phi:= [b(1,1) 0 0 0 0 0 ] [ ] [ 0 b(2,2) 0 0 0 0 ] [ ] [b(3,1) b(3,2) b(3,3) 0 0 0 ] [ ] [b(4,1) b(4,2) 0 b(2,2)*b(1,1) 0 0 ] [ ] [ 2 ] [b(5,1) b(5,2) b(5,3) - b(4,1)*b(2,2) b(2,2) *b(1,1) 0 ] [ ] [b(6,1) b(6,2) b(6,3) b(3,2)*b(1,1) 0 b(3,3)*b(1,1)] 2 4 4 det(phi):=b(3,3) *b(2,2) *b(1,1) delta:=delta generic derivation : delta:= [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [xi(3,1) xi(3,2) 0 0 0 0] [ ] [xi(4,1) xi(4,2) 0 0 0 0] [ ] [xi(5,1) xi(5,2) xi(5,3) - xi(4,1) 0 0] [ ] [xi(6,1) xi(6,2) xi(6,3) xi(3,2) 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx1 := [ ] [0 1 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 1 0 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx2 := [ ] [-1 0 0 0 0 0] [ ] [0 0 0 1 0 0] [ ] [0 0 0 0 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx3 := [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [-1 0 0 0 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx4 := [ ] [0 0 0 0 0 0] [ ] [0 -1 0 0 0 0] [ ] [0 0 0 0 0 0] The nonzero ajoints are delta(4,1), delta(5,2), delta(6,1), and delta(4,2)-del\ ta(6,3) Hence delta(4,2) and delta(6,3) are adjoint related delta:= [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [xi(3,1) xi(3,2) 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [xi(5,1) 0 xi(5,3) 0 0 0] [ ] [ 0 xi(6,2) xi(6,3) xi(3,2) 0 0] We denote this delta by the shortform shortformdelta:={xi(3,1), xi(3,2), ss, xi(5,1), xi(5,3), ss, xi(6,2), xi(6,3)} paramindexeslist:={{3,1},{3,2},{5,1},{5,3},{6,2},{6,3}} With the generic automorphism one gets$ shortformdeltaprimemodadg:={(b(3,3)*xi(3,1))/b(1,1), (b(3,3)*xi(3,2))/b(2,2), ss, (b(5,3)*b(3,3)*xi(3,1) + b(3,3)*b(2,2)**2*b(1,1)*xi(5,1) - b(3,1)*b(2,2)**2*b(1, 1)*xi(5,3))/(b(3,3)*b(1,1)), (b(2,2)**2*b(1,1)*xi(5,3))/b(3,3), ss, (b(6,3)*b(2,2)*xi(3,2) - b(4,2)*b(3,3)*xi(3,2) + b(3,3)*b(2,2)*b(1,1)*xi(6,2) - b(3,2)*b(2,2)*b(1,1)*xi(6,3))/b(2,2)**2, b(1,1)*xi(6,3)}$ deltaprimemodg(3,1):=(b(3,3)*xi(3,1))/b(1,1)$ deltaprimemodg(3,2):=(b(3,3)*xi(3,2))/b(2,2)$ deltaprimemodg(5,1):=(b(5,3)*b(3,3)*xi(3,1) + b(3,3)*b(2,2)**2*b(1,1)*xi(5,1) - b(3,1)*b(2,2)**2*b(1,1)*xi(5,3))/(b(3,3)*b(1,1))$ deltaprimemodg(5,3):=(b(2,2)**2*b(1,1)*xi(5,3))/b(3,3)$ deltaprimemodg(6,2):=(b(6,3)*b(2,2)*xi(3,2) - b(4,2)*b(3,3)*xi(3,2) + b(3,3)*b(2 ,2)*b(1,1)*xi(6,2) - b(3,2)*b(2,2)*b(1,1)*xi(6,3))/b(2,2)**2$ deltaprimemodg(6,3):=b(1,1)*xi(6,3)$ det(AUTOM):=b(3,3)**2*b(2,2)**4*b(1,1)**4$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), b(3,3)*xi(3,1) b(3,3)*xi(3,2) (----------------,----------------,0,0,0,0), b(1,1) b(2,2) (0,0,0,0,0,0), 2 (b(3,3)*xi(5,1) - b(3,1)*xi(5,3))*b(2,2) *b(1,1) + b(5,3)*b(3,3)*xi(3,1) (--------------------------------------------------------------------------, b(3,3)*b(1,1) 2 b(2,2) *b(1,1)*xi(5,3) 0,------------------------,0,0,0), b(3,3) (0,((b(3,3)*xi(6,2) - b(3,2)*xi(6,3))*b(2,2)*b(1,1) - b(4,2)*b(3,3)*xi(3,2) 