D.6.3.5 InvariantRing

Procedure from library rinvar.lib (see rinvar_lib).

Usage:

InvariantRing(G, Gact [, opt]); ideal G, Gact; int opt

Purpose:

compute generators of the invariant ring of G w.r.t. the action ’Gact’

Assume:

G is a finite group and ’Gact’ is a linear action.

Return:

polynomial ring over a simple extension of the ground field of the basering (the extension might be trivial), containing the ideals ’invars’ and ’groupid’ and the poly ’newA’
- ’invars’ contains the algebra-generators of the invariant ring - ’groupid’ is the ideal of G in the new ring
- ’newA’ if the minpoly changes this is the new representation of the algebraic number, otherwise it is set to ’a’.

Note:

the delivered ring might have a different minimal polynomial

Example:

LIB "rinvar.lib";
ring B = 0, (s(1..2), t(1..2)), dp;
ideal G = -s(1)+s(2)^3, s(1)^4-1;
ideal action = s(1)*t(1), s(2)*t(2);
def R = InvariantRing(std(G), action);
setring R;
invars;
→ invars[1]=t(1)^4
→ invars[2]=t(1)^3*t(2)^3
→ invars[3]=t(1)^2*t(2)^6
→ invars[4]=t(1)*t(2)^9
→ invars[5]=t(2)^12