rreducparautommodg6_51xCN31.r The generic automorphism phi of C x g_{5,1} as computed by calculautom6_51xC.r\ ed : They fall into 4 kinds: The first kind (which contains the identity component) is: The parameters are subject to the supplementary conditions : b(2,2)*( - b(3,1)*b(1,3) + b(3,3)*b(1,1))neq 0 phi:= mat((b(1,1),b(1,2),b(1,3),b(1,4),0,0), ((b(3,1)*b(2,2)*b(1,4) - b(3,1)*b(2,4)*b(1,2) + b(3,2)*b(2,4)*b(1,1) - b(3,4)*b(2,2)*b(1,1))/( - b(3,1)*b(1,3) + b(3,3)*b(1,1)),b(2,2),( b(3,2)*b(2,4)*b(1,3) + b(3,3)*b(2,2)*b(1,4) - b(3,3)*b(2,4)*b(1,2) - b(3,4)*b(2,2)*b(1,3))/( - b(3,1)*b(1,3) + b(3,3)*b(1,1)),b(2,4),0,0), (b(3,1),b(3,2),b(3,3),b(3,4),0,0), 2 ((b(3,1) *b(1,3)*b(1,2) - b(3,2)*b(3,1)*b(1,3)*b(1,1) 2 - b(3,3)*b(3,1)*b(1,2)*b(1,1) + b(3,3)*b(3,2)*b(1,1) + b(4,2)*b(3,1)*b(2,2)*b(1,4) - b(4,2)*b(3,1)*b(2,4)*b(1,2) + b(4,2)*b(3,2)*b(2,4)*b(1,1) - b(4,2)*b(3,4)*b(2,2)*b(1,1))/(b(2,2) 2 *( - b(3,1)*b(1,3) + b(3,3)*b(1,1))),b(4,2),( - b(3,2)*b(3,1)*b(1,3) + b(3,3)*b(3,1)*b(1,3)*b(1,2) + b(3,3)*b(3,2)*b(1,3)*b(1,1) 2 - b(3,3) *b(1,2)*b(1,1) + b(4,2)*b(3,2)*b(2,4)*b(1,3) + b(4,2)*b(3,3)*b(2,2)*b(1,4) - b(4,2)*b(3,3)*b(2,4)*b(1,2) - b(4,2)*b(3,4)*b(2,2)*b(1,3))/(b(2,2) *( - b(3,1)*b(1,3) + b(3,3)*b(1,1))), - b(3,1)*b(1,3) + b(3,3)*b(1,1) + b(4,2)*b(2,4) --------------------------------------------------,0,0), b(2,2) (b(5,1),b(5,2),b(5,3),b(5,4), - b(3,1)*b(1,3) - b(3,2)*b(1,4) + b(3,3)*b(1,1) + b(3,4)*b(1,2),b(5,6)), (b(6,1),b(6,2),b(6,3),b(6,4),0,b(6,6))) 3 det(phi):=( - b(3,1)*b(1,3) - b(3,2)*b(1,4) + b(3,3)*b(1,1) + b(3,4)*b(1,2)) *b(6,6) condition11:=( - b(3,1)*b(1,3) + b(3,3)*b(1,1))*b(2,2) neq 0 generic derivation : delta:= mat((xi(1,1),xi(1,2),xi(1,3),xi(1,4),0,0), (xi(2,1),xi(2,2),xi(1,4),xi(2,4),0,0), (xi(3,1),xi(3,2),xi(3,3), - xi(2,1),0,0), (xi(3,2),xi(4,2), - xi(1,2), - ( - xi(3,3) - xi(1,1) + xi(2,2)),0,0), (xi(5,1),xi(5,2),xi(5,3),xi(5,4),xi(1,1) + xi(3,3),xi(5,6)), (xi(6,1),xi(6,2),xi(6,3),xi(6,4),0,xi(6,6))) [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx1 := [ ] [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 0] adx2 := [ ] [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 0 0 0] adx3 := [ ] [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 0] adx4 := [ ] [0 0 0 0 0 0] [ ] [0 -1 0 0 0 0] [ ] [0 0 0 0 0 0] The generic nilpotent derivation : the eigenvalues are 0 xi(3,3):= - xi(1,1) xi(6,6):=0 And the matrix A:= [xi(1,1) xi(1,2) xi(1,3) xi(1,4) ] [ ] [xi(2,1) xi(2,2) xi(1,4) xi(2,4) ] [ ] [xi(3,1) xi(3,2) - xi(1,1) - xi(2,1)] [ ] [xi(3,2) xi(4,2) - xi(1,2) - xi(2,2)] is nilpotent. [xi(1,1) xi(1,2) xi(1,3) xi(1,4) ] [ ] [xi(2,1) xi(2,2) xi(1,4) xi(2,4) ] m := [ ] [xi(3,1) xi(3,2) - xi(1,1) - xi(2,1)] [ ] [xi(3,2) xi(4,2) - xi(1,2) - xi(2,2)] we may suppose that A is an element of sp(4,C)+, that is: xi(1,1):=0 xi(2,1):=0 xi(2,2):=0 xi(3,1):=0 xi(3,2):=0 xi(4,2):=0 M**1:= [0 xi(1,2) xi(1,3) xi(1,4)] [ ] [0 0 xi(1,4) xi(2,4)] [ ] [0 0 0 0 ] [ ] [0 0 - xi(1,2) 0 ] M**2:= [0 0 0 xi(2,4)*xi(1,2)] [ ] [0 0 - xi(2,4)*xi(1,2) 0 ] [ ] [0 0 0 0 ] [ ] [0 0 0 0 ] M**3:= [ 2 ] [0 0 - xi(2,4)*xi(1,2) 0] [ ] [0 0 0 0] [ ] [0 0 0 0] [ ] [0 0 0 0] M**4:= [0 0 0 0] [ ] [0 0 0 0] [ ] [0 0 0 0] [ ] [0 0 0 0] Trace(M**1)/2:=0$ Trace(M**2)/2:=0$ Trace(M**3)/2:=0$ Trace(M**4)/2:=0$ By subtracting adjoints one then may suppose:$ xi(5,1):=0,xi(5,2):=0,xi(5,3):=0,xi(5,4):=0$ delta:= mat((0,xi(1,2),xi(1,3),xi(1,4),0,0),(0,0,xi(1,4),xi(2,4),0,0),(0,0,0,0,0,0),(0,0 , - xi(1,2),0,0,0),(0,0,0,0,0,xi(5,6)),(xi(6,1),xi(6,2),xi(6,3),xi(6,4),0,0))$ We denote this delta by the shortform$ shortformdelta:={xi(1,2), xi(1,3), xi(1,4), ss, xi(2,4), ss, xi(5,6), ss, xi(6,1), xi(6,2), xi(6,3), xi(6,4)}$ paramindexeslist:={{1,2},{1,3},{1,4},{2,4},{5,6},{6,1},{6,2},{6,3},{6,4}}$ ************************************************************************$ *******Suppose xi(1,2) = 0.$ ************************************************************************$ xi(1,2):=0$ ************************************************************************$ *******Suppose xi(1,3) NEQ 0.$ ************************************************************************$ Take the following values:$ b(1,3):=0$ b(1,2):=0$ b(1,4):=0$ b(3,2):=0$ b(4,2):=0$ b(3,1):=0$ b(3,3):=b(1,1)*xi(1,3)$ b(3,4):=b(1,1)*xi(1,4)$ b(6,1):=( - (b(6,2)*xi(1,4) + b(6,6)*xi(6,3)))/xi(1,3)$ With the first kind automorphism one gets$ shortformdeltaprimemodadg:={0, 1, 0, ss, (b(2,2)**2*( - xi(1,4)**2 + xi(2,4)*xi(1,3)))/(b(1,1)**2*xi(1,3)**2), ss, (b(1,1)**2*xi(5,6)*xi(1,3))/b(6,6), ss, (b(6,6)*(xi(6,1)*xi(1,3) + xi(6,2)*xi(1,4)))/(b(1,1)*xi(1,3)), (b(6,6)*xi(6,2))/b(2,2), 0, ( - b(6,2)*b(2,2)*xi(1,4)**2 + b(6,2)*b(2,2)*xi(2,4)*xi(1,3) - b(6,6)*b(2,2)*xi( 6,3)*xi(1,4) + b(6,6)*b(2,2)*xi(6,4)*xi(1,3) - b(6,6)*b(2,4)*xi(6,2)*xi(1,3))/(b (1,1)**2*xi(1,3)**2)}$ deltaprimemodg(1,2):=0$ deltaprimemodg(1,3):=1$ deltaprimemodg(1,4):=0$ deltaprimemodg(2,4):=(b(2,2)**2*( - xi(1,4)**2 + xi(2,4)*xi(1,3)))/(b(1,1)**2*xi (1,3)**2)$ deltaprimemodg(5,6):=(b(1,1)**2*xi(5,6)*xi(1,3))/b(6,6)$ deltaprimemodg(6,1):=(b(6,6)*(xi(6,1)*xi(1,3) + xi(6,2)*xi(1,4)))/(b(1,1)*xi(1,3 ))$ deltaprimemodg(6,2):=(b(6,6)*xi(6,2))/b(2,2)$ deltaprimemodg(6,3):=0$ deltaprimemodg(6,4):=( - b(6,2)*b(2,2)*xi(1,4)**2 + b(6,2)*b(2,2)*xi(2,4)*xi(1,3 ) - b(6,6)*b(2,2)*xi(6,3)*xi(1,4) + b(6,6)*b(2,2)*xi(6,4)*xi(1,3) - b(6,6)*b(2,4 )*xi(6,2)*xi(1,3))/(b(1,1)**2*xi(1,3)**2)$ det(AUTOM):=b(6,6)*b(1,1)**6*xi(1,3)**3$ DELTAPRIMEMODADG:= mat((0,0,1,0,0,0),(0,0,0,(( - xi(1,4)**2 + xi(2,4)*xi(1,3))*b(2,2)**2)/(b(1,1)** 2*xi(1,3)**2),0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,(b(1,1)**2*xi(5,6)*xi( 1,3))/b(6,6)),(((xi(6,1)*xi(1,3) + xi(6,2)*xi(1,4))*b(6,6))/(b(1,1)*xi(1,3)),(b( 6,6)*xi(6,2))/b(2,2),0,(( - xi(1,4)**2 + xi(2,4)*xi(1,3))*b(6,2)*b(2,2) + ( - b( 2,4)*xi(6,2)*xi(1,3) + ( - xi(6,3)*xi(1,4) + xi(6,4)*xi(1,3))*b(2,2))*b(6,6))/(b (1,1)**2*xi(1,3)**2),0,0))$ hence we may suppose :$ xi(1,3):=1$ xi(1,4):=0$ xi(6,3):=0$ With the first kind automorphism one gets$ shortformdeltaprimemodadg:={0, 1, 0, ss, (b(2,2)**2*xi(2,4))/b(1,1)**2, ss, (b(1,1)**2*xi(5,6))/b(6,6), ss, (b(6,6)*xi(6,1))/b(1,1), (b(6,6)*xi(6,2))/b(2,2), 0, (b(6,2)*b(2,2)*xi(2,4) + b(6,6)*b(2,2)*xi(6,4) - b(6,6)*b(2,4)*xi(6,2))/b(1,1)** 2}$ deltaprimemodg(1,2):=0$ deltaprimemodg(1,3):=1$ deltaprimemodg(1,4):=0$ deltaprimemodg(2,4):=(b(2,2)**2*xi(2,4))/b(1,1)**2$ deltaprimemodg(5,6):=(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):=(b(6,6)*xi(6,2))/b(2,2)$ deltaprimemodg(6,3):=0$ deltaprimemodg(6,4):=(b(6,2)*b(2,2)*xi(2,4) + b(6,6)*b(2,2)*xi(6,4) - b(6,6)*b(2 ,4)*xi(6,2))/b(1,1)**2$ det(AUTOM):=b(6,6)*b(1,1)**6$ DELTAPRIMEMODADG:= mat((0,0,1,0,0,0),(0,0,0,(b(2,2)**2*xi(2,4))/b(1,1)**2,0,0),(0,0,0,0,0,0),(0,0,0 ,0,0,0),(0,0,0,0,0,(b(1,1)**2*xi(5,6))/b(6,6)),((b(6,6)*xi(6,1))/b(1,1),(b(6,6)* xi(6,2))/b(2,2),0,( - ( - b(6,2)*b(2,2)*xi(2,4) + ( - b(2,2)*xi(6,4) + b(2,4)*xi (6,2))*b(6,6)))/b(1,1)**2,0,0))$ ************************************************************************$ *******Suppose xi(2,4) NEQ 0.