|
# This module is part of a GAP package MAPCLASS.
# It contains the function constructing
# the generators of the mapping class group.
#
# A. James, K. Magaard, S.Shpectorov 2011
#
# MappingClassGroupGenerators
# Function producing the generators of the mapping class group
#
InstallGlobalFunction(MappingClassGroupGenerators,function(OurG, OurR, AbsGens,
OurAlpha, OurBeta, OurGamma, p)
local i,j,y,s,z, OurAction;
OurAction:=[];
if OurR>1 then
# transforming the tuple partition...
y:=[p[1]];
for i in [2..Length(p)] do
y[i]:=y[i-1]+p[i];
od;
# getting the generators of the braid group
z:=BraidGroupGenerators(OurR,OurGamma);
# selecting the braid generators preserving the partition
for i in [1..OurR-1] do
for j in [i..OurR] do
if (i=j and not (i in y)) or
(i<j and (i in y) and (j-1 in y))
then
Append(OurAction,[Concatenation(AbsGens{[1..2*OurG]},z[i][j])]);
fi;
od;
od;
fi;
#
# getting the Dehn twists
# a's
z:=[];
y:=[];
if OurG>0 then
z[1]:=OurAlpha[1]^-1*OurBeta[1]^-1*OurAlpha[1];
y[1]:=z[1];
for i in [OurR,OurR-1..1] do
y[1]:=y[1]^(OurGamma[i]^-1);
od;
for i in [1..OurR] do
y[i+1]:=y[i]*OurGamma[i];
z[i+1]:=y[i+1];
for j in [i+1..OurR] do
z[i+1]:=z[i+1]^OurGamma[j];
od;
od;
for i in [1..OurR+1] do
s:=ShallowCopy(AbsGens);
s[1]:=OurAlpha[1]*z[i];
for j in [1..i-1] do
s[2*OurG+j]:=OurGamma[j]^y[i];
od;
Append(OurAction,[s]);
od;
fi;
# b's
if OurG>0 then
for i in [1..OurG] do
s:=ShallowCopy(AbsGens);
s[OurG+i]:=OurAlpha[i]^-1*OurBeta[i];
Append(OurAction,[s]);
od;
fi;
# c
if OurG>1 then
s:=ShallowCopy(AbsGens);
s[2]:=OurBeta[2]^-1*OurAlpha[2];
Append(OurAction,[s]);
fi;
# d's
if OurG>1 then
for i in [1..OurG-1] do
s:=ShallowCopy(AbsGens);
s[i]:=OurBeta[i]*OurAlpha[i+1]^-1*OurBeta[i+1]^-1*
OurAlpha[i+1]*OurAlpha[i];
s[i+1]:=OurBeta[i+1]*OurAlpha[i+1]*OurBeta[i]^-1;
s[OurG+i]:=OurBeta[i]^(OurAlpha[i+1]^-1*OurBeta[i+1]*
OurAlpha[i+1]*OurBeta[i]^-1);
Append(OurAction,[s]);
od;
fi;
return OurAction;
end);
InstallGlobalFunction(MappingClassGroupGeneratorsL,function(OurG, OurR, AbsGens,
OurAlpha, OurBeta, OurGamma, p)
local i,j,y,s,z, OurAction;
OurAction:=[];
if OurR>1 then
# transforming the tuple partition...
y:=[p[1]];
for i in [2..Length(p)] do
y[i]:=y[i-1]+p[i];
od;
# getting the generators of the braid group
z:=BraidGroupGenerators(OurR,OurGamma);
# selecting the braid generators preserving the partition
# KEY here is that i starts at 2 so the first element is not altered at all
for i in [1..OurR-1] do
for j in [i..OurR] do
if (i=j and not (i in y)) or
(i<j and (i in y) and (j-1 in y))
then
if OurG = 0 then
if z[i][j][1] = AbsGens[1] then
Append(OurAction,[Concatenation(AbsGens{[1..2*OurG]},z[i][j])]);
fi;
fi;
fi;
od;
od;
fi;
#
# getting the Dehn twists
# a's
z:=[];
y:=[];
if OurG>0 then
z[1]:=OurAlpha[1]^-1*OurBeta[1]^-1*OurAlpha[1];
y[1]:=z[1];
for i in [OurR,OurR-1..1] do
y[1]:=y[1]^(OurGamma[i]^-1);
od;
for i in [1..OurR] do
y[i+1]:=y[i]*OurGamma[i];
z[i+1]:=y[i+1];
for j in [i+1..OurR] do
z[i+1]:=z[i+1]^OurGamma[j];
od;
od;
for i in [1..OurR+1] do
s:=ShallowCopy(AbsGens);
s[1]:=OurAlpha[1]*z[i];
for j in [1..i-1] do
s[2*OurG+j]:=OurGamma[j]^y[i];
od;
Append(OurAction,[s]);
od;
fi;
# b's
if OurG>0 then
for i in [1..OurG] do
s:=ShallowCopy(AbsGens);
s[OurG+i]:=OurAlpha[i]^-1*OurBeta[i];
Append(OurAction,[s]);
od;
fi;
# c
if OurG>1 then
s:=ShallowCopy(AbsGens);
s[2]:=OurBeta[2]^-1*OurAlpha[2];
Append(OurAction,[s]);
fi;
# d's
if OurG>1 then
for i in [1..OurG-1] do
s:=ShallowCopy(AbsGens);
s[i]:=OurBeta[i]*OurAlpha[i+1]^-1*OurBeta[i+1]^-1*
OurAlpha[i+1]*OurAlpha[i];
s[i+1]:=OurBeta[i+1]*OurAlpha[i+1]*OurBeta[i]^-1;
s[OurG+i]:=OurBeta[i]^(OurAlpha[i+1]^-1*OurBeta[i+1]*
OurAlpha[i+1]*OurBeta[i]^-1);
Append(OurAction,[s]);
od;
fi;
return OurAction;
end);
# End
[ Dauer der Verarbeitung: 0.4 Sekunden
(vorverarbeitet)
]
|
