*********** CASE 2.2.3. The generic automorphism phi of C x g_{5,3} as computed by calculautom6_53xC.r\ ed : phi:= mat((b(1,1),0,0,0,0,0), (b(2,1),b(2,2),0,0,0,0), 2 (b(3,1),b(3,2),b(1,1) ,0,0,0), (b(4,1),b(4,2), - b(2,1)*b(1,1),b(2,2)*b(1,1),0,0), (b(5,1),b(5,2),b(5,3), - b(3,1)*b(2,2) + b(3,2)*b(2,1) + b(4,2)*b(1,1), 2 b(2,2)*b(1,1) ,b(5,6)), (b(6,1),b(6,2),b(6,3),0,0,b(6,6))) 3 6 det(phi):=b(6,6)*b(2,2) *b(1,1) generic derivation : delta:= mat((xi(1,1),0,0,0,0,0), (xi(2,1),xi(2,2),0,0,0,0), (xi(3,1),xi(3,2),2*xi(1,1),0,0,0), (xi(4,1),xi(4,2), - xi(2,1),xi(1,1) + xi(2,2),0,0), (xi(5,1),xi(5,2),xi(5,3), - xi(3,1) + xi(4,2),2*xi(1,1) + xi(2,2),xi(5,6)), (xi(6,1),xi(6,2),xi(6,3),0,0,xi(6,6))) [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 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] adx2 := [ ] [-1 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 0] adx3 := [ ] [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 0 0] adx4 := [ ] [0 0 0 0 0 0] [ ] [-1 0 0 0 0 0] [ ] [0 0 0 0 0 0] The generic nilpotent derivation : the eigenvalues are 0 xi(1,1):=0 xi(2,2):=0 xi(6,6):=0 by subtracting adjoints one then may suppose: xi(4,1):=0,xi(4,2):=0,xi(5,1):=0,xi(5,2):=0 delta:= [ 0 0 0 0 0 0 ] [ ] [xi(2,1) 0 0 0 0 0 ] [ ] [xi(3,1) xi(3,2) 0 0 0 0 ] [ ] [ 0 0 - xi(2,1) 0 0 0 ] [ ] [ 0 0 xi(5,3) - xi(3,1) 0 xi(5,6)] [ ] [xi(6,1) xi(6,2) xi(6,3) 0 0 0 ] We denote this delta by the shortform shortformdelta:={xi(2,1), ss, xi(3,1), xi(3,2), ss, xi(5,3), xi(5,6), ss, xi(6,1), xi(6,2), xi(6,3)} paramindexeslist:={{2,1},{3,1},{3,2},{5,3},{5,6},{6,1},{6,2},{6,3}} With the general automorphism one gets$ shortformdeltaprimemodadg:={(b(2,2)*xi(2,1))/b(1,1), ss, (b(3,2)*b(2,2)*xi(2,1) + ( - b(2,1)*xi(3,2) + b(2,2)*xi(3,1))*b(1,1)**2)/(b(2,2) *b(1,1)), (b(1,1)**2*xi(3,2))/b(2,2), ss, ( - b(6,3)*b(2,2)**2*b(1,1)**2*xi(5,6) + (b(5,6)*b(2,2)*xi(6,3) - 2*b(3,2)*b(2,2 )*b(2,1)*xi(2,1) + 2*b(3,1)*b(2,2)**2*xi(2,1) + (b(2,1)**2*xi(3,2) - 2*b(2,2)*b( 2,1)*xi(3,1) + b(2,2)**2*xi(5,3))*b(1,1)**2)*b(6,6))/(b(6,6)*b(2,2)*b(1,1)**2), (b(2,2)*b(1,1)**2*xi(5,6))/b(6,6), ss, ((b(6,2)*b(2,2)*xi(2,1) + ( - b(2,1)*xi(3,2) + b(2,2)*xi(3,1))*b(6,3))*b(1,1)**2 + (( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1))*b(1,1)**2 + ( - b(3,1)*b(2,2) + b(3,2)* b(2,1))*xi(6,3))*b(6,6))/(b(2,2)*b(1,1)**3), ( - ( - b(6,3)*b(1,1)**2*xi(3,2) + ( - b(1,1)**2*xi(6,2) + b(3,2)*xi(6,3))*b(6,6 )))/(b(2,2)*b(1,1)**2), (b(6,6)*xi(6,3))/b(1,1)**2}$ deltaprimemodg(2,1):=(b(2,2)*xi(2,1))/b(1,1)$ deltaprimemodg(3,1):=(b(3,2)*b(2,2)*xi(2,1) + ( - b(2,1)*xi(3,2) + b(2,2)*xi(3,1 ))*b(1,1)**2)/(b(2,2)*b(1,1))$ deltaprimemodg(3,2):=(b(1,1)**2*xi(3,2))/b(2,2)$ deltaprimemodg(5,3):=( - b(6,3)*b(2,2)**2*b(1,1)**2*xi(5,6) + (b(5,6)*b(2,2)*xi( 6,3) - 2*b(3,2)*b(2,2)*b(2,1)*xi(2,1) + 2*b(3,1)*b(2,2)**2*xi(2,1) + (b(2,1)**2* xi(3,2) - 2*b(2,2)*b(2,1)*xi(3,1) + b(2,2)**2*xi(5,3))*b(1,1)**2)*b(6,6))/(b(6,6 )*b(2,2)*b(1,1)**2)$ deltaprimemodg(5,6):=(b(2,2)*b(1,1)**2*xi(5,6))/b(6,6)$ deltaprimemodg(6,1):=((b(6,2)*b(2,2)*xi(2,1) + ( - b(2,1)*xi(3,2) + b(2,2)*xi(3, 1))*b(6,3))*b(1,1)**2 + (( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1))*b(1,1)**2 + ( - b( 3,1)*b(2,2) + b(3,2)*b(2,1))*xi(6,3))*b(6,6))/(b(2,2)*b(1,1)**3)$ deltaprimemodg(6,2):=( - ( - b(6,3)*b(1,1)**2*xi(3,2) + ( - b(1,1)**2*xi(6,2) + b(3,2)*xi(6,3))*b(6,6)))/(b(2,2)*b(1,1)**2)$ deltaprimemodg(6,3):=(b(6,6)*xi(6,3))/b(1,1)**2$ det(AUTOM):=b(6,6)*b(2,2)**3*b(1,1)**6$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), b(2,2)*xi(2,1) (----------------,0,0,0,0,0), b(1,1) 2 b(3,2)*b(2,2)*xi(2,1) + ( - b(2,1)*xi(3,2) + b(2,2)*xi(3,1))*b(1,1) (----------------------------------------------------------------------, b(2,2)*b(1,1) 2 b(1,1) *xi(3,2) -----------------,0,0,0,0), b(2,2) - b(2,2)*xi(2,1) (0,0,-------------------,0,0,0), b(1,1) 2 2 (0,0,( - b(6,3)*b(2,2) *b(1,1) *xi(5,6) + (b(5,6)*b(2,2)*xi(6,3) 2 - 2*b(3,2)*b(2,2)*b(2,1)*xi(2,1) + 2*b(3,1)*b(2,2) *xi(2,1) + 2 2 (b(2,1) *xi(3,2) - 2*b(2,2)*b(2,1)*xi(3,1) + b(2,2) *xi(5,3)) 2 2 *b(1,1) )*b(6,6))/(b(6,6)*b(2,2)*b(1,1) ), 2 - (b(3,2)*b(2,2)*xi(2,1) + ( - b(2,1)*xi(3,2) + b(2,2)*xi(3,1))*b(1,1) ) --------------------------------------------------------------------------- b(2,2)*b(1,1) 2 b(2,2)*b(1,1) *xi(5,6) ,0,------------------------), b(6,6) (((b(6,2)*b(2,2)*xi(2,1) + ( - b(2,1)*xi(3,2) + b(2,2)*xi(3,1))*b(6,3)) 2 2 *b(1,1) + (( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1))*b(1,1) 3 + ( - b(3,1)*b(2,2) + b(3,2)*b(2,1))*xi(6,3))*b(6,6))/(b(2,2)*b(1,1) ) 2 ,( - ( - b(6,3)*b(1,1) *xi(3,2) 2 2 + ( - b(1,1) *xi(6,2) + b(3,2)*xi(6,3))*b(6,6)))/(b(2,2)*b(1,1) ), b(6,6)*xi(6,3) ----------------,0,0,0)) 2 b(1,1) ****************** CASE 2 : xi(2,1) = 0 *************************$ xi(2,1):=0$ With the general automorphism one gets$ shortformdeltaprimemodadg:={0, ss, (b(1,1)*( - b(2,1)*xi(3,2) + b(2,2)*xi(3,1)))/b(2,2), (b(1,1)**2*xi(3,2))/b(2,2), ss, ( - b(6,3)*b(2,2)**2*b(1,1)**2*xi(5,6) + b(6,6)*b(2,1)**2*b(1,1)**2*xi(3,2) - 