$ ************************************************************************$ Then we get deltaprime(2,4)=1 and deltaprimemodg(6,4)=0 by taking$ b(1,1):=sqrt(xi(2,4))*b(2,2)$ b(6,2):=(( - b(2,2)*xi(6,4) + b(2,4)*xi(6,2))*b(6,6))/(b(2,2)*xi(2,4))$ With the first kind automorphism one gets$ shortformdeltaprimemodadg:={0, 1, 0, ss, 1, ss, (b(2,2)**2*xi(5,6)*xi(2,4))/b(6,6), ss, (b(6,6)*xi(6,1))/(sqrt(xi(2,4))*b(2,2)), (b(6,6)*xi(6,2))/b(2,2), 0, 0}$ deltaprimemodg(1,2):=0$ deltaprimemodg(1,3):=1$ deltaprimemodg(1,4):=0$ deltaprimemodg(2,4):=1$ deltaprimemodg(5,6):=(b(2,2)**2*xi(5,6)*xi(2,4))/b(6,6)$ deltaprimemodg(6,1):=(b(6,6)*xi(6,1))/(sqrt(xi(2,4))*b(2,2))$ deltaprimemodg(6,2):=(b(6,6)*xi(6,2))/b(2,2)$ deltaprimemodg(6,3):=0$ deltaprimemodg(6,4):=0$ det(AUTOM):=b(6,6)*b(2,2)**6*xi(2,4)**3$ DELTAPRIMEMODADG:= mat((0,0,1,0,0,0),(0,0,0,1,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,(b(2,2)** 2*xi(5,6)*xi(2,4))/b(6,6)),((b(6,6)*xi(6,1))/(sqrt(xi(2,4))*b(2,2)),(b(6,6)*xi(6 ,2))/b(2,2),0,0,0,0))$ Hence we may suppose:$ xi(2,4):=1$ for i:=1:6 do for j:=1:6 do <>$ Now we use some third kind automorphism.$ We take as third kind automorphism:$ isom:= [i 0 0 - i 0 0] [ ] [1 0 0 1 0 0] [ ] [0 i i 0 0 0] [ ] [0 1 -1 0 0 0] [ ] [0 0 0 0 -2 0] [ ] [0 0 0 0 0 1] which is indeed of the third kind as Delta^(1,3)_(2,4):=-1 and isom(3,2):=i In fact, we use rather than isom its inverse chi chi:= [ - i 1 ] [------ --- 0 0 0 0] [ 2 2 ] [ ] [ - i 1 ] [ 0 0 ------ --- 0 0] [ 2 2 ] [ ] [ - i - 1 ] [ 0 0 ------ ------ 0 0] [ 2 2 ] [ ] [ i 1 ] [ --- --- 0 0 0 0] [ 2 2 ] [ ] [ - 1 ] [ 0 0 0 0 ------ 0] [ 2 ] [ ] [ 0 0 0 0 0 1] - 1 For chi, one has Delta^(1,3)_(2,4):=------ 4 and chi(3,2):=0 - 1 For chi, one has Delta^(2,4)_(2,4):=------ 4 and chi(2,2):=0 Hence chi is of the second and fourth kind, not the third. With the fourth kind automorphism one gets shortformdeltaprimemodadg:={1, 0, 0, ss, 0, ss, - xi(5,6) ------------, 2 ss, i*xi(6,1) + xi(6,2), xi(6,4), - xi(6,4), - i*xi(6,1) + xi(6,2)} deltaprimemodg(1,2):=1 deltaprimemodg(1,3):=0 deltaprimemodg(1,4):=0 deltaprimemodg(2,4):=0 - xi(5,6) deltaprimemodg(5,6):=------------ 2 deltaprimemodg(6,1):=i*xi(6,1) + xi(6,2) deltaprimemodg(6,2):=xi(6,4) deltaprimemodg(6,3):= - xi(6,4) deltaprimemodg(6,4):= - i*xi(6,1) + xi(6,2) - 1 det(AUTOM):=------ 8 DELTAPRIMEMODADG:= mat((0,1,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,-1,0,0,0),(0,0,0,0,0,( - xi(5 ,6))/2),(i*xi(6,1) + xi(6,2),xi(6,4), - xi(6,4), - i*xi(6,1) + xi(6,2),0,0))$ Hence in the present case where xi(1,2):=0 and xi(1,3) neq 0 and xi(2,4) neq 0,$ with chi we are back to the case where xi(1,2) neq 0, xi(2,4):=0.$