off echo$
out "rreducparautommodg6_nxC3case6N1.r"$
write "rreducparautommodg6_nxC3case6N1.r"$

operator b$
ON REVPRI$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic automorphism phi of n x C**3 :"$


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

write "generic derivation : delta:=",delta;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The nonzero adjoint derivations
matrix adx1(6,6)$
adx1:=
sub({
xi(1,1)=0,xi(1,2)=0,
xi(2,1)=0,xi(2,2)=0,
xi(3,1)=0,xi(3,2)=1,xi(3,4)=0,xi(3,5)=0,xi(3,6)=0,
xi(4,1)=0,xi(4,2)=0,xi(4,4)=0,xi(4,5)=0,xi(4,6)=0,
xi(5,1)=0,xi(5,2)=0,xi(5,4)=0,xi(5,5)=0,xi(5,6)=0,
xi(6,1)=0,xi(6,2)=0,xi(6,4)=0,xi(6,5)=0,xi(6,6)=0}, delta)$
matrix adx2(6,6)$
adx2:=
sub({
xi(1,1)=0,xi(1,2)=0,
xi(2,1)=0,xi(2,2)=0,
xi(3,1)=-1,xi(3,2)=0,xi(3,4)=0,xi(3,5)=0,xi(3,6)=0,
xi(4,1)=0,xi(4,2)=0,xi(4,4)=0,xi(4,5)=0,xi(4,6)=0,
xi(5,1)=0,xi(5,2)=0,xi(5,4)=0,xi(5,5)=0,xi(5,6)=0,
xi(6,1)=0,xi(6,2)=0,xi(6,4)=0,xi(6,5)=0,xi(6,6)=0}, delta)$
%matrix adx3(6,6)$
%adx3:=
%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$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic nilpotent derivation : "$
%write "The generic nilpotent derivation : the eigenvalues are 0"$
write "The matrices MATA:=((xi(1,1),xi(1,2)),(xi(2,1),xi(2,2))) is nilpotent"$
MATA:=mat((xi(1,1),xi(1,2)),(xi(2,1),xi(2,2)))$

write "And  MATB:=((xi(4,4),xi(4,5),xi(4,6)),(xi(5,4),xi(5,5),xi(5,6)),(xi(6,4),xi(6,5),xi(6,6))) are nilpotent"$
MATB:=mat((xi(4,4),xi(4,5),xi(4,6)),(xi(5,4),xi(5,5),xi(5,6)),(xi(6,4),xi(6,5),xi(6,6)))$
MATC:=mat((xi(3,4)),(xi(3,5)),(xi(3,6)))$
MATD:=mat((xi(4,1),xi(4,2)),(xi(5,1),xi(5,2)),(xi(6,1),xi(6,2)))$
write "(WE denote MATA,MATB, since B is already for the entries of phi)"$

write "****** We consider here the case 6 where MATA has nilpotent order 1"$
write " and MATB has nilpotent order 2.***"$

write "MATB:= ",MATB;
write "MATC:= ",MATC;
write "MATD:= ",MATD;
write "In block matrices, we get :"$
write "phi:=mat((MATU,0,0),(MATR,det(MATU),MATS),(MATT,0,MATV))"$
write "where MATU is in GL(2), MATV in GL(3)."$
MATU:=mat((b(1,1),b(1,2)),(b(2,1),b(2,2)))$
MATV:=mat((b(4,4),b(4,5),b(4,6)),(b(5,4),b(5,5),b(5,6)),(b(6,4),b(6,5),b(6,6)))$
write "delta:=mat((MATA,0,0),(0,0,MATC),(MATD,0,MATB))"$
write "Under the action of phi one gets:"$
write "phi*delta*phi**(-1):=mat((MATU*MATA*MATU**(-1),0,0),(**,0,MATCprime),(MATDprime,0,MATV*MATB*MATV**(-1)))"$


%write "In that case, one may suppose MATA:=((0,1),(0,0))."$
write " As MATA has nilpotent order 1, we may suppose : "$
xi(1,1):=0$
xi(1,2):=1$
xi(2,1):=0$
xi(2,2):=0$
for i:=1:2 do for j:=1:2 do <<write "xi(",i,",",j,"):= ",xi(i,j)>>$

write " As MATB has nilpotent order 2, we may suppose : "$
for i:=4:6 do for j:=4:6 do xi(i,j):=0$

clear xi(4,5)$
xi(4,5):=1$

clear xi(5,6)$
xi(5,6):=1$

write "MATB:= ", MATB$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% by subtracting adjoints one then may suppose
xi(3,1):=0$
xi(3,2):=0$
write "by subtracting adjoints one then may suppose:"$
write "xi(3,1):=0,xi(3,2):=0."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "delta:=",delta;
write "We denote this delta by the shortform"$
shortformdelta:=
{xi(4,1),xi(4,2),SS,xi(5,1),xi(5,2),SS,xi(6,1),xi(6,2),SS,xi(3,4),xi(3,5),xi(3,6) }$
paramindexeslist:=
{{4,1},{4,2},{5,1},{5,2},{6,1},{6,2},{3,4},{3,5},{3,6}}$

