Symplectic-Conditions

3113 days ago by jimlb

var('a11,a12,a21,a22,c11,c12,c21,c22,b11,b12,b21,b22,d11,d12,d21,d22,x,y,z') 
       
g=Matrix(SR,[[a11,a12,b11,b12],[a21,a22,b21,b22],[c11,c12,d11,d12],[c21,c22,d21,d22]]);g; 
       
J=Matrix(SR,[[0,0,1,0],[0,0,0,1],[-1,0,0,0],[0,-1,0,0]]);J; 
       

The general conditions are given by setting the following matrix equal to $J$.

g*J*g.transpose() 
       

Here is the calculation of the determinant from group 4.  I don't know how to take the symplectic conditions into account in SAGE. I can probably do it in Maple if necessary.

A=Matrix(SR,[[-b12,-b11],[-b22,-b21]]);A 
       
B=Matrix(SR,[[b11*y + b12*z + a12,b11*x+b12*y+a11],[b21*y+b22*z + a22,b21*x+b22*y+a21]]);B 
       
C=Matrix(SR,[[-d12,-d11],[-d22,-d21]]);C 
       
Cinv=Matrix(SR,[[-d21/(d12*d21-d11*d22),d11/(d12*d21-d11*d22)],[d22/(d12*d21-d11*d22),-d12/(d12*d21-d11*d22)]]);Cinv 
       
D=Matrix(SR,[[d11*y+d12*z,d11*x+d12*y],[d21*y+d22*z, d21*x+d22*y]]);D 
       
E=-A*Cinv*D+B;E 
       
E.det()