D.6.1.25 secondary_charp

Procedure from library finvar.lib (see finvar_lib).

Usage:

secondary_charp(P,REY,ringname[,v]);
P: a 1xn <matrix> with primary invariants, REY: a gxn <matrix> representing the Reynolds operator, ringname: a <string> giving the name of a ring of characteristic 0 where the Molien series is stored, 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(), ‘ringname‘ is a ring of char 0 that has been created by molien() or reynolds_molien() or primary_invariants()

Return:

secondary invariants of the invariant ring (type <matrix>) and irreducible secondary invariants (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 or their power products such that they are linearly independent modulo the primary invariants (see paper "Some Algorithms in Invariant Theory of Finite Groups" by Kemper and Steel (1997)).

Example:

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