Group Actions on Polynomials

This section describes in detail the actions which permutation and matrix groups have on multivariate polynomial rings.

Subsections

• Permutation Group Actions on Polynomials
• Matrix Group Actions on Polynomials

Permutation Group Actions on Polynomials

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.

f ^ G : RngMPolElt, GrpPerm -> { RngMPolElt }

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.

IsInvariant(f, g) : RngMPolElt, GrpElt -> BoolElt

Given a polynomial f belonging to a polynomial ring having n indeterminates, and a permutation g of degree n or an element of a matrix group of degree n whose coefficient ring is the same as that of f, return whether f is an invariant of g, i.e., whether f^g = f.

IsInvariant(f, G) : RngMPolElt, Grp -> BoolElt

Given a polynomial f belonging to a polynomial ring having n indeterminates, and a permutation group G of degree n or a matrix group of degree n whose coefficient ring is the same as that of f, return whether f is an invariant of G, i.e., whether f^g = f for all g in G.

Matrix Group Actions on Polynomials

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.

f ^ G : RngMPolElt, GrpMat -> { RngMPolElt }

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.

Example RngInvar_GroupActions (H78E1)

> 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
}