%off echo,nat$
off echo$
out "rreducparautommodg6_56xCN3.r"$
operator b$
ON REVPRI$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic automorphism phi of C x g_{5,6} as computed by calculautom6_cx56.red :"$


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

write "phi:=",phi;
on factor$
write "det(phi):=",det(phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generic derivation as computed by geneLplus.tex :
operator xi$


delta:=
mat((xi(1,1),0,0,0,0,0),(xi(2,1),2*xi(1,1),0,0,0,0),(xi(3,1),xi(3,2),3*xi(
1,1),0,0,0),(xi(4,1),xi(4,2),xi(3,2),4*xi(1,1),0,0),(xi(5,1),xi(5,2),xi(4
,2) - xi(3,1),xi(3,2) + xi(2,1),5*xi(1,1),xi(5,6)),(xi(6,1),xi(6,2),0,0,0
,xi(6,6)))$

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

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

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

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$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "****************** CASE 2 : xi(2,1) = 0 *************************"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%write "Then one can suppose xi(2,1):=1"$
xi(2,1):=0$
write "xi(2,1):=",xi(2,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "****************** SUBCASE 2.1 : xi(6,2) NEQ 0 *************************"$
write "Then one can suppose xi(6,2):=1"$
xi(6,2):=1$
write "xi(6,2):=",xi(6,2)$
write "and one keeps deltaprimemodg(6,2):=k by taking :"$
b(6,6):= k*b(1,1)**2$
write "b(6,6):=",b(6,6)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then one gets deltaprimemodg(5,2)=0 by taking : "$
%deltaprimemodg(5,2):=(b(1,1)**5*xi(5,2)*k + 2*b(2,1)*b(1,1)**3*xi(4,2)*k + b(5,6)*k - b(6,2)*b(1,1)**3*xi(5,6))/(b(1,1)**2*k)$
b(5,6):=-(b(1,1)**5*xi(5,2)*k + 2*b(2,1)*b(1,1)**3*xi(4,2)*k - b(6,2)*b(1,1)**3*xi(5,6))
/(k)$
write "b(5,6):=",b(5,6)$
write "and one gets deltaprimemodg(6,1)=0 by taking : "$
%deltaprimemodg(6,1):=(k*(b(1,1)**2*xi(6,1) - b(2,1)))/b(1,1)$
b(2,1):=b(1,1)**2*xi(6,1)$
write "b(2,1):=",b(2,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then if xi(4,2) NEQ 0 one gets deltaprimemodg(4,2)=k by taking : "$
b(1,1):=sqrt(k/xi(4,2))$
write "b(1,1):=",b(1,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then if xi(5,6) NEQ 0 one gets deltaprimemodg(5,6)=k by taking : "$
%deltaprimemodg(5,6):=(sqrt(k/xi(4,2))*xi(5,6))/xi(4,2)$
k:=xi(5,6)**2/xi(4,2)**3$
write "k:=",k$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence, we are reduced in the subcase 2.1. under consideration to:"$
write "shortformdeltaprime ={0,SS,epsilon,SS,0,eta,SS,0,1}"$
write "where epsilon=xi(4,2)=0,1 and eta=xi(5,6)=0,1."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "****************** SUBCASE 2.2 : xi(6,2) = 0 *************************"$
xi(6,2):=0$
write "xi(6,2):=",xi(6,2)$
clear b(6,6),b(5,6),b(2,1),b(1,1),k$
write "clear b(6,6),b(5,6),b(2,1),b(1,1),k"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "****************** SUBSUBCASE 2.2.1 : xi(4,2) NEQ 0 *************************"$
write "Then one can suppose xi(4,2):=1"$
xi(4,2):=1$
write "xi(4,2):=",xi(4,2)$
write "and one keeps deltaprimemodg(4,2):=k by taking :"$
%deltaprimemodg(4,2):=b(1,1)**2*xi(4,2)$
b(1,1):= sqrt(k/xi(4,2))$
write "b(1,1):=",b(1,1)$
write "and one gets deltaprimemodg(5,2)=0 by taking : "$
%deltaprimemodg(5,2):=(b(1,1)*( - b(6,2)*b(1,1)**2*xi(5,6) + b(6,6)*b(1,1)**2*xi(5,2) + 2*b(6,6)*b(2,1)*xi(4,2)))/b(6,6)$
b(2,1):=-( - b(6,2)*b(1,1)**2*xi(5,6) + b(6,6)*b(1,1)**2*xi(5,2))/( 2*b(6,6)*xi(4,2))$
write "b(2,1):=",b(2,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "If xi(6,1) neq 0, we get deltaprime(6,1):=k by taking :"$
%deltaprimemodg(6,1):=(b(6,6)*xi(6,1))/sqrt(k)$
b(6,6):=k*sqrt(k)/xi(6,1)$
write "b(6,6):=",b(6,6)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence, we are reduced in that case to:"$
write "shortformdeltaprime ={0,SS,1,SS,0,a,SS,1,0}"$
write "where a=xi(5,6) is any complex number."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "If  xi(6,1) = 0, we get :"$
xi(6,1):=0$
write "xi(6,1):=",xi(6,1)$
clear b(6,6)$
write "clear b(6,6)"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence, we are reduced in that case  to:"$
write "shortformdeltaprime ={0,SS,1,SS,0,epsilon,SS,0,0}"$
write "where epsilon=xi(5,6)=0,1 "$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear xi(6,1)$
write "clear xi(6,1)"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "****************** SUBSUBCASE 2.2.2 : xi(4,2) = 0 *************************"$
xi(4,2):=0$
write "xi(4,2):=",xi(4,2)$
clear b(2,1),b(1,1)$
write "clear b(2,1),b(1,1)"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Suppose first xi(5,6) NEQ 0"$
write "Then one can suppose xi(5,6):=1"$
xi(5,6):=1$
write "xi(5,6):=",xi(5,6)$
write "and one keeps deltaprimemodg(5,6):=k by taking :"$
b(6,6):=b(1,1)**5*xi(5,6)/k$
write "b(6,6):=",b(6,6)$
write "and one gets deltaprimemodg(5,2)=0 by taking : "$
%deltaprimemodg(5,2):=(b(1,1)**3*( - b(6,2)*xi(5,6) + b(6,6)*xi(5,2)))/b(6,6)$
b(6,2):= b(6,6)*xi(5,2)/xi(5,6)$
write "b(6,2):=",b(6,2)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence, we are reduced in that case to:"$
write "shortformdeltaprime ={0,SS,0,SS,0,1,SS,eta,0}"$
write "where  eta=xi(6,1)=0,1."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Suppose now xi(5,6) = 0"$
xi(5,6):=0$
write "xi(5,6):=",xi(5,6)$
clear b(6,6),b(6,2)$
write "clear b(6,6),b(6,2)"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence, we are reduced in that  to either :"$
write "shortformdeltaprime ={0,SS,0,SS,1,0,SS,epsilon,0}"$
write "where epsilon=xi(6,1)=0,1"$
write "if xi(5,2) NEQ 0"$
write "or "$
write "shortformdeltaprime ={0,SS,0,SS,0,0,SS,1,0}"$
write "if xi(5,2) = 0."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

bye$