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#############################################################################
##
#W ckforms.gi NoCK Package
##
## Installation file for functions of the NoCK package.
##
InstallGlobalFunction( AllZeroDH, function(arg)
local G, typeG, rankG, idRealFormG, A, rankRG, dimG, dimPG, K, dimK, rankK, pi, qi, permPi, t, Pt, j, i, List1, List2, List3,List4,ListD,deg,isOK,Coeff;
typeG:=arg[1];
rankG:=arg[2];
idRealFormG:=arg[3];
G:=RealFormById(typeG,rankG,idRealFormG);
A:=CartanSubspace(G);
rankRG:=Dimension(A);
dimG:=Dimension(G);
K:=CartanDecomposition( G ).K;
dimK:= Dimension(K);
rankK:= Dimension(CartanSubalgebra(K));
pi:=PCoefficients(typeG,rankG);
if typeG="A" then
if rankG=1 and idRealFormG=2 then
qi:=PCoefficients("A",0);
elif idRealFormG=2 then
qi:=PCoefficients("A",rankG-idRealFormG+1);
Append(qi,[1]);
elif RemInt(rankG,2)=0 then
if idRealFormG>2 and idRealFormG<=rankG/2+1 then
qi:=PCoefficients("A",idRealFormG-2);
Append(qi,PCoefficients("A",rankG-idRealFormG+1));
Append(qi,[1]);
elif idRealFormG=rankG/2+2 then
qi:=PCoefficients("B",rankG/2);
fi;
else
if idRealFormG>2 and idRealFormG<=(rankG+3)/2 then
qi:=PCoefficients("A",idRealFormG-2);
Append(qi,PCoefficients("A",rankG-idRealFormG+1));
Append(qi,[1]);
elif idRealFormG=(rankG+5)/2 then
qi:=PCoefficients("C",(rankG+1)/2);
elif idRealFormG=(rankG+7)/2 then
qi:=PCoefficients("D",(rankG+1)/2);
fi;
fi;
elif typeG="B" then
qi:=PCoefficients("D",idRealFormG-1);
if idRealFormG<=rankG then
Append(qi,PCoefficients("B",rankG-idRealFormG+1));
fi;
elif typeG="C" then
if RemInt(rankG,2)=0 then
if idRealFormG>=2 and idRealFormG<=rankG/2+1 then
qi:=PCoefficients("C",idRealFormG-1);
Append(qi,PCoefficients("C",rankG-idRealFormG+1));
elif idRealFormG=rankG/2+2 then
qi:=PCoefficients("A",rankG-1);
Append(qi,[1]);
fi;
else
if idRealFormG>=2 and idRealFormG<=(rankG+1)/2 then
qi:=PCoefficients("C",idRealFormG-1);
Append(qi,PCoefficients("C",rankG-idRealFormG+1));
elif idRealFormG=(rankG+3)/2 then
qi:=PCoefficients("A",rankG-1);
Append(qi,[1]);
fi;
fi;
elif typeG="D" then
if rankG=4 then
if idRealFormG=2 then
qi:=PCoefficients("A",3);
Append(qi,[1]);
elif idRealFormG=3 then
qi:=PCoefficients("D",2);
Append(qi,PCoefficients("D",2));
elif idRealFormG=4 then
qi:=PCoefficients("B",1);
Append(qi,PCoefficients("B",2));
elif idRealFormG=5 then
qi:=PCoefficients("B",3);
fi;
elif RemInt(rankG,2)=0 then
if idRealFormG>=2 and idRealFormG<=rankG/2+1 then
qi:=PCoefficients("D",idRealFormG-1);
Append(qi,PCoefficients("D",rankG+1-idRealFormG));
elif idRealFormG=rankG/2+2 then
qi:=PCoefficients("A",rankG-1);
Append(qi,[1]);
elif idRealFormG=rankG/2+3 then
qi:=PCoefficients("B",rankG-1);
elif idRealFormG>=rankG/2+4 and idRealFormG<=rankG+2 then
qi:=PCoefficients("B",idRealFormG-rankG/2-3);
Append(qi,PCoefficients("B",3*rankG/2+2-idRealFormG));
fi;
else
if idRealFormG>=2 and idRealFormG<=(rankG+1)/2 then
qi:=PCoefficients("D",idRealFormG-1);
Append(qi,PCoefficients("D",rankG+1-idRealFormG));
elif idRealFormG=(rankG+3)/2 then
qi:=PCoefficients("A",rankG-1);
Append(qi,[1]);
elif idRealFormG=(rankG+5)/2 then
qi:=PCoefficients("B",rankG-1);
elif idRealFormG>=(rankG+7)/2 and idRealFormG<=rankG+2 then
