|
|
D.7.4.4 finiterep
Procedure from library invar.lib (see invar_lib).
- Usage:
- finiterep(<list>), <list> must be a list of matrices
- Returns:
- finiterep(m) gives a matrix with coefficients in the ring 'group'
which represents the action of the finite group where the elements
of the finite group act as m[1],m[2],...m[size(m)].
Example:
| | 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),
==> 900*x(2)^2-1496*x(1)*x(3)-596*x(2)*x(3)-596*x(3)^2-900*x(2)*y(1)+596*x(3)\
*y(1)+2076*x(1)*y(2)+1176*x(2)*y(2)+2672*x(3)*y(2)-1176*y(1)*y(2)-2076*y(\
2)^2-187*x(1)*y(3)-1087*x(2)*y(3)+409*x(3)*y(3)+1087*y(1)*y(3)-989*y(2)*y\
(3)+187*y(3)^2,
==> 28800*x(3)^3-25200*x(1)^2*y(1)+61920*x(1)*x(2)*y(1)+130320*x(2)^2*y(1)-11\
7360*x(1)*x(3)*y(1)+38160*x(2)*x(3)*y(1)-77760*x(3)^2*y(1)+25200*x(1)*y(1\
)^2-130320*x(2)*y(1)^2+48960*x(3)*y(1)^2+187965*x(1)^2*y(2)+171390*x(1)*x\
(2)*y(2)-54375*x(2)^2*y(2)+686997*x(1)*x(3)*y(2)+444657*x(2)*x(3)*y(2)+43\
2432*x(3)^2*y(2)-164205*x(1)*y(1)*y(2)-77385*x(2)*y(1)*y(2)-384912*x(3)*y\
(1)*y(2)+44640*y(1)^2*y(2)-427905*x(1)*y(2)^2-185565*x(2)*y(2)^2-701172*x\
(3)*y(2)^2+203580*y(1)*y(2)^2+239940*y(2)^3-147165*x(1)^2*y(3)-229950*x(1\
)*x(2)*y(3)-59385*x(2)^2*y(3)-428789*x(1)*x(3)*y(3)-341009*x(2)*x(3)*y(3)\
-287024*x(3)^2*y(3)+204045*x(1)*y(1)*y(3)-39255*x(2)*y(1)*y(3)+367664*x(3\
)*y(1)*y(3)+11520*y(1)^2*y(3)-163470*x(1)*y(2)*y(3)-8910*x(2)*y(2)*y(3)-2\
96878*x(3)*y(2)*y(3)+92430*y(1)*y(2)*y(3)+324810*y(2)^2*y(3)+445455*x(1)*\
y(3)^2+357675*x(2)*y(3)^2+556514*x(3)*y(3)^2-306570*y(1)*y(3)^2-152220*y(\
2)*y(3)^2-298290*y(3)^3
==>
==> Zero Fiber Ideal:
==> x(1)+x(2)+x(3),
==> 225*x(2)^2-374*x(1)*x(3)-149*x(2)*x(3)-149*x(3)^2,
==> x(3)^3
==>
==> Generating Invariants:
==> x(1)+x(2)+x(3),
==> 76/3*x(1)^2-523/3*x(1)*x(2)+76/3*x(2)^2-523/3*x(1)*x(3)-523/3*x(2)*x(3)+7\
6/3*x(3)^2,
==> 1/3*x(1)^3+1/3*x(2)^3+1/3*x(3)^3
|
|
