rreducparautommodg6_g4xC2case2N5.r The generic automorphism phi of g_{4} x C**2 as computed by calculautom6_g4xC\ 2.red : phi:= [b(1,1) 0 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) 0 0 0 ] [ ] [ 2 ] [b(4,1) b(4,2) b(3,2)*b(1,1) b(2,2)*b(1,1) b(4,5) b(4,6)] [ ] [b(5,1) b(5,2) 0 0 b(5,5) b(5,6)] [ ] [b(6,1) b(6,2) 0 0 b(6,5) b(6,6)] 3 4 det(phi):=( - b(6,5)*b(5,6) + b(6,6)*b(5,5))*b(2,2) *b(1,1) generic derivation : delta:= [xi(1,1) 0 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) 0 0 0 ] [ ] [xi(4,1) xi(4,2) xi(3,2) 2*xi(1,1) + xi(2,2) xi(4,5) xi(4,6)] [ ] [xi(5,1) xi(5,2) 0 0 xi(5,5) xi(5,6)] [ ] [xi(6,1) xi(6,2) 0 0 xi(6,5) xi(6,6)] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 1 0 0 0 0] adx1 := [ ] [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] [ ] [-1 0 0 0 0 0] adx2 := [ ] [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 0] [ ] [0 0 0 0 0 0] adx3 := [ ] [-1 0 0 0 0 0] [ ] [0 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 And the matrix A:=(xi(5,5),xi(5,6)),(xi(6,5),xi(6,6)) is nilpotent We hence get 2 cases according to whether A neq 0 or A=0. We consider here the case 2 where A := 0. xi(5,5):=0 xi(5,6):=0 xi(6,5):=0 xi(6,6):=0 by subtracting adjoints one then may suppose: xi(3,1):=0,xi(3,2):=0,xi(4,1):=0 delta:= [ 0 0 0 0 0 0 ] [ ] [xi(2,1) 0 0 0 0 0 ] [ ] [ 0 0 0 0 0 0 ] [ ] [ 0 xi(4,2) 0 0 xi(4,5) xi(4,6)] [ ] [xi(5,1) xi(5,2) 0 0 0 0 ] [ ] [xi(6,1) xi(6,2) 0 0 0 0 ] We denote this delta by the shortform shortformdelta:={xi(2,1), ss, xi(4,2), xi(4,5), xi(4,6), ss, xi(5,1), xi(5,2), ss, xi(6,1), xi(6,2)} paramindexeslist:={{2,1},{4,2},{4,5},{4,6},{5,1},{5,2},{6,1},{6,2}} With the generic automorphism one gets$ shortformdeltaprimemodadg:={(b(2,2)*xi(2,1))/b(1,1), ss, ( - b(6,2)*b(5,5)*b(2,2)*b(1,1)**2*xi(4,6) + b(6,2)*b(5,6)*b(2,2)*b(1,1)**2*xi(4 ,5) + b(6,5)*b(5,2)*b(2,2)*b(1,1)**2*xi(4,6) - b(6,5)*b(5,6)*b(2,2)*b(1,1)**2*xi (4,2) - b(6,5)*b(5,6)*b(4,5)*xi(5,2) - b(6,5)*b(5,6)*b(4,6)*xi(6,2) - b(6,6)*b(5 ,2)*b(2,2)*b(1,1)**2*xi(4,5) + b(6,6)*b(5,5)*b(2,2)*b(1,1)**2*xi(4,2) + b(6,6)*b (5,5)*b(4,5)*xi(5,2) + b(6,6)*b(5,5)*b(4,6)*xi(6,2))/(b(2,2)*( - b(6,5)*b(5,6) + b(6,6)*b(5,5))), (b(2,2)*b(1,1)**2*( - b(6,5)*xi(4,6) + b(6,6)*xi(4,5)))/( - b(6,5)*b(5,6) + b(6, 