D.6.3.4 ImageGroup

Procedure from library rinvar.lib (see rinvar_lib).

Usage:

ImageGroup(G, action); ideal G, action;

Purpose:

compute the ideal of the image of G in GL(m,K) induced by the linear action ’action’, where G is an algebraic group and ’action’ defines an action of G on K^m (size(action) = m).

Return:

ring, a polynomial ring over the same ground field as the basering, containing the ideals ’groupid’ and ’actionid’.
- ’groupid’ is the ideal of the image of G (order <= order of G) - ’actionid’ defines the linear action of ’groupid’ on K^m.

Note:

’action’ and ’actionid’ have the same orbits
all variables which give only rise to 0’s in the m x m matrices of G have been omitted.

Assume:

basering K[s(1..r),t(1..m)] has r + m variables, G is the ideal of an algebraic group and F is an action of G on K^m. G contains only the variables s(1)...s(r). The action ’action’ is given by polynomials f_1,...,f_m in basering, s.t. on the ring level we have K[t_1,...,t_m] –> K[s_1,...,s_r,t_1,...,t_m]/G
t_i –> f_i(s_1,...,s_r,t_1,...,t_m)

Example:

LIB "rinvar.lib";
ring B   = 0,(s(1..2), t(1..2)),dp;
ideal G = s(1)^3-1, s(2)^10-1;
ideal action = s(1)*s(2)^8*t(1), s(1)*s(2)^7*t(2);
def R = ImageGroup(G, action);
setring R;
groupid;
→ groupid[1]=-s(1)+s(2)^4
→ groupid[2]=s(1)^8-s(2)^2
→ groupid[3]=s(1)^7*s(2)^2-1
actionid;
→ actionid[1]=s(1)*t(1)
→ actionid[2]=s(2)*t(2)