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D.6.3.7 LinearizeAction
Procedure from library rinvar.lib (see rinvar_lib).
- Usage:
LinearizeAction(G,action,r); ideal G, action; int r
- Purpose:
linearize the group action ’action’ and find an equivariant embedding of K^m where m = size(action).
- Assume:
G contains only variables var(1..r) (r = nrs)
basering = K[s(1..r),t(1..m)], K = Q or K = Q(a) and minpoly != 0.- Return:
polynomial ring containing the ideals ’actionid’, ’embedid’, ’groupid’ - ’actionid’ is the ideal defining the linearized action of G - ’embedid’ is a parameterization of an equivariant embedding (closed) - ’groupid’ is the ideal of G in the new ring
- Note:
set printlevel > 0 to see a trace
Example:
LIB "rinvar.lib"; ring B = 0,(s(1..5), t(1..3)),dp; ideal G = s(3)-s(4), s(2)-s(5), s(4)*s(5), s(1)^2*s(4)+s(1)^2*s(5)-1, s(1)^2*s(5)^2-s(5), s(4)^4-s(5)^4+s(1)^2, s(1)^4+s(4)^3-s(5)^3, s(5)^5-s(1)^2*s(5); ideal action = -s(4)*t(1)+s(5)*t(1), -s(4)^2*t(2)+2*s(4)^2*t(3)^2+s(5)^2*t(2), s(4)*t(3)+s(5)*t(3); LinearActionQ(action, 5); → 0 def R = LinearizeAction(G, action, 5); setring R; R; → // characteristic : 0 → // number of vars : 9 → // block 1 : ordering dp → // : names s(1) s(2) s(3) s(4) s(5) t(1) t(2) t(3) t(\ 4) → // block 2 : ordering C actionid; → actionid[1]=-s(4)*t(1)+s(5)*t(1) → actionid[2]=-s(4)^2*t(2)+s(5)^2*t(2)+2*s(4)^2*t(4) → actionid[3]=s(4)*t(3)+s(5)*t(3) → actionid[4]=s(4)^2*t(4)+s(5)^2*t(4) embedid; → embedid[1]=t(1) → embedid[2]=t(2) → embedid[3]=t(3) → embedid[4]=t(3)^2 groupid; → groupid[1]=s(3)-s(4) → groupid[2]=s(2)-s(5) → groupid[3]=s(4)*s(5) → groupid[4]=s(1)^2*s(4)+s(1)^2*s(5)-1 → groupid[5]=s(1)^2*s(5)^2-s(5) → groupid[6]=s(4)^4-s(5)^4+s(1)^2 → groupid[7]=s(1)^4+s(4)^3-s(5)^3 → groupid[8]=s(5)^5-s(1)^2*s(5) LinearActionQ(actionid, 5); → 1 |