6)*b(5,5)), (b(2,2)*b(1,1)**2*(b(5,5)*xi(4,6) - b(5,6)*xi(4,5)))/( - b(6,5)*b(5,6) + b(6,6)* b(5,5)), ss, (b(5,2)*b(2,2)*xi(2,1) - b(5,5)*b(2,1)*xi(5,2) + b(5,5)*b(2,2)*xi(5,1) - b(5,6)* b(2,1)*xi(6,2) + b(5,6)*b(2,2)*xi(6,1))/(b(2,2)*b(1,1)), (b(5,5)*xi(5,2) + b(5,6)*xi(6,2))/b(2,2), ss, (b(6,2)*b(2,2)*xi(2,1) - b(6,5)*b(2,1)*xi(5,2) + b(6,5)*b(2,2)*xi(5,1) - 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(6,5)*xi(5,2) + b(6,6)*xi(6,2))/b(2,2)}$ deltaprimemodg(2,1):=(b(2,2)*xi(2,1))/b(1,1)$ deltaprimemodg(4,2):=( - b(6,2)*b(5,5)*b(2,2)*b(1,1)**2*xi(4,6) + b(6,2)*b(5,6)* b(2,2)*b(1,1)**2*xi(4,5) + b(6,5)*b(5,2)*b(2,2)*b(1,1)**2*xi(4,6) - b(6,5)*b(5,6 )*b(2,2)*b(1,1)**2*xi(4,2) - b(6,5)*b(5,6)*b(4,5)*xi(5,2) - b(6,5)*b(5,6)*b(4,6) *xi(6,2) - b(6,6)*b(5,2)*b(2,2)*b(1,1)**2*xi(4,5) + b(6,6)*b(5,5)*b(2,2)*b(1,1) **2*xi(4,2) + b(6,6)*b(5,5)*b(4,5)*xi(5,2) + b(6,6)*b(5,5)*b(4,6)*xi(6,2))/(b(2, 2)*( - b(6,5)*b(5,6) + b(6,6)*b(5,5)))$ deltaprimemodg(4,5):=(b(2,2)*b(1,1)**2*( - b(6,5)*xi(4,6) + b(6,6)*xi(4,5)))/( - b(6,5)*b(5,6) + b(6,6)*b(5,5))$ deltaprimemodg(4,6):=(b(2,2)*b(1,1)**2*(b(5,5)*xi(4,6) - b(5,6)*xi(4,5)))/( - b( 6,5)*b(5,6) + b(6,6)*b(5,5))$ deltaprimemodg(5,1):=(b(5,2)*b(2,2)*xi(2,1) - b(5,5)*b(2,1)*xi(5,2) + b(5,5)*b(2 ,2)*xi(5,1) - b(5,6)*b(2,1)*xi(6,2) + b(5,6)*b(2,2)*xi(6,1))/(b(2,2)*b(1,1))$ deltaprimemodg(5,2):=(b(5,5)*xi(5,2) + b(5,6)*xi(6,2))/b(2,2)$ deltaprimemodg(6,1):=(b(6,2)*b(2,2)*xi(2,1) - b(6,5)*b(2,1)*xi(5,2) + b(6,5)*b(2 ,2)*xi(5,1) - 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(6,5)*xi(5,2) + b(6,6)*xi(6,2))/b(2,2)$ det(AUTOM):=( - b(6,5)*b(5,6) + b(6,6)*b(5,5))*b(2,2)**3*b(1,1)**4$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), b(2,2)*xi(2,1) (----------------,0,0,0,0,0), b(1,1) (0,0,0,0,0,0), 2 (0,( - ( - b(5,2)*b(2,2)*b(1,1) *xi(4,6) 2 + (b(4,6)*xi(6,2) + b(2,2)*b(1,1) *xi(4,2) + b(4,5)*xi(5,2))*b(5,6) )*b(6,5) 2 + ( - b(5,5)*xi(4,6) + b(5,6)*xi(4,5))*b(6,2)*b(2,2)*b(1,1) + ( 2 - b(5,2)*b(2,2)*b(1,1) *xi(4,5) 2 + (b(4,6)*xi(6,2) + b(2,2)*b(1,1) *xi(4,2) + b(4,5)*xi(5,2))*b(5,5)) *b(6,6))/(( - b(6,5)*b(5,6) + b(6,6)*b(5,5))*b(2,2)),0,0, 2 ( - b(6,5)*xi(4,6) + b(6,6)*xi(4,5))*b(2,2)*b(1,1) -----------------------------------------------------, - b(6,5)*b(5,6) + b(6,6)*b(5,5) 2 - ( - b(5,5)*xi(4,6) + b(5,6)*xi(4,5))*b(2,2)*b(1,1) --------------------------------------------------------), - b(6,5)*b(5,6) + b(6,6)*b(5,5) ((( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1))*b(5,6) + b(5,2)*b(2,2)*xi(2,1) + ( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1))*b(5,5))/(b(2,2)*b(1,1)), b(5,5)*xi(5,2) + b(5,6)*xi(6,2) ---------------------------------,0,0,0,0), b(2,2) ((( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1))*b(6,6) + b(6,2)*b(2,2)*xi(2,1) + ( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1))*b(6,5))/(b(2,2)*b(1,1)), b(6,5)*xi(5,2) + b(6,6)*xi(6,2) ---------------------------------,0,0,0,0)) b(2,2) ****************** SUBCASE 2 : xi(4,5),xi(4,6) BOTH 0 *********************$ xi(4,5):=0$ xi(4,6):=0$ With the generic automorphism one gets$ shortformdeltaprimemodadg:={(b(2,2)*xi(2,1))/b(1,1), ss, (b(2,2)*b(1,1)**2*xi(4,2) + b(4,5)*xi(5,2) + b(4,6)*xi(6,2))/b(2,2), 0, 0, ss, (b(5,2)*b(2,2)*xi(2,1) - b(5,5)*b(2,1)*xi(5,2) + b(5,5)*b(2,2)*xi(5,1) - b(5,6)* b(2,1)*xi(6,2) + b(5,6)*b(2,2)*xi(6,1))/(b(2,2)*b(1,1)), (b(5,5)*xi(5,2) + b(5,6)*xi(6,2))/b(2,2), ss, (b(6,2)*b(2,2)*xi(2,1) - b(6,5)*b(2,1)*xi(5,2) + b(6,5)*b(2,2)*xi(5,1) - 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(6,5)*xi(5,2) + b(6,6)*xi(6,2))/b(2,2)}$ deltaprimemodg(2,1):=(b(2,2)*xi(2,1))/b(1,1)$ deltaprimemodg(4,2):=(b(2,2)*b(1,1)**2*xi(4,2) + b(4,5)*xi(5,2) + b(4,6)*xi(6,2) )/b(2,2)$ deltaprimemodg(4,5):=0$ deltaprimemodg(4,6):=0$ deltaprimemodg(5,1):=(b(5,2)*b(2,2)*xi(2,1) - b(5,5)*b(2,1)*xi(5,2) + b(5,5)*b(2 ,2)*xi(5,1) - b(5,6)*b(2,1)*xi(6,2) + b(5,6)*b(2,2)*xi(6,1))/(b(2,2)*b(1,1))$ deltaprimemodg(5,2):=(b(5,5)*xi(5,2) + b(5,6)*xi(6,2))/b(2,2)$ deltaprimemodg(6,1):=(b(6,2)*b(2,2)*xi(2,1) - b(6,5)*b(2,1)*xi(5,2) + b(6,5)*b(2 ,2)*xi(5,1) - 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(6,5)*xi(5,2) + b(6,6)*xi(6,2))/b(2,2)$ det(AUTOM):=( - b(6,5)*b(5,6) + b(6,6)*b(5,5))*b(2,2)**3*b(1,1)**4$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), b(2,2)*xi(2,1) (----------------,0,0,0,0,0), b(1,1) (0,0,0,0,0,0), 2 b(4,6)*xi(6,2) + b(2,2)*b(1,1) *xi(4,2) + b(4,5)*xi(5,2) (0,----------------------------------------------------------,0,0,0,0), b(2,2) ((( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1))*b(5,6) + b(5,2)*b(2,2)*xi(2,1) + ( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1))*b(5,5))/(b(2,2)*b(1,1)), b(5,5)*xi(5,2) + b(5,6)*xi(6,2) ---------------------------------,0,0,0,0), b(2,2) ((( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1))*b(6,6) + b(6,2)*b(2,2)*xi(2,1) + ( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1))*b(6,5))/(b(2,2)*b(1,1)), b(6,5)*xi(5,2) + b(6,6)*xi(6,2) ---------------------------------,0,0,0,0)) b(2,2) *************** SUBCASE 