%off echo,nat$
off echo$
out "rreducparautommodg6_52xCcase2N2.r"$
write "rreducparautommodg6_52xCcase2N2.r"$
operator b$
ON REVPRI$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic automorphism phi of C x g_{5,2} as computed by calculautom6_52xC.red :"$

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

write "generic derivation : delta:=",delta;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%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(3,1)=0,xi(3,2)=0,xi(3,3)=0,xi(4,1)=0,xi(4,2)=1,xi(4,3)=0,xi(4,6)=0,xi(5,1)=0,xi(5,2)=0,xi(5,3)=1,xi(5,6)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=0,xi(6,6)=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(3,1)=0,xi(3,2)=0,xi(3,3)=0,xi(4,1)=-1,xi(4,2)=0,xi(4,3)=0,xi(4,6)=0,xi(5,1)=0,xi(5,2)=0,xi(5,3)=0,xi(5,6)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=0,xi(6,6)=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(3,1)=0,xi(3,2)=0,xi(3,3)=0,xi(4,1)=0,xi(4,2)=0,xi(4,3)=0,xi(4,6)=0,xi(5,1)=-1,xi(5,2)=0,xi(5,3)=0,xi(5,6)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=0,xi(6,6)=0}, delta)$
%matrix adx4(6,6)$
%adx4:=
%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$
write "xi(1,1):=",xi(1,1)$
xi(6,6):=0$
write "xi(6,6):=",xi(6,6)$

write "And the matrix A:=(xi(2,2),xi(2,3)),(xi(3,2),xi(3,3)) is nilpotent"$
write "We hence get 2 cases according to whether A neq 0 or A=0."$
write "We consider here the case 2 where A = 0."$
xi(2,2):=0$
xi(2,3):=0$
xi(3,2):=0$
xi(3,3):=0$
for i:=2:3 do for j:=2:3 do <<write "xi(",i,",",j,"):=",xi(i,j)>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% by subtracting adjoints one then may suppose
xi(4,1):=0$
xi(4,2):=0$
xi(5,1):=0$
write "by subtracting adjoints one then may suppose:"$
write "xi(4,1):=0,xi(4,2):=0,xi(5,1):=0"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "delta:=",delta;
write "We denote this delta by the shortform"$
shortformdelta:=
{xi(2,1),xi(3,1),xi(6,1),SS,xi(5,2),xi(6,2),SS,xi(4,3),xi(5,3),xi(6,3),SS,xi(4,6),xi(5,6)}$
paramindexeslist:=
{{2,1},{3,1},{6,1},{5,2},{6,2},{4,3},{5,3},{6,3},{4,6},{5,6}}$

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

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

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 "************ SUBCASE 1 : xi(2,1),xi(3,1) NOT SIMULTANEOUSLY ZERO *************"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then one may suppose :"$
xi(2,1):=1$
xi(3,1):=0$
write "xi(3,1):=",xi(3,1)$
write "xi(2,1):=",xi(2,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then one keeps deltaprime(2,1)=k and deltaprime(3,1)=0 by taking :"$
b(2,2):=k*b(1,1)$
b(3,2):=0$
write "b(2,2):=",b(2,2)$
write "b(3,2):=",b(3,2)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then one gets deltaprime(6,1)=0 by taking :"$
comment
deltaprimemodg(6,1):=(b(6,2)*b(3,3)*b(1,1)*k - b(6,6)*b(3,1)*b(1,1)*xi(6,3)*k +
b(6,6)*b(3,1)*b(2,3)*xi(6,2) + b(6,6)*b(3,3)*b(1,1)*xi(6,1)*k - b(6,6)*b(3,3)*b(
2,1)*xi(6,2))/(b(3,3)*b(1,1)**2*k)$

b(6,2):=-( - b(6,6)*b(3,1)*b(1,1)*xi(6,3)*k +
b(6,6)*b(3,1)*b(2,3)*xi(6,2) + b(6,6)*b(3,3)*b(1,1)*xi(6,1)*k - b(6,6)*b(3,3)*b(
2,1)*xi(6,2))/(b(3,3)*b(1,1)*k)$
write "b(6,2):=",b(6,2)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "***************    SUBSUBCASE 1.2 : xi(6,2) := 0 *************************"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
xi(6,2):=0$
write "xi(6,2):=",xi(6,2)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "***************    SUBSUBSUBCASE 1.2.1. : xi(5,6) NEQ 0 ***********************"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then one gets deltaprime(5,6)=k by taking :"$
%deltaprimemodg(5,6):=(b(3,3)*b(1,1)*xi(5,6))/b(6,6)$
b(6,6):=(b(3,3)*b(1,1)*xi(5,6))/k$
write "b(6,6):=",b(6,6)$
write "Hence we may suppose xi(5,6):=1 and we keep deltaprime(5,6):=k by our choice."$
xi(5,6):=1$
write "xi(5,6):=",xi(5,6)$

