Matrix Group Actions on Polynomials

If P is a polynomial ring in n indeterminates x1, ..., xn, over the ring S, then GL(n, S) acts on P as follows: Let x denote the vector (x1, ..., xn). Then the image g of a polynomial f of P under the action of a matrix a of GL(n, S) is defined by g((x)) = f((x) * a).

f ^ a : RngMPolElt, GrpMatElt -> RngMPolElt
Given a polynomial f belonging to a polynomial ring having n indeterminates and coefficient ring S, and a matrix a belonging subgroup G of GL(n, S), return the image of f under a.
f ^ G : RngMPolElt, GrpMat -> { RngMPolElt }
Given a polynomial f belonging to a polynomial ring having n indeterminates and coefficient ring S, and a to a subgroup of GL(n, S), return the orbit of f under G.

Example RngInvar_GroupActions (H120E1)

We act on the polynomial ring in two indeterminates over the field K = Q(√2), by a cyclic subgroup of GL(2, K). 
> K := QuadraticField(2);
> Aq := [ x / K.1 : x in [1, 1, -1, 1]];
> G := MatrixGroup<2, K | Aq>;
> P<x, y> := PolynomialRing(K, 2);
> f := x^2 + x * y + y^2;
> g := f^G.1;
> g;
1/2*x^2 + 3/2*y^2
> f^G;
{
   1/2*x^2 + 3/2*y^2,
   x^2 - x*y + y^2,
   x^2 + x*y + y^2,
   3/2*x^2 + 1/2*y^2
}
V2.29, 21 October 2025