Permutation Group Actions on Polynomials

If \(P\) is a polynomial ring in \(n\) indeterminates \(x_1, \ldots, x_n\), over any coefficient ring, \({\operatorname{Sym}}(n)\) acts on \(P\) by permuting the indices of the indeterminates. Thus, the polynomial \(f(x_1, \ldots, x_n)\) is mapped into the polynomial \(f(x_{g(1)}, \ldots, x_{g(n)})\).

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 \({\operatorname{Sym}}(\lbrace 1, \ldots, n \rbrace)\), 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 \({\operatorname{Sym}}(\lbrace 1, \ldots, n \rbrace)\), 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 \(g\in G\).