This section describes in detail the actions which permutation and matrix groups have on multivariate polynomial rings.
If P is a polynomial ring in n indeterminates x_1, ..., x_n,
over any coefficient ring, Sym(n) acts on P by permuting the indices
of the indeterminates. Thus, the polynomial f(x_1, ..., x_n) is
mapped into the polynomial f(x_(g(1)), ..., x_(g(n))).
f ^ g : RngMPolElt, GrpPermElt -> RngMPolElt
Given a polynomial f belonging to a polynomial ring having n indeterminates, and a permutation g belonging to a subgroup of Sym({ 1, ..., n }), return the image of f under g.
Given a polynomial f belonging to a polynomial ring having n indeterminates, and a permutation group G contained in Sym({ 1, ..., n }), return the orbit of f under G.
If P is a polynomial ring in n indeterminates x_1, ..., x_n,
over the ring S, then GL(n, S) acts on P as follows: Let x
denote the vector (x_1, ..., x_n). 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 subgroup G of GL(n, S), return the image of f under a.
Given a polynomial f belonging to a polynomial ring having n indeterminates and coefficient ring S, and a matrix a belonging to a subgroup of GL(n, S), return the orbit of f under G.
> 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 }