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#(C) Graham Ellis, 2005-2006
#####################################################################
InstallGlobalFunction(ResolutionArtinGroup_spherical,
function(D,K)
local
#We asume that D defines a sperical Artin group G and then, from the
#point of view of computations, treat G as a Coxeter group.
Dimension,
Boundary,
Contraction, #not yet used
EltsG, EltsG1,
Vertices,
G, gensG, Glist, G1, M, #G is the Artin group of D. We treat G as a
GhomG1, gensG1, gensM, #free group. However, we output a copy G1 of
#G with relators.
W, Wgens, GhomW, #W is the Coxeter group of D
ResGens,
BoundaryCoeff,
PseudoBoundary,
BoundaryRecord,
m, n, i, S, SD,R, iso, epi, x, y, F, gensF;
if not CoxeterDiagramIsSpherical(D) then return fail; fi;
Vertices:=CoxeterDiagramVertices(D);
Glist:=CoxeterDiagramFpArtinGroup(D);
G1:=Glist[1]/Glist[2]; #Take care for this not to cause Knuth-Bendix
gensG1:=GeneratorsOfGroup(G1); #to start up later on!
M:=CoxeterDiagramMatCoxeterGroup(D);
gensM:=GeneratorsOfGroup(M);
iso:=IsomorphismPermGroup(M);
gensG:=List(gensM,x->Image(iso,x));
G:=Group(gensG);
#G:=Image(iso);
F:=Glist[1];
gensF:=GeneratorsOfGroup(F);
epi:=GroupHomomorphismByImages(F,G,gensF,gensG);
GhomG1:=GroupHomomorphismByImagesNC(G,G1,gensG,gensG1);
EltsG:=Enumerator(G);;
ResGens:=[];
ResGens[1]:=[[]];
for n in [1..K] do
ResGens[n+1]:=[];
for S in Combinations(Vertices,n) do
SD:=CoxeterSubDiagram(D,S);
AddSet(ResGens[n+1],S);
od;
od;
#####################################################################
Dimension:=function(n)
if n=0 then return 1;
else return Length(ResGens[n+1]); fi;
end;
#####################################################################
BoundaryRecord:=[];
for n in [1..K] do
BoundaryRecord[n]:=[];
for m in [1..Dimension(n)] do
BoundaryRecord[n][m]:=true;
od;
od;
#####################################################################
PseudoBoundary:=function(S) #S is a subset of vertices with finite
#Coxeter group WS.
local T, bndry, a;
bndry:=[];
for T in Combinations(S,Length(S)-1) do
a:=Difference(S,T)[1];
Append(bndry,[ [T,BoundaryCoeff(S,T),Position(S,a)] ]);
od;
return bndry;
end;
#####################################################################
####################################################################
BoundaryCoeff:=function(S,T) #S is a set of vertices generating a
#finite Coxeter group WS. T is a
#subset of S, and WT is the corresponding
#subgroup of WS.
local SD, WS, gensWS,
WT, gensWT,
Trans,
WShomG1, WShomG, Ggens, G1gens,
x,y;
SD:=CoxeterSubDiagram(D,S);
WS:=CoxeterDiagramFpCoxeterGroup(SD);
WS:=WS[1]/WS[2];
G1gens:=List(S,x->gensG1[Position(Vertices,x)]);
gensWS:=GeneratorsOfGroup(WS);
WShomG1:=GroupHomomorphismByImagesNC(WS,G1,gensWS,G1gens);
Ggens:=List(S,x->gensG[Position(Vertices,x)]);
WShomG:=GroupHomomorphismByImagesNC(WS,G,gensWS,Ggens);
gensWT:=List(T,x->gensWS[Position(S,x)]);
if Length(T)>0 then WT:=Group(gensWT);
else WT:=Group(Identity(WS)); fi;
#Trans:=List(Elements(RightTransversal(WS,WT)),x->x^-1);
Trans:=RightTransversal(WS,WT);
return List(Trans,x->Image(WShomG,x^-1));
end;
#####################################################################
#####################################################################
Boundary:=function(n,kk)
local B, B1, FreeGWord, x, y, k;
#n:=AbsoluteValue(m);
if n<1 then return 0; fi;
k:=AbsoluteValue(kk);
if not BoundaryRecord[n][k]=true then
if kk>0 then return BoundaryRecord[n][k];
else return NegateWord(BoundaryRecord[n][k]);fi;
fi;
B:=PseudoBoundary(ResGens[n+1][k]);
B1:=List(B,x->[Position(ResGens[n],x[1]),
List(x[2],y->(-1)^(Length(Image(GhomG1,y))+x[3])*Position(EltsG,y)) ]);
FreeGWord:=[];
for x in B1 do
for y in x[2] do
Append(FreeGWord,[ [SignInt(y)*x[1],AbsoluteValue(y)] ]);
od;
od;
BoundaryRecord[n][k]:=FreeGWord;
if kk>0 then return FreeGWord;
else return NegateWord(FreeGWord); fi;
end;
#####################################################################
EltsG1:=[];
R:= Objectify(HapResolution,
rec(
dimension:=Dimension,
boundary:=Boundary,
homotopy:=fail,
elts:=EltsG1,
group:=G1,
resGens:=ResGens,
properties:=
[["length",n],
["characteristic",0],
["type","resolution"],
["reduced",true]] ));
for n in [1..Length(R)] do
for i in [1..R!.dimension(n)] do
x:=R!.boundary(n,i);
for y in x do
EltsG1[y[2]]:= Image(GhomG1,EltsG[y[2]]) ;
od;
od;od;
R!.elts:=EltsG1;
return R;
end);
#####################################################################
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