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

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

IF AUTOM=phi THEN  <<write "With the generic 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$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "MATU*MATA-MATA*MATU:=",MATU*MATA-k*MATA*MATU;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
comment
MATU*MATA-MATA*MATU:=
mat(( - b(2,1)*k, - ( - b(1,1) + b(2,2)*k)),(0,b(2,1)))$

write "to keep in MATA deltaprimemodadg(1,2):=k, and the others zero, one has to take :"$
b(2,1):=0$
b(1,1):=k*b(2,2)$
write "MATU:= ", MATU$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
%DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "MATV*MATB-MATB*MATV:=",MATV*MATB-k*MATB*MATV;
comment
MATV*MATB-MATB*MATV:=
mat(( - b(5,4)*k,b(4,4) - b(5,5)*k,b(4,5) - b(5,6)*k),( - b(6,4)*k,b(5,4) - b(6,
5)*k,b(5,5) - b(6,6)*k),(0,b(6,4),b(6,5)))$

write "to keep in MATB deltaprimemodadg(4,5):=k,deltaprimemodadg(5,6):=k,"$
write "and the others zero, one has to take :"$

b(5,4):=0$
write "b(5,4):=",b(5,4)$

b(4,4) :=b(5,5)*k$
write "b(4,4):=",b(4,4)$

b(4,5) :=b(5,6)*k$
write "b(4,5):=",b(4,5)$

b(6,4):=0$
write "b(6,4):=",b(6,4)$

b(6,5):=0$
write "b(6,5):=",b(6,5)$

b(5,5) :=b(6,6)*k$
write "b(5,5):=",b(5,5)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "One gets deltaprime(4,1):=0 ,deltaprime(5,1):=0 if we take :"$
comment
deltaprimemodg(4,1):=(b(4,6)*xi(6,1) - b(5,1)*k + b(5,6)*xi(5,1)*k + b(6,6)*xi(4,1)*k**2)/(b(2,2)*k)$
b(5,1):=(b(4,6)*xi(6,1) + b(5,6)*xi(5,1)*k + b(6,6)*xi(4,1)*k**2)/(k)$


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

write "b(5,1):=",b(5,1)$
write "b(6,1):=",b(6,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "One gets deltaprime(4,2):=0 as well if we take :"$
comment
%deltaprimemodg(4,2):=(b(4,1) + b(4,6)*xi(6,2) - b(5,2)*k + b(5,6)*xi(5,2)*k + b(6,6)*xi(4,2)*k**2)/b(2,2)$
b(4,1):=-( b(4,6)*xi(6,2) - b(5,2)*k + b(5,6)*xi(5,2)*k + b(6,6)*xi(4,2)*k**2)$
write "b(4,1):=",b(4,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "One gets moreover deltaprime(5,2):=0  and "$
write "deltaprime(3,6):=0 if we take :"$
%deltaprimemodg(5,2):=(b(4,6)*xi(6,1) + b(5,6)*xi(5,1)*k + b(5,6)*xi(6,2)*k - b(6,2)*k**2 + b(6,6)*xi(4,1)*k**2 + b(6,6)*xi(5,2)*k**2)/(b(2,2)*k)$

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


comment
deltaprimemodg(3,6):=(b(5,6)**2*b(2,2)**2*xi(3,4) - b(6,6)*b(4,6)*b(2,2)**2*xi(3
,4) - b(6,6)*b(5,6)*b(2,2)**2*xi(3,5)*k - b(6,6)*b(5,6)*b(3,4) + b(6,6)**2*b(2,2
)**2*xi(3,6)*k**2 + b(6,6)**2*b(3,5)*k)/(b(6,6)**3*k)$

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "One gets deltaprime(3,5):=0 as well if we take :"$
%deltaprimemodg(3,5):=( - b(5,6)*b(2,2)**2*xi(3,4) + b(6,6)*b(2,2)**2*xi(3,5)*k +  b(6,6)*b(3,4))/(b(6,6)**2*k)$
b(3,4):= - ( - b(5,6)*b(2,2)**2*xi(3,4) + b(6,6)*b(2,2)**2*xi(3,5)*k )/(b(6,6))$
write "b(3,4):=",b(3,4)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "************** SUBCASE 1 :xi(6,1) neq 0. ***********************************"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then we get deltaprimemodadg(6,1)=k by taking :"$
%deltaprimemodg(6,1):=(b(6,6)*xi(6,1))/(b(2,2)*k)$
b(6,6):=(b(2,2)*k**2)/xi(6,1)$
write "b(6,6):=",b(6,6)$
write "and deltaprimemodadg(6,2)=0 by taking :"$

comment
deltaprimemodg(6,2):=(b(5,6)*b(2,2)*xi(6,1) - b(6,6)*b(1,2)*xi(6,1) + b(6,6)*b(2
,2)*xi(5,1)*k + b(6,6)*b(2,2)*xi(6,2)*k)/(b(2,2)**2*k)$

b(1,2):=(b(5,6)*b(2,2)*xi(6,1)  + b(6,6)*b(2
,2)*xi(5,1)*k + b(6,6)*b(2,2)*xi(6,2)*k)/(b(6,6)*xi(6,1))$
write "b(1,2):=",b(1,2)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence we are reduced in that subcase 1 to :"$
write "deltaprimemodg:=(0,0;0,0;1,0;epsilon,0,0)_6."$
write "where epsilon=xi(3,4)=0,1."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "************** SUBCASE 2 :xi(6,1) = 0. ***********************************"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear b(1,2), b(6,6)$
write "clear b(1,2), b(6,6)"$
xi(6,1):=0$
write "xi(6,1):=",xi(6,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%deltaprimemodg(6,2):=((xi(5,1) + xi(6,2))*b(6,6))/b(2,2)$

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

write "Hence we are reduced in that subcase 2 to :"$
write "deltaprimemodg:=(0,0;0,0;0,epsilon;eta,0,0)_6."$
write "where epsilon=xi(6,2)=0,1 and eta=xi(3,4)=0,1."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


bye$