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# This module is part of a GAP package MAPCLASS.
# It contains the function constructing the generators
# of the braid group and pure braid group.
#
# A. James, K. Magaard, S. Shpectorov 2011
# Function producing the generators of the pure braid group
#
# BraidGroupGenerators
#
InstallGlobalFunction(BraidGroupGenerators,function(OurR, OurGamma)
local i,j,k,s,Q,Qinv,PQ,PQcopy;
# getting the generators of the braid group
Q:=[];
for i in [1..OurR-1] do
Q[i]:=[];
for j in [1..i-1] do
Q[i][j]:=OurGamma[j];
od;
Q[i][i]:=OurGamma[i]*OurGamma[i+1]*OurGamma[i]^-1;
Q[i][i+1]:=OurGamma[i];
for j in [i+2..OurR] do
Q[i][j]:=OurGamma[j];
od;
od;
# and their inverses
Qinv:=[];
for i in [1..OurR-1] do
Qinv[i]:=[];
for j in [1..i-1] do
Qinv[i][j]:=OurGamma[j];
od;
Qinv[i][i]:=OurGamma[i+1];
Qinv[i][i+1]:=OurGamma[i+1]^-1*OurGamma[i]*OurGamma[i+1];
for j in [i+2..OurR] do
Qinv[i][j]:=StructuralCopy(OurGamma[j]);
od;
od;
# finally, the pure braid group generators
PQ:=[];
for i in [1..OurR-1] do
PQ[i]:=[];
PQ[i][i]:=ShallowCopy(Q[i]);
for j in [i+1..OurR] do
PQ[i][j]:=List(OurGamma);
for k in [i..j-2] do
PQcopy:=ShallowCopy(PQ[i][j]);
for s in [1..OurR] do
PQ[i][j][s]:=MappedWord(Q[k][s],OurGamma,PQcopy);
od;
od;
PQcopy:=ShallowCopy(PQ[i][j]);
for s in [1..OurR] do
PQ[i][j][s]:=MappedWord(Qinv[j-1][s],OurGamma,PQcopy);
od;
PQcopy:=ShallowCopy(PQ[i][j]);
for s in [1..OurR] do
PQ[i][j][s]:=MappedWord(Qinv[j-1][s],OurGamma,PQcopy);
od;
for k in [i..j-2] do
PQcopy:=ShallowCopy(PQ[i][j]);
for s in [1..OurR] do
PQ[i][j][s]:=MappedWord(Qinv[j-2+i-k][s],OurGamma,PQcopy);
od;
od;
od;
od;
# we normalize the off-diagonal generators so that they don't change
# the first entry
# for i in [2..OurR] do
# for j in [1..OurR] do
# PQ[1][i][j]:=OurGamma[i]*PQ[1][i][j]*OurGamma[i]^-1;
# od;
# od;
return PQ;
end);
# End
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