2.2 : xi(5,2),xi(6,2) BOTH 0 *********************$ xi(5,2):=0$ xi(6,2):=0$ With the generic automorphism one gets$ shortformdeltaprimemodadg:={(b(2,2)*xi(2,1))/b(1,1), ss, b(1,1)**2*xi(4,2), 0, 0, ss, (b(5,2)*xi(2,1) + b(5,5)*xi(5,1) + b(5,6)*xi(6,1))/b(1,1), 0, ss, (b(6,2)*xi(2,1) + b(6,5)*xi(5,1) + b(6,6)*xi(6,1))/b(1,1), 0}$ deltaprimemodg(2,1):=(b(2,2)*xi(2,1))/b(1,1)$ deltaprimemodg(4,2):=b(1,1)**2*xi(4,2)$ deltaprimemodg(4,5):=0$ deltaprimemodg(4,6):=0$ deltaprimemodg(5,1):=(b(5,2)*xi(2,1) + b(5,5)*xi(5,1) + b(5,6)*xi(6,1))/b(1,1)$ deltaprimemodg(5,2):=0$ deltaprimemodg(6,1):=(b(6,2)*xi(2,1) + b(6,5)*xi(5,1) + b(6,6)*xi(6,1))/b(1,1)$ deltaprimemodg(6,2):=0$ det(AUTOM):=( - b(6,5)*b(5,6) + b(6,6)*b(5,5))*b(2,2)**3*b(1,1)**4$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), b(2,2)*xi(2,1) (----------------,0,0,0,0,0), b(1,1) (0,0,0,0,0,0), 2 (0,b(1,1) *xi(4,2),0,0,0,0), b(5,6)*xi(6,1) + b(5,2)*xi(2,1) + b(5,5)*xi(5,1) (--------------------------------------------------,0,0,0,0,0), b(1,1) b(6,6)*xi(6,1) + b(6,2)*xi(2,1) + b(6,5)*xi(5,1) (--------------------------------------------------,0,0,0,0,0)) b(1,1) **** Suppose xi(2,1) neq 0.$ ******* Then if xi(2,1) neq 0, we get$ deltaprime(2,1)=k, and$ deltaprime(5,1)=0, and$ deltaprime(6,1)=0 by taking:$ b(2,2):=(b(1,1)*k)/xi(2,1)$ b(5,2):=( - (b(5,5)*xi(5,1) + b(5,6)*xi(6,1)))/xi(2,1)$ b(6,2):=( - (b(6,5)*xi(5,1) + b(6,6)*xi(6,1)))/xi(2,1)$ With the generic automorphism one gets$ shortformdeltaprimemodadg:={k, ss, b(1,1)**2*xi(4,2), 0, 0, ss, 0, 0, ss, 0, 0}$ deltaprimemodg(2,1):=k$ deltaprimemodg(4,2):=b(1,1)**2*xi(4,2)$ deltaprimemodg(4,5):=0$ deltaprimemodg(4,6):=0$ deltaprimemodg(5,1):=0$ deltaprimemodg(5,2):=0$ deltaprimemodg(6,1):=0$ deltaprimemodg(6,2):=0$ det(AUTOM):=(( - b(6,5)*b(5,6) + b(6,6)*b(5,5))*b(1,1)**7*k**3)/xi(2,1)**3$ DELTAPRIMEMODADG:= [0 0 0 0 0 0] [ ] [k 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [ 2 ] [0 b(1,1) *xi(4,2) 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] Hence in that case C x_5 +C x_6 would be a direct factor of gtildedelta.$ Hence we dismiss that case.$ **** Suppose xi(2,1) = 0.