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###############################################
###############################################
InstallGlobalFunction(RadicalSeriesOfResolution,
function(R)
local G, ordG, FG, FGdims, Tdims, T,L,prime,
Dimension, FDimension, Boundary, BoundRec, IntToPair, one,
PairToInt, cnt, Mult, invT, n, i,g, j, k, r,
WordToVectorList, VectorListToWord ;
G:=R!.group;
ordG:=Order(G);
prime:=Factors(ordG)[1];
###################
###################
if not (IsPrime(EvaluateProperty(R,"characteristic"))
and
IsPrimePowerInt(ordG)) then
Print("The resolution must have prime characteristic for a prime-power group\n");
return fail;
fi;
###################
###################
one:=One(GF(prime));
###################
Mult:=function(g,h)
return Position(R!.elts,R!.elts[g]*R!.elts[h]);
end;
###################
###################
FG:=GroupAlgebraAsFpGModule(G);
L:=RadicalSeriesOfFpGModule(FG);;
L:=L{[1..Length(L)-1]};
L:=Reversed(L);
T:=List(L,x->GeneratorsOfFpGModule(x));
FGdims:=List(T,Length);
Tdims:=1*FGdims;
for i in [2..Length(FGdims)] do
FGdims[i]:=FGdims[i]+FGdims[i-1];
od;
T:=Concatenation(T); ##So T is the new basis
invT:=T^-1;
###################
###################
Dimension:=function(n)
if n>Length(R) then return 0; fi;
return ordG*R!.dimension(n);
end;
###################
###################
FDimension:=function(k,n)
if n>Length(R) then return 0; fi;
return FGdims[k]*R!.dimension(n);
end;
###################
##################################
IntToPair:=List([0..Length(R)],i->[]);
PairToInt:=List([0..Length(R)],i->List([1..R!.dimension(i)], j->[]));
# so i --> [r,g] where r is the FG summand and g is the (new basis) position
# in summand
for n in [0..Length(R)] do
cnt:=0;
for j in [1..Length(Tdims)] do
for g in [1..Tdims[j]] do
for r in [1..R!.dimension(n)] do
cnt:=cnt+1;
IntToPair[n+1][cnt]:=[r,g];
if j>1 then IntToPair[n+1][cnt][2]:=IntToPair[n+1][cnt][2]+FGdims[j-1]; fi;
PairToInt[n+1][IntToPair[n+1][cnt][1]][IntToPair[n+1][cnt][2]]:=cnt;
od;
od;
od;
od;
################################
#####################################################################
WordToVectorList:=function(w,k) #w is a FG-word in R_k.
local v,x,r; #v is a list of vectors mod p.
v:=List([1..R!.dimension(k)],i->List([1..ordG],j->0*one) );
for x in w do
r:=AbsInt(x[1]);
v[r][x[2]]:=v[r][x[2]] + SignInt(x[1])*one;
od;
return v ;
end;
#####################################################################
#####################################################################
VectorListToWord:=function(v,n) #returns a sparse word over F.
local w, r, g, vv;
w:=[];
for r in [1..Length(v)] do
for g in [1..Length(v[r])] do
if not IsZero(v[r][g]) then Add(w,[PairToInt[n+1][r][g],v[r][g]]); fi;
od;
od;
return w;
end;
#####################################################################
BoundRec:=List([1..Length(R)],i->[]);;
###################################
Boundary:=function(n,k)
local w, x,b, bnd, pr, gg,i;
if IsBound(BoundRec[n][AbsInt(k)]) then
if SignInt(k)>0 then return 1*BoundRec[n][k];
else
return NegateWord(BoundRec[n][k]);
fi;
fi;
pr:=IntToPair[n+1][AbsInt(k)];
r:=pr[1];
g:=pr[2];
w:=T[g];
gg:=[];
for i in [1..Length(w)] do
for j in [1..IntFFE(w[i])] do
Add(gg,i);
od;
od;
w:=R!.boundary(n,r); #This is an FG-word
bnd:=[];
for i in gg do
b:=List(w, x->[x[1],Mult(i,x[2])]);
Append(bnd,b);
od;
bnd:=WordToVectorList(bnd,n-1);
for i in [1..Length(bnd)] do
if not IsZero(bnd[i]) then
bnd[i]:=bnd[i]*invT;
fi;
od;
bnd:=VectorListToWord(bnd,n-1);
BoundRec[n][AbsInt(k)]:=bnd;
if SignInt(k)=1 then return 1*bnd;
else return NegateWord(bnd); fi;
end;
###################
return
Objectify(HapFilteredSparseChainComplex,
rec(
dimension:=Dimension,
boundary:=Boundary,
filteredDimension:=FDimension,
properties:=[
["length",Length(R)],
["filtration_length",Length(FGdims)],
["type","FilteredChainComplex"],
["characteristic",prime]]
));
end);
###############################################
###############################################
|
