D.6.3.11 NullCone

Procedure from library rinvar.lib (see rinvar_lib).

Usage:

NullCone(G, action); ideal G, action

Purpose:

compute the ideal of the null cone of the linear action of G on K^n, given by ’action’, by means of Derksen’s algorithm

Assume:

basering = K[s(1..r),t(1..n)], K = Q or K = Q(a) and minpoly != 0, G is an ideal of a reductive algebraic group in K[s(1..r)], ’action’ is a linear group action of G on K^n (n = ncols(action))

Return:

ideal of the null cone of G.

Note:

the generators of the null cone are homogeneous, but i.g. not invariant

Example:

LIB "rinvar.lib";
ring R = 0, (s(1..2), x, y), dp;
ideal G = -s(1)+s(2)^3, s(1)^4-1;
ideal action = s(1)*x, s(2)*y;
ideal inv = NullCone(G, action);
inv;
→ inv[1]=x^4
→ inv[2]=x^3*y^3
→ inv[3]=x^2*y^6
→ inv[4]=x*y^9
→ inv[5]=y^12