Matrix Group Actions on Polynomials#
If \(P\) is a polynomial ring in \(n\) indeterminates \(x_1, \ldots, x_n\), over the ring \(S\), then \({\operatorname{GL}}(n, S)\) acts on \(P\) as follows: Let x denote the vector \((x_1, \ldots, x_n)\). Then the image \(g\) of a polynomial \(f\) of \(P\) under the action of a matrix \(a\) of \({\operatorname{GL}}(n, S)\) is defined by \(g({\bf x}) = f({\bf x}*a)\).
- f ^ a: RngMPolElt, GrpMatElt -> RngMPolElt#
Given a polynomial \(f\) belonging to a polynomial ring having \(n\) indeterminates and coefficient ring \(S\), and a matrix \(a\) belonging subgroup \(G\) of \({\operatorname{GL}}(n, S)\), 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 to a subgroup of \({\operatorname{GL}}(n, S)\), return the orbit of \(f\) under \(G\).
- Example: Group Actions (ex-df3c71)#
We act on the polynomial ring in two indeterminates over the field \(K = Q(\sqrt 2)\), by a cyclic subgroup of \({\operatorname{GL}}(2, K)\).
> 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 }