D.6.1.26 secondary_no_molien

Procedure from library finvar.lib (see finvar_lib).

Usage:

secondary_no_molien(P,REY[,deg_vec,v]);
P: a 1xn <matrix> with primary invariants, REY: a gxn <matrix> representing the Reynolds operator, deg_vec: an optional <intvec> listing some degrees where no non-trivial homogeneous invariants can be found, v: an optional <int>

Assume:

n is the number of variables of the basering, g the size of the group, REY is the 1st return value of group_reynolds(), reynolds_molien() or the second one of primary_invariants(), deg_vec is the second return value of primary_char0_no_molien(), primary_charp_no_molien(), primary_char0_no_molien_random() or primary_charp_no_molien_random()

Return:

secondary invariants of the invariant ring (type <matrix>)

Display:

information if v does not equal 0

Theory:

Secondary invariants are calculated by finding a basis (in terms of monomials) of the basering modulo primary invariants, mapping those to invariants with the Reynolds operator and using these images as candidates for secondary invariants.

Example:

LIB "finvar.lib";
ring R=0,(x,y,z),dp;
matrix A[3][3]=0,1,0,-1,0,0,0,0,-1;
list L=primary_invariants(A,intvec(1,1,0));
matrix S=secondary_no_molien(L[1..3]);
print(S);
→ 1,xyz,x2z-y2z,x3y-xy3