%off echo,nat$
off echo$
out "rreducparautommodg6_4N2.r"$
operator b$



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generic automorphism of g_{6,4} as in (art ijac1 section5) :
phi:=mat((b(1,1),0,0,0,0,0),
         (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),0,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), 0),
         (b(6,1),b(6,2),b(6,3), b(3,2)*b(1,1), 0, b(3,3)*b(1,1)))$

write "phi:=",phi;
write "det(phi):=",det(phi);
write "delta:=",delta;

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "delta:=",delta;
write "We denote this delta by the shortform"$
shortformdelta:={xi(3,1),xi(3,2),SS,xi(5,1),xi(5,3),SS,xi(6,2),xi(6,3)}$
paramindexeslist:={{3,1},{3,2},{5,1},{5,3},{6,2},{6,3}}$

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

off nat$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE SHORTFORM(M0)$
BEGIN$
M:=M0$
WS:=
{M(3,1),M(3,2),SS,M(5,1),M(5,3),SS,M(6,2),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(6,1)*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);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%case under consideration here  for reduction by automorphism
write "******* Suppose xi(3,1) = 0 and xi(3,2) neq 0"$
xi(3,1):=0$
write "write xi(3,1):=", xi(3,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "we get deltaprime(3,2)=k (k nonzero) and deltaprime(6,2)  = 0 if we take"$
%deltaprime(3,2):=(b(3,3)*xi(3,2))/b(2,2)$
%deltaprimemodg(3,2):=(b(3,3)*xi(3,2))/b(2,2)$
b(3,3):=k*b(2,2)/xi(3,2)$
write "write b(3,3):=", b(3,3)$


%deltaprimemodg(6,2):=(b(6,3)*b(2,2)*xi(3,2) - b(4,2)*b(3,3)*xi(3,2) + b(3,3)*b(2,2)*b(1,1)*xi(6,2) - b(3,2)*b(2,2)*b(1,1)*xi(6,3))/b(2,2)**2$
%deltaprime(6,2):=(b(6,3)*b(2,2)*xi(3,2) - b(4,2)*b(3,3)*xi(3,2) + b(3,3)*b(2,2)*b(1,1)*xi(6,2) - b(3,2)*b(2,2)*b(1,1)*xi(6,3))/b(2,2)**2$
b(6,3):= -(- b(4,2)*b(3,3)*xi(3,2) + b(3,3)*b(2,2)*b(1,1)*xi(6,2) - b(3,2)*b(2,2)*b(1,1)*xi(6,3))/(b(2,2)*xi(3,2))$
write "write b(6,3):=", b(6,3)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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


%deltaprimemodg(5,3):=(b(2,2)*b(1,1)*xi(5,3)*xi(3,2))/k$
b(2,2):=k**2/(b(1,1)*xi(5,3)*xi(3,2))$
write "write b(3,1):=", b(3,1)$
write "write b(2,2):=", b(2,2)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "shortformdeltaprime:=", {0,1,ss,0,1,ss,0,epsilon}$
write "clear b(3,1),b(2,2)"$
clear b(3,1),b(2,2)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "******  Suppose xi(5,3) = 0"$
xi(5,3):=0$
write "xi(5,3):=",ws$
for each U in paramindexeslist do  <<write "deltaprime(" ,part(U,1), ",", part(U,2), "):=" ,           deltaprime(part(U,1),part(U,2))>>$
write "deltaprime:=", deltaprime$
write "shortformdeltaprime:=", shortformdeltaprime$
write "Hence in the case xi(3,1)=0, xi(3,2) neq 0, xi(5,3) = 0, we are reduced to"$
write "shortformdeltaprime:=", {0,1,ss,eta,0,ss,0,epsilon}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
bye$
b(3,3):=k*b(1,1)$
write "b(3,3):=", b(3,3)$

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

b(5,3):=-(
%b(5,3)*b(3,3) +
b(3,3)*b(2,2)**2*b(1,1)*xi(5,1) - b(3,1)*b(2,2
)**2*b(1,1)*xi(5,3))/(b(3,3))$
write "b(5,3):=",ws;


for each U in paramindexeslist do  <<write "deltaprime(" ,part(U,1), ",", part(U,2), "):=" ,           deltaprime(part(U,1),part(U,2))>>$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "******* Suppose xi(3,2) neq 0"$
write "we get deltaprime(3,2)=k  if we take"$
%deltaprime(3,2):=(b(1,1)*xi(3,2)*k)/b(2,2)$
b(2,2):=(b(1,1)*xi(3,2))$
write "b(2,2):=",ws;


