:=[];
id:=pos(One(C!.group));
for i in [1..N+1] do
Cells[i]:=[];
od;
if IsBound(C!.Partition) then
Cells[N+1]:=StructuralCopy(C!.Partition);
else
for j in [1..Dims[N+1]] do
Add(Cells[N+1],[j,id]);
od;
fi;
# Construct the list of cells and the corresponding coboundary of those cells
i:=N;
while i>0 do
for k in [1..Length(Cells[i+1])] do
x:=Cells[i+1][k];
w:=StructuralCopy(C!.boundary(i,AbsInt(x[1])));
w:=mult(w,x[2]);
w:=List(w,a->[AbsInt(a[1]),Rep(i-1,AbsInt(a[1]),a[2])]);
ListUnion(Cells[i],w);
od;
i:=i-1;
od;
##################################################################
nrCells:=[];
for i in [1..N] do
nrCells[i]:=[];
for j in [1..Dims[i+1]] do
if i=1 then nrCells[i][j]:=2;
else
nrCells[i][j]:=0;
bdry:=C!.boundary(i,j);
for x in bdry do
nrCells[i][j]:=nrCells[i][j]+nrCells[i-1][AbsInt(x[1])];
od;
fi;
od;
od;
d:=[];
for i in [1..N] do
d[i]:=0;
for x in Cells[i+1] do
d[i]:=d[i]+nrCells[i][AbsInt(x[1])];
od;
od;
return d;
end);
###########################################################################
DeclareGlobalFunction("CountingNumberOfCellsInBaryCentricSubdivision");
InstallGlobalFunction(CountingNumberOfCellsInBaryCentricSubdivision,
function(C)
local Cells,N,i,Dims,pos,j,x,y,w,id,t,k,ck,c,s,a,v,g,b,L,intst,d,nr,
Elts,Rep,mult,ListUnion, AddReturn,NCells,nrCells,bdry,nrRec,bdryRec,
Orbit, R, Mat, Chains, A, nrChains, originalChains, fn, newChains, AdjMat;
Elts:=C!.elts;
##################################################################
# If g in Elts return the position of g in the list,
# otherwise, add g to Elts and return the position.
pos:=function(g)
local posit;
posit:=Position(Elts,g);
if posit=fail then
Add(Elts,g);
return Length(Elts);
else
return posit;
fi;
end;
##################################################################
# returns a "canonical" representative of the right coset
# Elts[g]*Stab[i+1][j]
Rep:=function(i,j,g)
return pos(CanonicalRightCountableCosetElement
(C!.stabilizer(i,j),Elts[g]^-1)^-1);
end;
##################################################################
# AddReturn:=function(a,g)
# local b;
# b:=StructuralCopy(a);
# Add(b,g);
# return b;
# end;
##################################################################
mult:=function(L,g)
return List(L,a->[a[1],pos(Elts[g]*Elts[a[2]])]);
end;
##################################################################
ListUnion:=function(x,y)
local a,p;
p:=[];
for a in y do
if not a in x then
Add(x,a);
fi;
Add(p,Position(x,a));
od;
return p;
end;
##################################################################
Dims:=[];
for i in [0..Length(C)] do
if C!.dimension(i)=0 then N:=i-1; break; fi;
Dims[i+1]:=C!.dimension(i);
od;
Cells:=[];
id:=pos(One(C!.group));
for i in [1..N+1] do
Cells[i]:=[];
od;
if IsBound(C!.Partition) then
Cells[N+1]:=StructuralCopy(C!.Partition);
else
for j in [1..Dims[N+1]] do
Add(Cells[N+1],[j,id]);
od;
fi;
# Construct the list of cells and the corresponding coboundary of those cells
i:=N;
bdryRec:=[];
while i>0 do
bdryRec[i]:=[];
for k in [1..Length(Cells[i+1])] do
x:=Cells[i+1][k];
w:=StructuralCopy(C!.boundary(i,AbsInt(x[1])));
w:=mult(w,x[2]);
w:=List(w,a->[AbsInt(a[1]),Rep(i-1,AbsInt(a[1]),a[2])]);
#Print("Cells[i]=",[i,Cells[i]],"\n");
#Print("w=",w,"\n");
bdryRec[i][k]:=ListUnion(Cells[i],w);
#Print("bdryRec[i+1][k]=",[i,bdryRec[i+1][k]],"\n");
od;
i:=i-1;
od;
# Construct the associated matrices for each level
Mat:=[];
for i in [1..N] do
Mat[i]:=[];
for j in [1..Length(Cells[N-i+2])] do
Mat[i][j]:=[];
for k in [1..Length(Cells[N-i+1])] do
if k in bdryRec[N-i+1][j] then Mat[i][j][k]:=1;
else Mat[i][j][k]:=0;
fi;
od;
od;
od;
##################################################################
Chains:=[];
originalChains:=[];
for i in [1..N] do
originalChains[i]:=[];
A:=Mat[i];
Add(originalChains[i],A);
for j in [i+1..N] do
A:=A*Mat[j];
Add(originalChains[i],A);
od;
# nrChains[i]:=List(Chains[i],a->Sum(Sum(a)));
od;
fn:=function(A)
local i,j,B;
B:=[];
for i in [1..Length(A)] do
B[i]:=[];
for j in [1..Length(A[i])] do
if A[i][j]=0 then B[i][j]:=0;
else B[i][j]:=1;
fi;
od;
od;
return B;
end;
newChains:=[];
for i in [1..N] do
newChains[i]:=[];
for j in [1..Length(originalChains[i])] do
newChains[i][j]:=fn(originalChains[i][j]);
od;
od;
# Construct the adjacency matrix
d:=Reversed(List(Cells,a->Length(a)));
Add(d,0,1);
nr:=Sum(d);
AdjMat:=NullMat(nr,nr);
for i in [1..N+1] do
for j in [1..N+1] do
if i>=j then
AdjMat{[Sum(d{[1..i]})+1..Sum(d{[1..i+1]})]}{[Sum(d{[1..j]})+1..Sum(d{[1..j+1]})]}:=NullMat(d[i+1],d[j+1]);
else
AdjMat{[Sum(d{[1..i]})+1..Sum(d{[1..i+1]})]}{[Sum(d{[1..j]})+1..Sum(d{[1..j+1]})]}:=newChains[i][j-i];
fi;
od;
od;
A:=StructuralCopy(AdjMat);
nrChains:=[];
Add(nrChains,Sum(d));
Add(nrChains,Sum(Sum(A)));
for i in [1..N-1] do
A:=A*AdjMat;
Add(nrChains,Sum(Sum(A)));
od;
##################################################################
return nrChains;
end);
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