Singular

LIB "invar.lib";
finite(6);                              // The symmetric group S_3
matrix id=unitmat(3);                   // identity matrix
matrix m3[3][3]=0,1,0,0,0,1,1,0,0;      // corresponds with (1 2 3)
matrix m2[3][3]=0,1,0,1,0,0,0,0,1;      // corresponds with (1 2)
list a=id,m3,m3*m3,m2,m2*m3,m2*m3*m3;   // all elements of S_3
matrix rep=finiterep(a);                // compute matrix of standard repr.
invar(rep);                             // compute the invariant ring
==> 
==> Ideal B:
==> x(1)+x(2)+x(3)-y(1)-y(2)-y(3),
==> 36*x(2)^2-115*x(1)*x(3)-79*x(2)*x(3)-79*x(3)^2+24*x(1)*y(1)-12*x(2)*y(1)+\
   103*x(3)*y(1)-24*y(1)^2-36*x(2)*y(2)+79*x(3)*y(2)+12*y(1)*y(2)+310*x(1)*y\
   (3)+274*x(2)*y(3)+389*x(3)*y(3)-298*y(1)*y(3)-274*y(2)*y(3)-310*y(3)^2,
==> 1296*x(3)^3+1584*x(1)*x(3)*y(1)+1584*x(2)*x(3)*y(1)+288*x(3)^2*y(1)-1584*\
   x(3)*y(1)^2+2112*x(1)^2*y(2)+4224*x(1)*x(2)*y(2)+2112*x(2)^2*y(2)+6648*x(\
   1)*x(3)*y(2)+6648*x(2)*x(3)*y(2)+3240*x(3)^2*y(2)-2112*x(1)*y(1)*y(2)-211\
   2*x(2)*y(1)*y(2)-4824*x(3)*y(1)*y(2)-2112*x(1)*y(2)^2-2112*x(2)*y(2)^2-45\
   36*x(3)*y(2)^2+5248*x(1)^2*y(3)+13645*x(1)*x(2)*y(3)+8397*x(2)^2*y(3)+448\
   95*x(1)*x(3)*y(3)+48044*x(2)*x(3)*y(3)+38351*x(3)^2*y(3)-6832*x(1)*y(1)*y\
   (3)-9981*x(2)*y(1)*y(3)-41519*x(3)*y(1)*y(3)+1584*y(1)^2*y(3)-16120*x(1)*\
   y(2)*y(3)-19269*x(2)*y(2)*y(3)-51647*x(3)*y(2)*y(3)+9048*y(1)*y(2)*y(3)+8\
   760*y(2)^2*y(3)-26722*x(1)*y(3)^2-29871*x(2)*y(3)^2-61121*x(3)*y(3)^2+230\
   58*y(1)*y(3)^2+30234*y(2)*y(3)^2+21474*y(3)^3
==> 
==> Zero Fiber Ideal:
==> x(1)+x(2)+x(3),
==> 36*x(2)^2-115*x(1)*x(3)-79*x(2)*x(3)-79*x(3)^2,
==> x(3)^3
==> 
==> Generating Invariants:
==> x(1)+x(2)+x(3),
==> -43/3*x(1)^2-194/3*x(1)*x(2)-43/3*x(2)^2-194/3*x(1)*x(3)-194/3*x(2)*x(3)-\
   43/3*x(3)^2,
==> 1/3*x(1)^3+1/3*x(2)^3+1/3*x(3)^3