2 b(3,3)*xi(3,2) + b(6,3)*b(2,2)*xi(3,2))/b(2,2) ,b(1,1)*xi(6,3),----------------,0,0)) b(2,2) ******* Suppose xi(3,1) = 0 and xi(3,2) neq 0$ write xi(3,1):=0$ With the generic automorphism one gets$ shortformdeltaprimemodadg:={0, (b(3,3)*xi(3,2))/b(2,2), ss, (b(2,2)**2*(b(3,3)*xi(5,1) - b(3,1)*xi(5,3)))/b(3,3), (b(2,2)**2*b(1,1)*xi(5,3))/b(3,3), ss, (b(6,3)*b(2,2)*xi(3,2) - b(4,2)*b(3,3)*xi(3,2) + b(3,3)*b(2,2)*b(1,1)*xi(6,2) - b(3,2)*b(2,2)*b(1,1)*xi(6,3))/b(2,2)**2, b(1,1)*xi(6,3)}$ deltaprimemodg(3,1):=0$ deltaprimemodg(3,2):=(b(3,3)*xi(3,2))/b(2,2)$ deltaprimemodg(5,1):=(b(2,2)**2*(b(3,3)*xi(5,1) - b(3,1)*xi(5,3)))/b(3,3)$ deltaprimemodg(5,3):=(b(2,2)**2*b(1,1)*xi(5,3))/b(3,3)$ deltaprimemodg(6,2):=(b(6,3)*b(2,2)*xi(3,2) - b(4,2)*b(3,3)*xi(3,2) + b(3,3)*b(2 ,2)*b(1,1)*xi(6,2) - b(3,2)*b(2,2)*b(1,1)*xi(6,3))/b(2,2)**2$ deltaprimemodg(6,3):=b(1,1)*xi(6,3)$ det(AUTOM):=b(3,3)**2*b(2,2)**4*b(1,1)**4$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), b(3,3)*xi(3,2) (0,----------------,0,0,0,0), b(2,2) (0,0,0,0,0,0), 2 2 (b(3,3)*xi(5,1) - b(3,1)*xi(5,3))*b(2,2) b(2,2) *b(1,1)*xi(5,3) (-------------------------------------------,0,------------------------,0,0, b(3,3) b(3,3) 0), (0,((b(3,3)*xi(6,2) - b(3,2)*xi(6,3))*b(2,2)*b(1,1) - b(4,2)*b(3,3)*xi(3,2) 2 b(3,3)*xi(3,2) + b(6,3)*b(2,2)*xi(3,2))/b(2,2) ,b(1,1)*xi(6,3),----------------,0,0)) b(2,2) we get deltaprime(3,2)=k (k nonzero) and deltaprime(6,2) = 0 if we take$ write b(3,3):=(b(2,2)*k)/xi(3,2)$ write b(6,3):=((b(3,2)*xi(6,3)*xi(3,2) - b(2,2)*xi(6,2)*k)*b(1,1) + b(4,2)*xi(3, 2)*k)/xi(3,2)**2$ ****** Suppose xi(5,3) neq 0$ we get deltaprime(5,3)=k (k nonzero) and deltaprime(5,1)=0 if we take$ write b(3,1):=(xi(5,1)*k**3)/(b(1,1)*xi(5,3)**2*xi(3,2)**2)$ write b(2,2):=k**2/(b(1,1)*xi(5,3)*xi(3,2))$ With the generic automorphism one gets$ shortformdeltaprimemodadg:={0, k, ss, 0, k, ss, 0, b(1,1)*xi(6,3)}$ deltaprimemodg(3,1):=0$ deltaprimemodg(3,2):=k$ deltaprimemodg(5,1):=0$ deltaprimemodg(5,3):=k$ deltaprimemodg(6,2):=0$ deltaprimemodg(6,3):=b(1,1)*xi(6,3)$ det(AUTOM):=k**14/(b(1,1)**2*xi(5,3)**6*xi(3,2)**8)$ DELTAPRIMEMODADG:= [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 k 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 k 0 0 0] [ ] [0 0 b(1,1)*xi(6,3) k 0 0] shortformdeltaprime:={0,1,ss,0,1,ss,0,epsilon}$ clear b(3,1),b(2,2)$ ****** Suppose xi(5,3) = 0$ xi(5,3):=0$ deltaprime(3,1):=deltaprime(3,1)$ deltaprime(3,2):=deltaprime(3,2)$ deltaprime(5,1):=deltaprime(5,1)$ deltaprime(5,3):=deltaprime(5,3)$ deltaprime(6,2):=deltaprime(6,2)$ deltaprime(6,3):=deltaprime(6,3)$ deltaprime:=deltaprime$ shortformdeltaprime:=shortformdeltaprime$ Hence in the case xi(3,1)=0, xi(3,2) neq 0, xi(5,3) = 0, we are reduced to$ shortformdeltaprime:={0,1,ss,eta,0,ss,0,epsilon}$