2* b(6,6)*b(2,2)*b(2,1)*b(1,1)**2*xi(3,1) + b(6,6)*b(2,2)**2*b(1,1)**2*xi(5,3) + b( 6,6)*b(5,6)*b(2,2)*xi(6,3))/(b(6,6)*b(2,2)*b(1,1)**2), (b(2,2)*b(1,1)**2*xi(5,6))/b(6,6), ss, ( - b(6,3)*b(2,1)*b(1,1)**2*xi(3,2) + b(6,3)*b(2,2)*b(1,1)**2*xi(3,1) - b(6,6)*b (2,1)*b(1,1)**2*xi(6,2) + b(6,6)*b(2,2)*b(1,1)**2*xi(6,1) - b(6,6)*b(3,1)*b(2,2) *xi(6,3) + b(6,6)*b(3,2)*b(2,1)*xi(6,3))/(b(2,2)*b(1,1)**3), (b(6,3)*b(1,1)**2*xi(3,2) + b(6,6)*b(1,1)**2*xi(6,2) - b(6,6)*b(3,2)*xi(6,3))/(b (2,2)*b(1,1)**2), (b(6,6)*xi(6,3))/b(1,1)**2}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=(b(1,1)*( - b(2,1)*xi(3,2) + b(2,2)*xi(3,1)))/b(2,2)$ deltaprimemodg(3,2):=(b(1,1)**2*xi(3,2))/b(2,2)$ deltaprimemodg(5,3):=( - b(6,3)*b(2,2)**2*b(1,1)**2*xi(5,6) + b(6,6)*b(2,1)**2*b (1,1)**2*xi(3,2) - 2*b(6,6)*b(2,2)*b(2,1)*b(1,1)**2*xi(3,1) + b(6,6)*b(2,2)**2*b (1,1)**2*xi(5,3) + b(6,6)*b(5,6)*b(2,2)*xi(6,3))/(b(6,6)*b(2,2)*b(1,1)**2)$ deltaprimemodg(5,6):=(b(2,2)*b(1,1)**2*xi(5,6))/b(6,6)$ deltaprimemodg(6,1):=( - b(6,3)*b(2,1)*b(1,1)**2*xi(3,2) + b(6,3)*b(2,2)*b(1,1) **2*xi(3,1) - b(6,6)*b(2,1)*b(1,1)**2*xi(6,2) + b(6,6)*b(2,2)*b(1,1)**2*xi(6,1) - b(6,6)*b(3,1)*b(2,2)*xi(6,3) + b(6,6)*b(3,2)*b(2,1)*xi(6,3))/(b(2,2)*b(1,1)**3 )$ deltaprimemodg(6,2):=(b(6,3)*b(1,1)**2*xi(3,2) + b(6,6)*b(1,1)**2*xi(6,2) - b(6, 6)*b(3,2)*xi(6,3))/(b(2,2)*b(1,1)**2)$ deltaprimemodg(6,3):=(b(6,6)*xi(6,3))/b(1,1)**2$ det(AUTOM):=b(6,6)*b(2,2)**3*b(1,1)**6$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), 2 b(1,1)*( - b(2,1)*xi(3,2) + b(2,2)*xi(3,1)) b(1,1) *xi(3,2) (---------------------------------------------,-----------------,0,0,0,0), b(2,2) b(2,2) (0,0,0,0,0,0), 2 2 2 2 (0,0,( - b(6,3)*b(2,2) *b(1,1) *xi(5,6) + b(6,6)*b(2,1) *b(1,1) *xi(3,2) 2 - 2*b(6,6)*b(2,2)*b(2,1)*b(1,1) *xi(3,1) 2 2 + b(6,6)*b(2,2) *b(1,1) *xi(5,3) + b(6,6)*b(5,6)*b(2,2)*xi(6,3))/( 2 b(1,1)*(b(2,1)*xi(3,2) - b(2,2)*xi(3,1)) b(6,6)*b(2,2)*b(1,1) ),------------------------------------------,0, b(2,2) 2 b(2,2)*b(1,1) *xi(5,6) ------------------------), b(6,6) 2 2 (( - b(6,3)*b(2,1)*b(1,1) *xi(3,2) + b(6,3)*b(2,2)*b(1,1) *xi(3,1) 2 2 - b(6,6)*b(2,1)*b(1,1) *xi(6,2) + b(6,6)*b(2,2)*b(1,1) *xi(6,1) - b(6,6)*b(3,1)*b(2,2)*xi(6,3) + b(6,6)*b(3,2)*b(2,1)*xi(6,3))/(b(2,2) 3 *b(1,1) ), 2 2 b(6,3)*b(1,1) *xi(3,2) + b(6,6)*b(1,1) *xi(6,2) - b(6,6)*b(3,2)*xi(6,3) -------------------------------------------------------------------------, 2 b(2,2)*b(1,1) b(6,6)*xi(6,3) ----------------,0,0,0)) 2 b(1,1) ****************** SUBCASE 2.2 : xi(3,2) = 0 *************************$ xi(3,2):=0$ With the general automorphism one gets$ shortformdeltaprimemodadg:={0, ss, b(1,1)*xi(3,1), 0, ss, ( - b(6,3)*b(2,2)*b(1,1)**2*xi(5,6) - 2*b(6,6)*b(2,1)*b(1,1)**2*xi(3,1) + b(6,6) *b(2,2)*b(1,1)**2*xi(5,3) + b(6,6)*b(5,6)*xi(6,3))/(b(6,6)*b(1,1)**2), (b(2,2)*b(1,1)**2*xi(5,6))/b(6,6), ss, (b(6,3)*b(2,2)*b(1,1)**2*xi(3,1) - b(6,6)*b(2,1)*b(1,1)**2*xi(6,2) + b(6,6)*b(2, 2)*b(1,1)**2*xi(6,1) - b(6,6)*b(3,1)*b(2,2)*xi(6,3) + b(6,6)*b(3,2)*b(2,1)*xi(6, 3))/(b(2,2)*b(1,1)**3), (b(6,6)*(b(1,1)**2*xi(6,2) - b(3,2)*xi(6,3)))/(b(2,2)*b(1,1)**2), (b(6,6)*xi(6,3))/b(1,1)**2}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=b(1,1)*xi(3,1)$ deltaprimemodg(3,2):=0$ deltaprimemodg(5,3):=( - b(6,3)*b(2,2)*b(1,1)**2*xi(5,6) - 2*b(6,6)*b(2,1)*b(1,1 )**2*xi(3,1) + b(6,6)*b(2,2)*b(1,1)**2*xi(5,3) + b(6,6)*b(5,6)*xi(6,3))/(b(6,6)* b(1,1)**2)$ deltaprimemodg(5,6):=(b(2,2)*b(1,1)**2*xi(5,6))/b(6,6)$ deltaprimemodg(6,1):=(b(6,3)*b(2,2)*b(1,1)**2*xi(3,1) - b(6,6)*b(2,1)*b(1,1)**2* xi(6,2) + b(6,6)*b(2,2)*b(1,1)**2*xi(6,1) - b(6,6)*b(3,1)*b(2,2)*xi(6,3) + b(6,6 )*b(3,2)*b(2,1)*xi(6,3))/(b(2,2)*b(1,1)**3)$ deltaprimemodg(6,2):=(b(6,6)*(b(1,1)**2*xi(6,2) - b(3,2)*xi(6,3)))/(b(2,2)*b(1,1 )**2)$ deltaprimemodg(6,3):=(b(6,6)*xi(6,3))/b(1,1)**2$ det(AUTOM):=b(6,6)*b(2,2)**3*b(1,1)**6$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), (b(1,1)*xi(3,1),0,0,0,0,0), (0,0,0,0,0,0), 2 2 (0,0,( - b(6,3)*b(2,2)*b(1,1) *xi(5,6) - 2*b(6,6)*b(2,1)*b(1,1) *xi(3,1) 2 + b(6,6)*b(2,2)*b(1,1) *xi(5,3) + b(6,6)*b(5,6)*xi(6,3))/(b(6,6) 2 2 b(2,2)*b(1,1) *xi(5,6) *b(1,1) ), - b(1,1)*xi(3,1),0,------------------------), b(6,6) 2 2 ((b(6,3)*b(2,2)*b(1,1) *xi(3,1) - b(6,6)*b(2,1)*b(1,1) *xi(6,2) 2 + b(6,6)*b(2,2)*b(1,1) *xi(6,1) - b(6,6)*b(3,1)*b(2,2)*xi(6,3) 3 + b(6,6)*b(3,2)*b(2,1)*xi(6,3))/(b(2,2)*b(1,1) ), 2 b(6,6)*(b(1,1) *xi(6,2) - b(3,2)*xi(6,3)) b(6,6)*xi(6,3) -------------------------------------------,----------------,0,0,0)) 2 2 b(2,2)*b(1,1) b(1,1) ****************** SUBSUBCASE 2.2.1 : xi(6,3) NEQ 0 *************************$ see N3$ ****************** SUBSUBCASE 2.2.2 : xi(6,3) = 0 *************************$ ********************* and xi(3,1) neq 0 .********************************$ see N4$ We consider here the remaining subsubcase 2.2.3. :$ ****************** SUBSUBCASE 2.2.3 : xi(6,3) = 0 *************************$ ********************* and xi(3,1) = 0 .********************************$ Suppose now xi(6,3) = 0.