$ xi(2,1):=0$ clear b(2,2),b(5,2),b(6,2)$ With the generic automorphism one gets$ shortformdeltaprimemodadg:={0, ss, b(1,1)**2*xi(4,2), 0, 0, ss, (b(5,5)*xi(5,1) + b(5,6)*xi(6,1))/b(1,1), 0, ss, (b(6,5)*xi(5,1) + b(6,6)*xi(6,1))/b(1,1), 0}$ deltaprimemodg(2,1):=0$ deltaprimemodg(4,2):=b(1,1)**2*xi(4,2)$ deltaprimemodg(4,5):=0$ deltaprimemodg(4,6):=0$ deltaprimemodg(5,1):=(b(5,5)*xi(5,1) + b(5,6)*xi(6,1))/b(1,1)$ deltaprimemodg(5,2):=0$ deltaprimemodg(6,1):=(b(6,5)*xi(5,1) + b(6,6)*xi(6,1))/b(1,1)$ deltaprimemodg(6,2):=0$ det(AUTOM):=( - b(6,5)*b(5,6) + b(6,6)*b(5,5))*b(2,2)**3*b(1,1)**4$ DELTAPRIMEMODADG:= [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 2 ] [ 0 b(1,1) *xi(4,2) 0 0 0 0] [ ] [ b(5,5)*xi(5,1) + b(5,6)*xi(6,1) ] [--------------------------------- 0 0 0 0 0] [ b(1,1) ] [ ] [ b(6,5)*xi(5,1) + b(6,6)*xi(6,1) ] [--------------------------------- 0 0 0 0 0] [ b(1,1) ] In that case xi(5,1),xi(6,1) cannot be both zero, since then$ C x_5 +C x_6 would again be a direct factor of gtildedelta.$ *************** Hence we suppose: xi(5,1),xi(6,1) NOT BOTH 0 *****************$ In that subcase, one may choose the invertible matrix$ ((xi(5,5),xi(5,6)),(xi(6,5),xi(6,6)) such that$ deltaprime(5,1):=0, deltaprime(6,1):=1.$ Hence we may suppose:$ xi(5,1):=0$ xi(6,1):=1$ With the generic automorphism one gets$ shortformdeltaprimemodadg:={0, ss, b(1,1)**2*xi(4,2), 0, 0, ss, b(5,6)/b(1,1), 0, ss, b(6,6)/b(1,1), 0}$ deltaprimemodg(2,1):=0$ deltaprimemodg(4,2):=b(1,1)**2*xi(4,2)$ deltaprimemodg(4,5):=0$ deltaprimemodg(4,6):=0$ deltaprimemodg(5,1):=b(5,6)/b(1,1)$ deltaprimemodg(5,2):=0$ deltaprimemodg(6,1):=b(6,6)/b(1,1)$ deltaprimemodg(6,2):=0$ det(AUTOM):=( - b(6,5)*b(5,6) + b(6,6)*b(5,5))*b(2,2)**3*b(1,1)**4$ DELTAPRIMEMODADG:= [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 2 ] [ 0 b(1,1) *xi(4,2) 0 0 0 0] [ ] [ b(5,6) ] [-------- 0 0 0 0 0] [ b(1,1) ] [ ] [ b(6,6) ] [-------- 0 0 0 0 0] [ b(1,1) ] We keep deltaprime(5,1):=0 and deltaprime(6,1):=k by taking:$ b(5,6):=0$ b(6,6):=b(1,1)*k$ With the generic automorphism one gets$ shortformdeltaprimemodadg:={0, ss, b(1,1)**2*xi(4,2), 0, 0, ss, 0, 0, ss, k, 0}$ deltaprimemodg(2,1):=0$ deltaprimemodg(4,2):=b(1,1)**2*xi(4,2)$ deltaprimemodg(4,5):=0$ deltaprimemodg(4,6):=0$ deltaprimemodg(5,1):=0$ deltaprimemodg(5,2):=0$ deltaprimemodg(6,1):=k$ deltaprimemodg(6,2):=0$ det(AUTOM):=b(5,5)*b(2,2)**3*b(1,1)**5*k$ DELTAPRIMEMODADG:= [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [ 2 ] [0 b(1,1) *xi(4,2) 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [k 0 0 0 0 0] Hence in that case C x_5 would be a direct factor of gtildedelta.$ Hence we dismiss that case as well.$