If P is a polynomial ring in n indeterminates x1, ..., xn, over any coefficient ring, Sym(n) acts on P by permuting the indices of the indeterminates. Thus, the polynomial f(x1, ..., xn) is mapped into the polynomial f(xg(1), ..., xg(n)).
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.
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 fg = f.
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 fg = f for all g∈G.