%off echo,nat$
off echo$
out "rreducparautommodg6_10N3.r"$
operator b$
ON REVPRI$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic automorphism phi of g_{6,10} as computed by calculautom6_10.red :"$


phi:=
mat((b(1,1),0,0,0,0,0),(b(2,1),b(3,3)**2/b(1,1),b(2,3),0,0,0),(( - b(2,3)*b(1,1)
)/b(3,3),0,b(3,3),0,0,0),(b(4,1),b(4,2),b(4,3),b(3,3)**2,b(2,3)*b(1,1),0),(b(5,1
),0,b(5,3),0,b(3,3)*b(1,1),0),(b(6,1),b(6,2),b(6,3),b(4,2)*b(1,1),( - ((b(5,1)*b
(3,3) - b(4,3)*b(1,1))*b(3,3) + b(5,3)*b(2,3)*b(1,1)))/b(3,3),b(3,3)**2*b(1,1)))
$


write "phi:=",phi;
on factor$
write "det(phi):=",det(phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generic derivation as in (Cohomology tables section 4.2.4) :
operator xi$
delta:=
mat((xi(1,1),0,0,0,0,0),
    (xi(2,1),xi(2,2),xi(2,3),0,0,0),
    (-xi(2,3),0,(xi(1,1)+xi(2,2))/2,0,0,0),
    (xi(4,1),xi(4,2),xi(4,3),xi(1,1)+xi(2,2),xi(2,3),0),
    (xi(5,1),0,xi(5,3),0,(3*xi(1,1)+xi(2,2))/2,0),
    (xi(6,1),xi(6,2),xi(6,3),xi(4,2),xi(4,3)-xi(5,1),2*xi(1,1)+xi(2,2)))$
write "generic derivation : delta:=",delta;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The nonzero adjoint derivations
matrix adx1(6,6)$
adx1:=
sub({xi(1,1)=0,xi(2,1)=0,xi(2,2)=0,xi(2,3)=0,xi(4,1)=0,xi(4,2)=1,xi(4,3)=0,xi(5,1)=0,xi(5,3)=1,xi(6,1)=0,xi(6,2)=0,xi(6,3)=0}, delta)$
matrix adx2(6,6)$
adx2:=
sub({xi(1,1)=0,xi(2,1)=0,xi(2,2)=0,xi(2,3)=0,xi(4,1)=-1,xi(4,2)=0,xi(4,3)=0,xi(5,1)=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(1,1)=0,xi(2,1)=0,xi(2,2)=0,xi(2,3)=0,xi(4,1)=0,xi(4,2)=0,xi(4,3)=0,xi(5,1)=-1,xi(5,3)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=0}, delta)$
matrix adx4(6,6)$
adx4:=
sub({xi(1,1)=0,xi(2,1)=0,xi(2,2)=0,xi(2,3)=0,xi(4,1)=0,xi(4,2)=0,xi(4,3)=0,xi(5,1)=0,xi(5,3)=0,xi(6,1)=-1,xi(6,2)=0,xi(6,3)=0}, delta)$
matrix adx5(6,6)$
adx5:=
sub({xi(1,1)=0,xi(2,1)=0,xi(2,2)=0,xi(2,3)=0,xi(4,1)=0,xi(4,2)=0,xi(4,3)=0,xi(5,1)=0,xi(5,3)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=-1}, delta)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on nat$
write adx1:=adx1$
write adx2:=adx2$
write adx3:=adx3$
write adx4:=adx4$
write adx5:=adx5$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic nilpotent derivation : the eigenvalues are 0"$
xi(1,1):=0$
xi(2,2):=0$
write "xi(1,1):=",xi(1,1)$
write "xi(2,2):=",xi(2,2)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% by subtracting adjoints one then may suppose
xi(4,1):=0$
xi(4,2):=0$
xi(5,1):=0$
xi(6,1):=0$
xi(6,3):=0$
write "by subtracting adjoints one then may suppose:"$
write "xi(4,1):=0,xi(4,2):=0,xi(5,1):=0,xi(6,1):=0,xi(6,3):=0"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "phi:=",phi;
on factor$
write "det(phi):=",det(phi);
write "delta:=",delta;
write "We denote this delta by the shortform"$
shortformdelta:=
{xi(2,1),xi(2,3),SS,xi(4,3),SS,xi(5,3),SS,xi(6,2)}$
paramindexeslist:=
{ {2,1},{2,3},{4,3},{5,3},{6,2}}$

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

off nat$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE SHORTFORM(M0)$
BEGIN$
M:=M0$
WS:=  {M(2,1),M(2,3),SS,M(4,3),SS,M(5,3),SS,M(6,2)}$
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(5,1)*adx3 + M(6,1)*adx4 + M(6,3)*adx5$