$ xi(6,3):=0$ With the general automorphism one gets$ shortformdeltaprimemodadg:={0, ss, b(1,1)*xi(3,1), 0, ss, ( - b(6,3)*b(2,2)*xi(5,6) - 2*b(6,6)*b(2,1)*xi(3,1) + b(6,6)*b(2,2)*xi(5,3))/b(6 ,6), (b(2,2)*b(1,1)**2*xi(5,6))/b(6,6), ss, (b(6,3)*b(2,2)*xi(3,1) - b(6,6)*b(2,1)*xi(6,2) + b(6,6)*b(2,2)*xi(6,1))/(b(2,2)* b(1,1)), (b(6,6)*xi(6,2))/b(2,2), 0}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=b(1,1)*xi(3,1)$ deltaprimemodg(3,2):=0$ deltaprimemodg(5,3):=( - b(6,3)*b(2,2)*xi(5,6) - 2*b(6,6)*b(2,1)*xi(3,1) + b(6,6 )*b(2,2)*xi(5,3))/b(6,6)$ deltaprimemodg(5,6):=(b(2,2)*b(1,1)**2*xi(5,6))/b(6,6)$ deltaprimemodg(6,1):=(b(6,3)*b(2,2)*xi(3,1) - b(6,6)*b(2,1)*xi(6,2) + b(6,6)*b(2 ,2)*xi(6,1))/(b(2,2)*b(1,1))$ deltaprimemodg(6,2):=(b(6,6)*xi(6,2))/b(2,2)$ deltaprimemodg(6,3):=0$ det(AUTOM):=b(6,6)*b(2,2)**3*b(1,1)**6$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), (b(1,1)*xi(3,1),0,0,0,0,0), (0,0,0,0,0,0), (0,0,( - b(6,3)*b(2,2)*xi(5,6) - 2*b(6,6)*b(2,1)*xi(3,1) + b(6,6)*b(2,2)*xi(5,3))/b(6,6), - b(1,1)*xi(3,1),0, 2 b(2,2)*b(1,1) *xi(5,6) ------------------------), b(6,6) b(6,3)*b(2,2)*xi(3,1) - b(6,6)*b(2,1)*xi(6,2) + b(6,6)*b(2,2)*xi(6,1) (-----------------------------------------------------------------------, b(2,2)*b(1,1) b(6,6)*xi(6,2) ----------------,0,0,0,0)) b(2,2) If xi(3,1) = 0, one gets :$ xi(3,1):=0$ With the general automorphism one gets$ shortformdeltaprimemodadg:={0, ss, 0, 0, ss, (b(2,2)*( - b(6,3)*xi(5,6) + b(6,6)*xi(5,3)))/b(6,6), (b(2,2)*b(1,1)**2*xi(5,6))/b(6,6), ss, (b(6,6)*( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1)))/(b(2,2)*b(1,1)), (b(6,6)*xi(6,2))/b(2,2), 0}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=0$ deltaprimemodg(3,2):=0$ deltaprimemodg(5,3):=(b(2,2)*( - b(6,3)*xi(5,6) + b(6,6)*xi(5,3)))/b(6,6)$ deltaprimemodg(5,6):=(b(2,2)*b(1,1)**2*xi(5,6))/b(6,6)$ deltaprimemodg(6,1):=(b(6,6)*( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1)))/(b(2,2)*b(1,1 ))$ deltaprimemodg(6,2):=(b(6,6)*xi(6,2))/b(2,2)$ deltaprimemodg(6,3):=0$ det(AUTOM):=b(6,6)*b(2,2)**3*b(1,1)**6$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), b(2,2)*( - b(6,3)*xi(5,6) + b(6,6)*xi(5,3)) (0,0,---------------------------------------------,0,0, b(6,6) 2 b(2,2)*b(1,1) *xi(5,6) ------------------------), b(6,6) b(6,6)*( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1)) b(6,6)*xi(6,2) (---------------------------------------------,----------------,0,0,0,0)) b(2,2)*b(1,1) b(2,2) Then if xi(6,2) neq 0, we get deltaprime(6,2)=k by taking :$ b(6,6):=(b(2,2)*k)/xi(6,2)$ and one gets deltaprime(6,1)=0 by taking :$ b(2,1):=(b(2,2)*xi(6,1))/xi(6,2)$ With the general automorphism one gets$ shortformdeltaprimemodadg:={0, ss, 0, 0, ss, (b(2,2)*xi(5,3)*k - b(6,3)*xi(6,2)*xi(5,6))/k, (b(1,1)**2*xi(6,2)*xi(5,6))/k, ss, 0, k, 0}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=0$ deltaprimemodg(3,2):=0$ deltaprimemodg(5,3):=(b(2,2)*xi(5,3)*k - b(6,3)*xi(6,2)*xi(5,6))/k$ deltaprimemodg(5,6):=(b(1,1)**2*xi(6,2)*xi(5,6))/k$ deltaprimemodg(6,1):=0$ deltaprimemodg(6,2):=k$ deltaprimemodg(6,3):=0$ det(AUTOM):=(b(2,2)**4*b(1,1)**6*k)/xi(6,2)$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), b(2,2)*xi(5,3)*k - b(6,3)*xi(6,2)*xi(5,6) (0,0,-------------------------------------------,0,0, k 2 b(1,1) *xi(6,2)*xi(5,6) -------------------------), k (0,k,0,0,0,0)) and if moreover xi(5,6) neq 0, we get deltaprime(5,6)=k $ and deltaprime(5,3)=0 by taking :$ b(1,1):=k/sqrt(xi(6,2)*xi(5,6))$ b(6,3):=(b(2,2)*xi(5,3)*k)/(xi(6,2)*xi(5,6))$ With the general automorphism one gets$ shortformdeltaprimemodadg:={0, ss, 0, 0, ss, 0, k, ss, 0, k, 0}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=0$ deltaprimemodg(3,2):=0$ deltaprimemodg(5,3):=0$ deltaprimemodg(5,6):=k$ deltaprimemodg(6,1):=0$ deltaprimemodg(6,2):=k$ deltaprimemodg(6,3):=0$ det(AUTOM):=(b(2,2)**4*k**7)/(xi(6,2)**4*xi(5,6)**3)$ DELTAPRIMEMODADG:= [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 0 0 0 k] [ ] [0 k 0 0 0 0] Hence, we are reduced in the subcase 2.2., and if moreover xi(6,3) = 0,$ and xi(3,1) = 0 (that is : in the subsubcase 2.2.3.)$ and moreover xi(6,2) *xi(5,6) neq 0$ to:$ shortformdeltaprime ={0,SS,0,0,SS,0,1,SS,0,1,0}$ Suppose now xi(6,2) neq 0 and xi(5,6) = 0 :$ clear b(1,1),b(6,3)$ xi(5,6):=0$ With the general automorphism one gets$ shortformdeltaprimemodadg:={0, ss, 0, 0, ss, b(2,2)*xi(5,3), 0, ss, 0, k, 0}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=0$ deltaprimemodg(3,2):=0$ deltaprimemodg(5,3):=b(2,2)*xi(5,3)$ deltaprimemodg(5,6):=0$ deltaprimemodg(6,1):=0$ deltaprimemodg(6,2):=k$ deltaprimemodg(6,3):=0$ det(AUTOM):=(b(2,2)**4*b(1,1)**6*k)/xi(6,2)$ DELTAPRIMEMODADG:= [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 b(2,2)*xi(5,3) 0 0 0] [ ] [0 k 0 0 0 0] Hence, we are reduced in the subcase 2.2.3$ if moreover xi(6,2) neq 0 , xi(5,6) = 0$ to:$ shortformdeltaprime ={0,SS,0,0,SS,epsilon,0,SS,0,1,0}$ where epsilon=xi(5,3)=0,1.$ Finally, suppose xi(6,2)=0.