Algebraic Group Actions on Polynomials

In the invariant theory package of Magma, a linear algebraic group \(G\) is given by polynomials, say in variables \(t_1,\ldots,t_m\), defined over some field \(K\) that is representable in Magma, as the affine variety over the algebraic closure \(\bar{K}\) of \(K\) given by these polynomials. A \(G\)-module is given by a matrix \(A \in K[t_1,\ldots,t_m]^{n \times n}\) such that a group element \((\eta_1,\ldots,\eta_m) \in G\) acts on \(\bar{K}^n\) by the matrix obtained by substituting \((\eta_1,\ldots,\eta_m)\) into the polynomials occurring in the matrix \(A\).

\(G\) then also acts on the ring of polynomials on \(\bar{K}^n\) by

\[\sigma(f) = f \circ \sigma^{-1}\]

for \(\sigma \in G\) and \(f \in \bar{K}[x_1,\ldots,x_n]\). Since the algorithms in Magma do not work with the algebraic closure, single group elements are never dealt with. In fact, all relevant algorithms ony involve field elements of \(K\), the field of definition.