%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))>>$
write "det(AUTOM):=",det(AUTOM);
DELTAPRIMEMODADG:=M$
on nat$
write "DELTAPRIMEMODADG:=", M$
off nat$
CLEAR M$
%RETURN DELTAPRIMEMODADG$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "************     SUBCASE 2 : we suppose xi(2,3) = 0   *********************"$
xi(2,3):=0$
write "xi(2,3):=",xi(2,3)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "************    SUBSUBCASE 2.1 : we suppose xi(5,3) NEQ 0  *********************"$
write "If xi(5,3) NEQ 0 we get deltaprime(5,3)=k by taking:"$

%deltaprimemodg(4,3):=(b(2,3)*b(1,1)*xi(5,3) + b(3,3)**2*xi(4,3))/b(3,3)$
%deltaprimemodg(5,3):=b(1,1)*xi(5,3)$
%deltaprimemodg(6,2):=b(1,1)**2*xi(6,2)$

b(1,1):=k/xi(5,3)$
write "b(1,1):=",b(1,1)$

write "and we get deltaprime(4,3)=0 by taking"$
b(2,3):=- b(3,3)**2*xi(4,3)/(b(1,1)*xi(5,3))$
write "b(2,3):=",b(2,3)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "If xi(6,2) NEQ 0 we get deltaprime(6,2)=k by taking:"$
%deltaprimemodg(2,1):=(b(3,3)**2*xi(5,3)**2*xi(2,1))/k**2$
%deltaprimemodg(6,2):=(xi(6,2)*k**2)/xi(5,3)**2$
k:=(xi(6,2)/xi(5,3)**2)**(-1)$
write "k:=",k$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence, we are reduced in that case to:"$
write "shortformdeltaprime ={epsilon,0,SS,0,SS,1,SS,eta}"$
write "where etam =xi(6,2) =0,1 and epsilon=xi(2,1):=0,1"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "************    SUBSUBCASE 2.2 : we suppose xi(5,3) = 0  *********************"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "If xi(5,3) = 0,  we get :"$
xi(5,3):=0$
write "xi(5,3):=",xi(5,3)$
clear b(1,1),b(2,3),k$
write "clear b(1,1),b(2,3),k"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%deltaprimemodg(2,1):=(b(3,3)**2*xi(2,1))/b(1,1)**2$
%deltaprimemodg(4,3):=b(3,3)*xi(4,3)$
%deltaprimemodg(6,2):=b(1,1)**2*xi(6,2)$
write "*********  Suppose first  xi(4,3) NEQ 0. *********************************"$
write "Then we get deltaprime(4,3)=k by taking:"$
b(3,3):=k/xi(4,3)$
write "b(3,3):=",b(2,3)$
write "****  If moreover xi(6,2) NEQ 0 we get deltaprime(6,2)=k by taking:"$
b(1,1):=sqrt(k/xi(6,2))$
write "b(1,1):=",b(1,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence, we are reduced if that case  to:"$
write "shortformdeltaprime ={a,0,SS,1,SS,0,SS,1}"$
write "where a =xi(2,1) is any complex"$
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write " ***** If  xi(6,2) = 0 we get :"$
xi(6,2):=0$
write "xi(6,2):=",xi(6,2)$
clear b(1,1)$
write "clear b(1,1)"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then we are reduced   to:"$
write "shortformdeltaprime ={epsilon,0,SS,1,SS,0,SS,0}"$
write "where epsilon =xi(2,1)=0,1."$
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "*********  Suppose now  xi(4,3) = 0. *********************************"$
xi(4,3):=0$
write "xi(4,3):=",xi(4,3)$
clear b(3,3),xi(6,2)$
write "clear b(3,3),xi(6,2)"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "****  If moreover xi(6,2) NEQ 0 we get deltaprime(6,2)=k by taking:"$
b(1,1):=sqrt(k/xi(6,2))$
write "b(1,1):=",b(1,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence, we are reduced if that case  to:"$
write "shortformdeltaprime ={epsilon,0,SS,0,SS,0,SS,1}"$
write "where epsilon =xi(2,1)=0,1 "$
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "****  If finally xi(6,2) = 0 we get :"$
xi(6,2):=0$
write "xi(6,2):=",xi(6,2)$
clear b(1,1)$
write "clear b(1,1)"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "As we suppose delta NEQ 0, we have xi(2,1) neq 0 in  that case:"$
write "Hence, we are reduced if that case  to:"$
write "shortformdeltaprime ={1,0,SS,0,SS,0,SS,0}"$
bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%