$ clear b(6,6),b(2,1),xi(5,6)$ xi(6,2):=0$ With the general automorphism one gets$ shortformdeltaprimemodadg:={0, ss, 0, 0, ss, (b(2,2)*( - b(6,3)*xi(5,6) + b(6,6)*xi(5,3)))/b(6,6), (b(2,2)*b(1,1)**2*xi(5,6))/b(6,6), ss, (b(6,6)*xi(6,1))/b(1,1), 0, 0}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=0$ deltaprimemodg(3,2):=0$ deltaprimemodg(5,3):=(b(2,2)*( - b(6,3)*xi(5,6) + b(6,6)*xi(5,3)))/b(6,6)$ deltaprimemodg(5,6):=(b(2,2)*b(1,1)**2*xi(5,6))/b(6,6)$ deltaprimemodg(6,1):=(b(6,6)*xi(6,1))/b(1,1)$ deltaprimemodg(6,2):=0$ deltaprimemodg(6,3):=0$ det(AUTOM):=b(6,6)*b(2,2)**3*b(1,1)**6$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), b(2,2)*( - b(6,3)*xi(5,6) + b(6,6)*xi(5,3)) (0,0,---------------------------------------------,0,0, b(6,6) 2 b(2,2)*b(1,1) *xi(5,6) ------------------------), b(6,6) b(6,6)*xi(6,1) (----------------,0,0,0,0,0)) b(1,1) ***** Suppose first xi(5,6) neq 0.$ Then one gets deltaprime(5,6)=k by taking :$ b(2,2):=(b(6,6)*k)/(b(1,1)**2*xi(5,6))$ and one gets deltaprime(5,3)=0 by taking :$ b(6,3):=(b(6,6)*xi(5,3))/xi(5,6)$ With the general automorphism one gets$ shortformdeltaprimemodadg:={0, ss, 0, 0, ss, 0, k, ss, (b(6,6)*xi(6,1))/b(1,1), 0, 0}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=0$ deltaprimemodg(3,2):=0$ deltaprimemodg(5,3):=0$ deltaprimemodg(5,6):=k$ deltaprimemodg(6,1):=(b(6,6)*xi(6,1))/b(1,1)$ deltaprimemodg(6,2):=0$ deltaprimemodg(6,3):=0$ det(AUTOM):=(b(6,6)**4*k**3)/xi(5,6)**3$ DELTAPRIMEMODADG:= [ 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 0 0 0 k] [ ] [ b(6,6)*xi(6,1) ] [---------------- 0 0 0 0 0] [ b(1,1) ] Hence, we are reduced in the subcase 2.2.3.$ if moreover xi(6,2) = 0 , xi(5,6) NEQ 0$ to:$ shortformdeltaprime ={0,SS,0,0,SS,0,1,SS,epsilon,0,0}$ where epsilon=xi(6,1)=0,1.$ ***** Suppose now xi(5,6) = 0 :$ clear b(6,3),b(2,2)$ xi(5,6):=0$ With the general automorphism one gets$ shortformdeltaprimemodadg:={0, ss, 0, 0, ss, b(2,2)*xi(5,3), 0, ss, (b(6,6)*xi(6,1))/b(1,1), 0, 0}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=0$ deltaprimemodg(3,2):=0$ deltaprimemodg(5,3):=b(2,2)*xi(5,3)$ deltaprimemodg(5,6):=0$ deltaprimemodg(6,1):=(b(6,6)*xi(6,1))/b(1,1)$ deltaprimemodg(6,2):=0$ deltaprimemodg(6,3):=0$ det(AUTOM):=b(6,6)*b(2,2)**3*b(1,1)**6$ DELTAPRIMEMODADG:= [ 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 b(2,2)*xi(5,3) 0 0 0] [ ] [ b(6,6)*xi(6,1) ] [---------------- 0 0 0 0 0] [ b(1,1) ] Then as we dismiss direct products, xi(6,1) neq 0.$ Hence, we are reduced in the subcase 2.2.3, and if moreover$ xi(6,2) = 0 , xi(5,6) = 0$ to:$ shortformdeltaprime ={0,SS,0,0,SS,epsilon,0,SS,1,0,0}$ where epsilon=xi(5,3)=0,1.$