qi:=PCoefficients("B",idRealFormG-(rankG-1)/2-3);
Append(qi,PCoefficients("B",3*(rankG-1)/2+3-idRealFormG));
fi;
fi;
elif typeG="E" then
if rankG=6 then
if idRealFormG=2 then
qi:=PCoefficients("C",4);
elif idRealFormG=3 then
qi:=PCoefficients("A",1);
Append(qi,PCoefficients("A",5));
elif idRealFormG=4 then
qi:=PCoefficients("D",5);
Append(qi,[1]);
elif idRealFormG=5 then
qi:=PCoefficients("F",4);
fi;
elif rankG=7 then
if idRealFormG=2 then
qi:=PCoefficients("A",7);
elif idRealFormG=3 then
qi:=PCoefficients("E",6);
Append(qi,[1]);
elif idRealFormG=4 then
qi:=PCoefficients("A",1);
Append(qi,PCoefficients("D",6));
fi;
elif rankG=8 then
if idRealFormG=2 then
qi:=PCoefficients("D",8);
elif idRealFormG=3 then
qi:=PCoefficients("A",1);
Append(qi,PCoefficients("E",7));
fi;
fi;
elif typeG="F" then
if idRealFormG=2 then
qi:=PCoefficients("A",1);
Append(qi,PCoefficients("C",3));
elif idRealFormG=3 then
qi:=PCoefficients("B",4);
fi;
elif typeG="G" then
qi:=PCoefficients("B",1);
Append(qi,PCoefficients("B",1));
fi;
permPi:=List(SymmetricGroup(rankG), p -> Permuted(pi, p));
t:=Indeterminate(Rationals,"t");
List1:=List(permPi, i -> PCalculate(i,qi));
List2:=List(List1, i -> IsPolynomial(i));
List3:=[];
for i in [1..Length(List1)] do
if List2[i]=true then
Add(List3,List1[i]);
fi;
od;
deg:=DegreeOfLaurentPolynomial(List3[1]);
ListD:=[];
isOK:=true;
for i in [1..deg] do
isOK:=true;
for j in [1..Length(List3)] do
Coeff:=CoefficientsOfUnivariatePolynomial(List3[j]);
if Coeff[i+1] <> 0 then
isOK:=false;
fi;
od;
if isOK=true then
Add(ListD,i);
fi;
od;
return ListD;
end);
###############################################################################
InstallGlobalFunction( PCoefficients, function(arg)
local typeG, rankG, pi, i;
typeG:=arg[1];
rankG:=arg[2];
if typeG = "A" then
pi:=[0];
for i in [1..rankG] do
pi[i]:= i+1;
od;
if rankG=0 then
pi:=[1];
fi;
return pi;
elif typeG = "B" then
pi:=[0];
for i in [1..rankG] do
pi[i]:= 2*i;
od;
return pi;
elif typeG = "C" then
pi:=[0];
for i in [1..rankG] do
pi[i]:= 2*i;
od;
return pi;
elif typeG = "D" then
pi:=[0];
for i in [1..rankG-1] do
pi[i]:= 2*i;
od;
pi[rankG]:=rankG;
return pi;
elif typeG = "E" then
if rankG=6 then
return [2, 5, 6, 8, 9, 12];
elif rankG=7 then
return [2, 6, 8, 10, 12, 14, 18];
elif rankG=8 then
return [2, 8, 12, 14, 18, 20, 24, 30];
fi;
elif typeG = "F" and rankG=4 then
return [2, 6, 8, 12];
elif typeG = "G" and rankG=2 then
return [2, 6];
fi;
end);
###############################################################################
InstallGlobalFunction( PCalculate, function(arg)
local pi,qi, Pt , t, i, j;
t:=Indeterminate(Rationals,"t");
Pt:=1;
pi:=arg[1];
qi:=arg[2];
for j in [Length(qi)+1..Length(pi)] do
Pt:=Pt*(1+t^(2*pi[j]-1));
od;
for i in [1..Length(qi)] do
Pt:=Pt*(1-t^(2*pi[i]))/(1-t^(2*qi[i]));
od;
return Pt;
end);
###############################################################################
InstallGlobalFunction( NonCompactDimension, function(arg)
local G, dimG, K, dimK, dG;
G:=arg[1];
dimG:=Dimension(G);
K:=CartanDecomposition( G ).K;
dimK:= Dimension(K);
dG:=dimG-dimK;
return dG;
end);
#E ckforms.gi . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here
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