write "One gets deltaprime(5,3)=0 and deltaprime(4,6)=0 by taking :"$
comment
deltaprimemodg(5,3):=( - b(6,3)*b(3,3)*b(1,1)*xi(5,6)*k + b(6,6)*b(3,1)*b(1,1)*
xi(6,3)*xi(4,6)*k + 2*b(6,6)*b(3,1)*b(2,3)*xi(6,3)*xi(5,6) + b(6,6)*b(3,3)*b(1,1
)*xi(5,3)*k - b(6,6)*b(3,3)*b(1,1)*xi(6,1)*xi(4,6)*k - 2*b(6,6)*b(3,3)*b(2,3)*xi
(5,2) - 2*b(6,6)*b(3,3)*b(2,3)*xi(6,1)*xi(5,6) + b(6,6)*b(5,6)*xi(6,3)*k)/(b(6,6
)*b(3,3)*k)$

b(6,3):=(  b(6,6)*b(3,1)*b(1,1)*
xi(6,3)*xi(4,6)*k + 2*b(6,6)*b(3,1)*b(2,3)*xi(6,3)*xi(5,6) + b(6,6)*b(3,3)*b(1,1
)*xi(5,3)*k - b(6,6)*b(3,3)*b(1,1)*xi(6,1)*xi(4,6)*k - 2*b(6,6)*b(3,3)*b(2,3)*xi
(5,2) - 2*b(6,6)*b(3,3)*b(2,3)*xi(6,1)*xi(5,6) + b(6,6)*b(5,6)*xi(6,3)*k)/
(b(3,3)*b(1,1)*xi(5,6)*k)$
write "b(6,3):=",b(6,3)$

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


write "Hence we may suppose xi(5,3):=0 and we keep deltaprime(5,3):=0 by our choice."$
write "and similarly xi(4,6):=0 and we keep deltaprime(4,6):=0 by our choice."$
xi(5,3):=0$
write "xi(5,3):=",xi(5,3)$
xi(4,6):=0$
write "xi(4,6):=",xi(4,6)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "**************  Suppose first that xi(6,3) neq 0."$
write "Then one gets deltaprime(6,3)=k ,"$
write "and one gets deltaprime(5,2)=0 and deltaprime(4,3)=0 by taking :"$

%deltaprimemodg(5,2):=( - b(3,1)*xi(6,3) + b(3,3)*xi(5,2) + b(3,3)*xi(6,1))/k$
b(3,1):= ( b(3,3)*xi(5,2) + b(3,3)*xi(6,1))/xi(6,3)$
write "b(3,1):=",b(3,1)$


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


%deltaprimemodg(6,3):=(b(1,1)*xi(6,3))/k$
b(1,1):=k**2/xi(6,3)$
write "b(1,1):=",b(1,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
shortformdeltaprimemodadg:={k,0,0,ss,0,0,ss,0,0,k,ss,0,k}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence, we are reduced in the subsubsubcase 1.2.1 if xi(6,3) neq 0 to:"$
write "shortformdeltaprime:={1,0,0,ss,0,0,ss,0,0,1,ss,0,1}"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "**************  Suppose now that xi(6,3) = 0."$
clear b(3,1),b(4,6),b(1,1)$
write "clear b(3,1),b(4,6),b(1,1)"$
xi(6,3):=0$
write "xi(6,3):=",xi(6,3)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
shortformdeltaprimemodadg:={k,0,0,ss,(b(3,3)*(xi(5,2) + xi(6,1)))/k,0,ss,(b(1,1)**2*xi(4,3)*k)/b(3,3),0,0,ss,0,k}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence, we are reduced in the subsubsubcase 1.2.1 if xi(6,3) := 0 to:"$
write "shortformdeltaprime:={1,0,0,ss,epsilon,0,ss,eta,0,0,ss,0,1}"$
write "where epsilon=xi(5,2)=0,1. and eta=xi(4,3)=0,1."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

bye$