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


for each U in paramindexeslist do  <<write "deltaprime(" ,part(U,1), ",", part(U,2), "):=" ,           deltaprime(part(U,1),part(U,2))>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "******* Suppose xi(5,3) neq 0"$
write "Then we get deltaprime(5,3)=k  if we take"$
%deltaprime(5,3):=(b(1,1)**2*xi(5,3)*xi(3,2)**2)/k$
b(1,1):=k/(sqrt(xi(5,3))*xi(3,2))$
write "b(1,1):=",ws;

for each U in paramindexeslist do  <<write "deltaprime(" ,part(U,1), ",", part(U,2), "):=" ,           deltaprime(part(U,1),part(U,2))>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence if xi(3,1)*xi(3,2)*xi(5,3) neq 0, we are reduced to the case "$
write "deltaprime:=",phi*delta*phi**(-1)$
write "shortformdeltaprime:=",shortformdeltaprime$
write "that is, to the case"$
write "shortformprimedelta:=",{1,1,0,1,0,a}," a in \Cmath"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear b(1,1)$
write "clear b(1,1)"$
write "******* Suppose xi(5,3) =0"$
xi(5,3):=0$

for each U in paramindexeslist do  <<write "deltaprime(" ,part(U,1), ",", part(U,2), "):=" ,           deltaprime(part(U,1),part(U,2))>>$
write "Then, if xi(6,3) neq 0 one may suppose xi(6,3)=1 by taking "$
b(1,1):=k/xi(6,3)$
write "b(1,1):=",b(1,1)$
write "one gets:"$
for each U in paramindexeslist do  <<write "deltaprime(" ,part(U,1), ",", part(U,2), "):=" ,           deltaprime(part(U,1),part(U,2))>>$
write "Hence if xi(3,1)*xi(3,2) neq 0 and xi(5,3)=0 , we are reduced for xi(6,3) neq 0 to the case "$
write "deltaprime:=",deltaprime$
write "shortformdeltaprime:=",shortformdeltaprime$
write "Hence if xi(3,1)*xi(3,2) neq 0 and xi(5,3)=0 , we are reduced  to the case "$
write "shortformprimedelta:=",{1,1,0,0,0,epsilon}," epsilon:=0,1"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear b(1,1),xi(5,3),b(2,2),b(6,3)$
write "clear b(1,1),xi(5,3),b(2,2),b(6,3)"$
write "******* Suppose finally xi(3,2) =0"$
xi(3,2):=0$


for each U in paramindexeslist do  <<write "deltaprime(" ,part(U,1), ",", part(U,2), "):=" ,           deltaprime(part(U,1),part(U,2))>>$

write "Then, if xi(6,3) neq 0 one may suppose xi(6,3)=1 and xi(6,2)=0 by taking "$
%deltaprime(6,2):=(b(1,1)*( - b(3,2)*xi(6,3) + b(1,1)*xi(6,2)*k))/b(2,2)$
 b(3,2) := b(1,1)*xi(6,2)*k/xi(6,3)$
write "b(3,2):=",ws;

%deltaprime(6,3):=b(1,1)*xi(6,3)$

b(1,1):=k/xi(6,3)$
write "b(1,1):=",ws;

write "deltaprime:=",deltaprime$
for each U in paramindexeslist do  <<write "deltaprime(" ,part(U,1), ",", part(U,2), "):=" ,           deltaprime(part(U,1),part(U,2))>>$
write "Hence if xi(3,2) = 0 and xi(6,3) neq 0, we are reduced to the case "$
write "shortformprimedelta:=",{1,0,ss,0,epsilon,ss,0,1}, " epsilon:=0,1"$
clear b(1,1),b(3,2)$
write "clear b(1,1),b(3,2)"$
write "If xi(3,2)= xi(6,3) = 0 one has:"$
xi(6,3):=0$
write "deltaprime:=",deltaprime$
for each U in paramindexeslist do  <<write "deltaprime(" ,part(U,1), ",", part(U,2), "):=" ,           deltaprime(part(U,1),part(U,2))>>$
write "Hence if xi(3,2) = xi(6,3)=0 , we are reduced  to the case "$
write "shortformprimedelta:=",{1,0,ss,0,epsilon,ss,eta,0}," epsilon,eta:=0,1"$
  bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%