Permutation Group Actions on Polynomials

If P is a polynomial ring in n indeterminates x₁, ..., x_n, over any coefficient ring, Sym(n) acts on P by permuting the indices of the indeterminates. Thus, the polynomial f(x₁, ..., 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∈G.