%off echo,nat$
off echo$
out "rtest3case2N6.r"$
write "This test verifies that [1,0;epsilon,a;0,0;0,0] is proj equiv to "$
write "some delta having xi(5,3)=xi(5,4)=0 and xi(6,3) neq 0,xi(6,4)=0."$
operator b$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generic automorphism of g_{6,1} as in (art ijac1 section6) :
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(1,1)*u,-b(1,2)*u,0,0),
         (b(4,1),b(4,2),-b(2,1)*u,b(2,2)*u,0,0),
         (b(5,1),b(5,2),b(5,3),b(5,4), b(2,2)*b(1,1)-b(2,1)*b(1,2), 0),
         (b(6,1),b(6,2),b(6,3),b(6,4), b(3,2)*b(2,1)-b(3,1)*b(2,2)-b(4,1)*b(1,2)+b(4,2)*b(1,1), (b(2,2)*b(1,1)-b(2,1)*b(1,2))*u))$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generic derivation as in (Cohomology tables section 4.3.1) :
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(3,3),-xi(1,2),0,0),
    (xi(4,1),xi(4,2),-xi(2,1),-xi(1,1)+xi(2,2)+xi(3,3),0,0),
    (xi(5,1),xi(5,2),xi(5,3),xi(5,4),xi(1,1)+xi(2,2),0),
    (xi(6,1),xi(6,2),xi(6,3),xi(6,4),-xi(3,1)+xi(4,2),xi(2,2)+xi(3,3)))$
write "generic derivation : delta:=",delta;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write
"The nonzero adjoints are delta(6,1), delta(6,2), and  delta(5,2)+delta(6,4),
-delta(5,1)+delta(6,3)"$
write "Hence -delta(5,2) and delta(6,4) are adjoint related"$
write "and so are delta(5,1) and delta(6,3) as well."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The nonzero adjoint derivations
matrix adx1(6,6)$
adx1(5,2):=1$
adx1(6,4):=1$
matrix adx2(6,6)$
adx2(5,1):=-1$
adx2(6,3):=1$
matrix adx3(6,6)$
adx3(6,2):=-1$
matrix adx4(6,6)$
adx4(6,1):=-1$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on nat$
write adx1:=adx1$
write adx2:=adx2$
write adx3:=adx3$
write adx4:=adx4$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generic nilpotent derivation as in (Cohomology tables page 104) : the eigenvalues
% are 0
write "delta  is nilpotent if and only if xi(2,2)=-xi(1,1) , xi(3,3)=xi(1,1) and
finally the matrix A:=mat((xi(1,1),xi(1,2)),(xi(2,1),xi(2,2))) is nilpotent"$
xi(2,2):=-xi(1,1)$
xi(3,3):=xi(1,1)$
write "generic nilpotent derivation : delta:=",delta;
A:=mat((xi(1,1),xi(1,2)),(xi(2,1),xi(2,2)))$
write "with A:=",A,  "nilpotent"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic automorphism of g_{6,1} as in (art ijac1 section6) :"$
write "phi:=",phi;
on factor$
write "det(phi):=",det(phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "We consider here in this case 2 the case where A = 0."$
xi(1,1):=0$
xi(2,1):=0$
xi(1,2):=0$
xi(2,2):=0$
write "A:=",A$
write "generic nilpotent derivation in the case 1 : delta:=",delta;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% by subtracting adjoints one then may suppose
xi(5,1):=0$
xi(5,2):=0$
xi(6,1):=0$
xi(6,2):=0$
write "by subtracting adjoints one then may suppose xi(5,1)=xi(5,2)=xi(6,1)=xi(6,2)=0"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "phi:=",phi;
on factor$
write "det(phi):=",det(phi);
write "delta:=",delta;
write "We denote this delta by the shortform"$
shortformdelta:=
{xi(3,1),xi(3,2),SS,xi(4,1),xi(4,2),SS,xi(5,3),xi(5,4),SS,xi(6,3),xi(6,4)}$
paramindexeslist:=
{ {3,1},{3,2},{4,1},{4,2},{5,3},{5,4},{6,3},{6,4} }$

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

off nat$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE SHORTFORM(M0)$
BEGIN$
M:=M0$
WS:=  {M(3,1),M(3,2),SS,M(4,1),M(4,2),SS,M(5,3),M(5,4),SS,M(6,3),M(6,4)}$
RETURN WS$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE DELTAPRIMEMODADG(M0)$
BEGIN $
M:=M0$
M:=phi*M*phi**(-1)$
M:=M-M(5,2)*adx1 +M(5,1)*adx2 +M(6,2)*adx3 +M(6,1)*adx4$
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(phi):=",det(phi);
DELTAPRIMEMODADG:=M$
write "DELTAPRIMEMODADG:=", M$
CLEAR M$
%RETURN DELTAPRIMEMODADG$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "********* Suppose that xi(5,3) and xi(5,4) are  both zero."$
xi(5,3):=0$
xi(5,4):=0$
write "xi(5,3):=",xi(5,3)$
write "xi(5,4):=",xi(5,4)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "******Suppose that xi(6,3) and xi(6,4) are  both zero:"$
xi(6,3):=0$
xi(6,4):=0$
write "xi(6,3):=",xi(6,3)$
write "xi(6,4):=",xi(6,4)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
xi(3,1):=1$
xi(3,2):=0$

%if xi(4,1)=1, one gets xi(6,3) neq 0 by"$
xi(4,1):=1$

b(2,1):=0$
b(5,3):=0$

%if xi(4,1)=0, one gets xi(6,3) neq 0 by"$
xi(4,1):=0$
clear b(2,1), b(5,3)$
%suppose first xi(4,2) neq 1/2."$
b(5,3):=0$
%Then we get xi(6,3) neq 0.
%suppose now xi(4,2) = 1/2."$
xi(4,2):=1/2$
clear b(5,3)$
%Then we get xi(6,3) neq 0 as well.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$ on exp$
DELTAPRIMEMODADG(delta);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%