%off echo,nat$
off echo$
out "rreducparautommodg6_6N5.r"$
write "rreducparautommodg6_6N5.r"$
operator b$



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generic automorphism of g_{6,6} as in (art ijac1 section7) :
phi:=mat((b(1,1),b(1,2),0,0,0,0),
         (0,b(2,2),0,0,0,0),
         (b(3,1),b(3,2),b(3,3),0,0,0),
         (b(4,1),b(4,2),b(4,3),b(2,2)*b(1,1),0,0),
         (b(5,1),b(5,2),b(5,3), -b(4,1)*b(2,2), b(2,2)**2*b(1,1), b(4,3)*b(2,2)),
         (b(6,1),b(6,2),b(6,3), -b(3,1)*b(2,2), 0, b(3,3)*b(2,2)))$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generic nilpotent derivation as in (Cohomology tables page 116) : the eigenvalues
% are 0
operator xi$
delta:=
mat((0,xi(1,2),0,0,0,0),
    (0,0,0,0,0,0),
    (xi(3,1),xi(3,2),0,0,0,0),
    (xi(4,1),xi(4,2),xi(4,3),0,0,0),
    (xi(5,1),xi(5,2),xi(5,3),-xi(4,1),0,xi(4,3)),
    (xi(6,1),xi(6,2),xi(6,3),-xi(3,1),0,0))$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "generic derivation : delta:=",delta;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The nonzero adjoint derivations
matrix adx1(6,6)$
adx1:=
sub({xi(1,2)=0,xi(3,1)=0,xi(3,2)=0,xi(4,1)=0,xi(4,2)=1,xi(4,3)=0,xi(5,1)=0,xi(5,2)=0,xi(5,3)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=0}, delta)$
matrix adx2(6,6)$
adx2:=
sub({xi(1,2)=0,xi(3,1)=0,xi(3,2)=0,xi(4,1)=-1,xi(4,2)=0,xi(4,3)=0,xi(5,1)=0,xi(5,2)=0,xi(5,3)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=1}, delta)$
matrix adx3(6,6)$
adx3:=
sub({xi(1,2)=0,xi(3,1)=0,xi(3,2)=0,xi(4,1)=0,xi(4,2)=0,xi(4,3)=0,xi(5,1)=0,xi(5,2)=0,xi(5,3)=0,xi(6,1)=0,xi(6,2)=-1,xi(6,3)=0}, delta)$
matrix adx4(6,6)$
adx4:=
sub({xi(1,2)=0,xi(3,1)=0,xi(3,2)=0,xi(4,1)=0,xi(4,2)=0,xi(4,3)=0,xi(5,1)=0,xi(5,2)=-1,xi(5,3)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=0}, delta)$
%matrix adx5(6,6)$
%adx5:=
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on nat$
write adx1:=adx1$
write adx2:=adx2$
write adx3:=adx3$
write adx4:=adx4$
%write adx5:=adx5$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% by subtracting adjoints one then may suppose
xi(4,1):=0$
xi(4,2):=0$
xi(5,2):=0$
xi(6,2):=0$
write "by subtracting adjoints one then may suppose:"$
write "xi(4,1):=0,xi(4,2):=0,xi(5,1):=0,xi(6,2):=0"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "delta:=",delta;
write "We denote this delta by the shortform"$
shortformdelta:=
{xi(1,2),SS,xi(3,1),xi(3,2),SS,xi(4,3),SS,xi(5,1),xi(5,3),SS,xi(6,1),xi(6,3)}$
paramindexeslist:=
{ {1,2},{3,1},{3,2},{4,3},{5,1},{5,3},{6,1},{6,3} }$

write "shortformdelta:=", shortformdelta$
write "paramindexeslist:=",paramindexeslist$

off nat$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE SHORTFORM(M0)$
BEGIN$
M:=M0$
WS:=
{M(1,2),SS,M(3,1),M(3,2),SS,M(4,3),SS,M(5,1),M(5,3),SS,M(6,1),M(6,3)}$
RETURN WS$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE DELTAPRIMEMODADG(M0,AUTOM)$
BEGIN $
M:=M0$
M:=AUTOM*M*AUTOM**(-1)$
M:=M-M(4,2)*adx1 +M(4,1)*adx2 + M(6,2)*adx3 + M(5,2)*adx4 $

IF AUTOM=phi THEN  <<write "With the generic automorphism one gets">>$
%IF AUTOM=phi THEN  <<write "With the first kind automorphism one gets">>$
IF AUTOM=psi THEN  <<write "With the second kind automorphism one gets">>$

write "shortformdeltaprimemodadg:=",shortform(M)$
for each U in paramindexeslist do <<write "deltaprimemodg(" ,part(U,1), ",", part(U,2), "):=" ,
                  M(part(U,1),part(U,2))>>$
on factor$
write "det(AUTOM):=",det(AUTOM);
DELTAPRIMEMODADG:=M$
on nat$
write "DELTAPRIMEMODADG:=", M$
off nat$
CLEAR M$
%RETURN DELTAPRIMEMODADG$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "******* Suppose xi(1,2) = 0"$
xi(1,2):=0$
write "xi(1,2):=", xi(1,2)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "***********SUBCASE  : xi(4,3) = 0."$
xi(4,3):=0$
write "xi(4,3):=", xi(4,3)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "*********SUBSUBCASE  : xi(3,1) = 0."$
xi(3,1):=0$
write "xi(3,1):=",xi(3,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "*********SUBSUBSUBCASE  : xi(6,3) NEQ 0."$
%deltaprimemodg(3,2):=(b(3,3)*xi(3,2))/b(2,2)$

write "Then on may suppose xi(6,3):=1"$
xi(6,3):=1$
write "xi(6,3):=",xi(6,3)$
write "and one keeps deltaprime(6,3):=k by taking"$
%deltaprimemodg(6,3):=b(2,2)*xi(6,3)$
b(2,2):=k/xi(6,3)$
write "b(2,2):=",b(2,2)$


write "and we get deltaprime(5,3)=0 if we take"$
%deltaprimemodg(5,3):=(b(2,2)*(b(4,3)*xi(6,3) + b(2,2)*b(1,1)*xi(5,3)))/b(3,3)$
b(4,3):=-( b(2,2)*b(1,1)*xi(5,3))/xi(6,3)$
write "b(4,3):=",b(4,3)$


write "and we get deltaprime(6,1)=0 if we take"$
%deltaprimemodg(6,1):=(b(2,2)*(b(3,3)*xi(6,1) - b(3,1)*xi(6,3)))/b(1,1)$
b(3,1):=b(3,3)*xi(6,1)/xi(6,3)$
write "b(3,1):=",b(3,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Thus  , for xi(6,3) neq 0 , we are reduced to:"$
write "shortformdeltaprime ={0,SS,0,epsilon,SS,0,SS,eta,0,SS,0,1}"$
write "where epsilon = xi(3,2) = 0,1 ,eta = xi(5,1) = 0,1."$
bye$